- Multiply the whole number by the fraction's numerator, then simplify. - Look for factors that can cancel before multiplying. - In part c), pay attention to the sign of the product. - In part d), check whether the fraction can be simplified before calculating.

1. For a), \(4 \cdot \frac{3}{8}=\frac{12}{8}=\frac{3}{2}\). 2. For b), \(5 \cdot \frac{7}{10}=\frac{35}{10}=\frac{7}{2}\). 3. For c), \(-6 \cdot \frac{1}{3}=-2\). 4. For d), \(11 \cdot \frac{2}{22}=1\).

a) \(\frac{3}{2}\) b) \(\frac{7}{2}\) c) \(-2\) d) \(1\)

- Separate the familiar whole-number fact from the decimal place-value decision. - State how many tenths, hundredths, or thousandths the decimal factor represents. - Determine the sign independently of decimal placement.

1. a) \(8\cdot4=32\). Since \(0.8\) is eight tenths, the product is \(3.2\). 2. b) \(5\cdot9=45\). The product is negative and \(0.5\) is five tenths, so the result is \(-4.5\). 3. c) \(12\cdot5=60\). Since \(1.2\) has one decimal place, the product is \(6\). 4. d) \(6\cdot7=42\). Since \(0.006\) is six thousandths, the product is \(0.042\).

a) \(8\cdot4=32\); \(0.8\cdot4=3.2\) b) \(5\cdot9=45\); sign negative; \(-0.5\cdot9=-4.5\) c) \(12\cdot5=60\); \(1.2\cdot5=6\) d) \(6\cdot7=42\); \(0.006\cdot7=0.042\)

- Determine the sign of each result before calculating. - What sign does a product or quotient have when the factors or terms have different signs? - What sign does the product of two negative numbers have?

1. For a), a negative number times a positive number is negative: \((-14)\cdot3=-42\). 2. For b), a positive number divided by a negative number is negative: \(35\div(-7)=-5\). 3. For c), the product of two negative numbers is positive: \((-4)\cdot(-12)=48\). 4. For d), a positive number divided by a negative number is negative: \(100\div(-5)=-20\).

a) \(-42\) b) \(-5\) c) \(48\) d) \(-20\)

- Identify the signed change that occurs once and the number of times it occurs. - Write one compact expression for those equal repeated changes. - Interpret the sign of the total in terms of the score.

1. Four equal changes of \(-3\) points are represented by \(4\cdot(-3)\). 2. \(4\cdot(-3)=-12\). 3. The negative sign means the penalties decrease the score by \(12\) points in total.

\(4\cdot(-3)=-12\). The team's score decreases by \(12\) points.

- Compare the results as you move down the pattern. - Check how the second factor changes in each row. - Keep the same change in the product for the last row.

1. Each time the second factor decreases by \(1\), the result increases by \(3\). 2. The next result is \(3\), so \((-3)\cdot(-1)=3\).

\(3\). The products rise by \(3\) each time; a negative times a negative is positive.

- Think of the number that produces the given negative amount when multiplied by the divisor. - Decide the sign before dividing the magnitudes. - Multiply the quotient by the divisor to check.

1. A negative divided by a positive is negative: \(-12\div3=-4\). 2. Check: \(3\cdot(-4)=-12\).

\(-4\). Check: \(3\cdot(-4)=-12\).

- Multiplying by a positive fraction smaller than one changes the size of a quantity. - Find the signed product before comparing distances from zero. - Don't mistake a less-negative value for a larger distance.

1. \(\frac{1}{3}\cdot(-6)=-2\). 2. \(|-2|=2\) and \(|-6|=6\), so \(-2\) is closer to zero.

\(\frac{1}{3}\cdot(-6)=-2\); since \(|-2|=2<6=|-6|\), the product is closer to zero.

- Division asks for a number that gives the dividend when multiplied by the divisor. - What does multiplying any number by zero produce? - Keep the dividend and divisor in the correct order.

1. \(0\div(-7)=0\), because \((-7)\cdot0=0\). 2. \((-7)\div0\) is undefined, because there is no number that multiplied by zero gives \(-7\).

a) Defined; \(0\) b) Undefined; no number multiplied by \(0\) equals \(-7\).

- How would you write \(6\) over \(3\) using a division sign? - What multiplication could you use to check the quotient you found?

1. The top number is \(\frac{3}{4}\), and the bottom number is \(2\). The bar means \(\frac{3}{4}\div2\). 2. Divide: \(\frac{3}{4}\div2=\frac{3}{4}\cdot\frac{1}{2}=\frac{3}{8}\).

\(\frac{3}{4}\div2=\frac{3}{8}\).

- Look for a nearby multiple of \(10\), \(100\), or a convenient divisible part. - Show the equivalent expression that supports the mental calculation. - Determine the sign before simplifying part b).

1. a) \(2.5(10+2)=25+5=30\). 2. b) \(14(-100-1)=-1400-14=-1414\). 3. c) \((700+28)\div7=100+4=104\).

a) \(2.5(10+2)=30\) b) \(14(-100-1)=-1414\) c) \((700+28)\div7=104\)

- Factor the whole number and denominator mentally before multiplying numerators. - Your response must show a reduced factor pair before the final multiplication. - Track the negative sign separately from cancellation.

1. a) \(15\cdot\frac{4}{25}=3\cdot\frac{4}{5}=\frac{12}{5}\). 2. b) \(24\cdot\frac{5}{16}=3\cdot\frac{5}{2}=\frac{15}{2}\). 3. c) \(-18\cdot\frac{7}{12}=-3\cdot\frac{7}{2}=-\frac{21}{2}\). 4. d) \(35\cdot\frac{3}{14}=5\cdot\frac{3}{2}=\frac{15}{2}\).

a) \(3\cdot\frac{4}{5}=\frac{12}{5}\) b) \(3\cdot\frac{5}{2}=\frac{15}{2}\) c) \(-3\cdot\frac{7}{2}=-\frac{21}{2}\) d) \(5\cdot\frac{3}{2}=\frac{15}{2}\)

- Keep track of the sign of each quotient. - Rewrite division as multiplication by a reciprocal. - Rewrite mixed numbers and decimals as fractions before dividing.

1. For a), \(-\frac{15}{16}\div5=-\frac{15}{16}\cdot\frac{1}{5}=-\frac{3}{16}\). 2. For b), \(\frac{7}{8}\div14=\frac{7}{8}\cdot\frac{1}{14}=\frac{1}{16}\). 3. For c), rewrite \(2 \frac{4}{7}=\frac{18}{7}\). Then \(\frac{18}{7}\div6=\frac{18}{7}\cdot\frac{1}{6}=\frac{3}{7}\). 4. For d), rewrite \(0.4=\frac{2}{5}\). Then \(\frac{2}{5}\div8=\frac{2}{5}\cdot\frac{1}{8}=\frac{1}{20}\).

a) \(-\frac{3}{16}\) b) \(\frac{1}{16}\) c) \(\frac{3}{7}\) d) \(\frac{1}{20}\)

- Multiply the fraction by the whole number, then simplify. - Rewrite a mixed number as an improper fraction before multiplying. - Look for opportunities to simplify before or after multiplying.

1. For a), \(-\frac{5}{12}\cdot8=-\frac{40}{12}=-\frac{10}{3}=-3 \frac{1}{3}\). 2. For b), rewrite \(2 \frac{3}{10}\) as \(\frac{23}{10}\). Then \(4\cdot\frac{23}{10}=\frac{92}{10}=\frac{46}{5}=9 \frac{1}{5}\).

a) \(-3 \frac{1}{3}\) b) \(9 \frac{1}{5}\)

- Rewrite the decimal as a fraction when useful. - Cancel common factors before multiplying. - A positive number times a negative number gives a negative product.

1. For a), \(\frac{3}{8}\cdot\frac{2}{3}=\frac{1}{4}\) after canceling common factors. 2. For b), rewrite \(1.2\) as \(\frac{6}{5}\). Then \(\frac{6}{5}\cdot\frac{5}{6}=1\). 3. For c), cancel common factors before multiplying: \(\frac{7}{10}\cdot\left(-\frac{5}{14}\right)=-\frac{1}{4}\).

a) \(\frac{1}{4}\) b) \(1\) c) \(-\frac{1}{4}\)

- Think about what happens when \(\frac{1}{2}\) is multiplied by a number less than \(1\) or greater than \(1\). - Calculate or estimate each product. - Rewrite the results with a common denominator if needed for comparison.

1. Calculate the values: \(A=\frac{1}{6}\), \(B=\frac{1}{3}\), \(C=\frac{3}{4}\), and \(D=\frac{1}{4}\). 2. Compare them, for example using denominator \(12\): \(\frac{2}{12}<\frac{3}{12}<\frac{4}{12}<\frac{9}{12}\). 3. Therefore \(A<D<B<C\), and C has the greatest value.

\(A<D<B<C\). Expression C has the greatest value.

- Rewrite decimals and percents as fractions when useful. - Rewrite a mixed number as an improper fraction before multiplying. - Cancel common factors before multiplying when possible.

1. For a), \(\frac{3}{8}\cdot\frac{4}{9}=\frac{12}{72}=\frac{1}{6}\). 2. For b), rewrite \(1 \frac{1}{4}=\frac{5}{4}\) and \(0.8=\frac{4}{5}\). Then \(\frac{5}{4}\cdot\frac{4}{5}=1\). 3. For c), rewrite \(75\%=\frac{3}{4}\). Then \(\frac{3}{4}\cdot\frac{2}{3}=\frac{1}{2}\).

a) \(\frac{1}{6}\) b) \(1\) c) \(\frac{1}{2}\)

- Cancel common factors before multiplying when possible. - Keep track of the sign of the product. - Rewrite the mixed number as an improper fraction.

1. For a), cancel common factors before multiplying: \(\frac{9}{14}\cdot\frac{7}{15}=\frac{3}{10}\). 2. For b), the product is negative. After canceling common factors, \(-\frac{3}{8}\cdot\frac{4}{9}=-\frac{1}{6}\). 3. For c), rewrite \(1 \frac{1}{5}\) as \(\frac{6}{5}\). Then \(\frac{5}{6}\cdot\frac{6}{5}=1\).

a) \(\frac{3}{10}\) b) \(-\frac{1}{6}\) c) \(1\)

- Rewrite the mixed number as an improper fraction. - Look for factors that cancel across the three fractions before multiplying. - Simplify the final product.

1. Rewrite \(3 \frac{3}{4}\) as \(\frac{15}{4}\). 2. Multiply \(\frac{15}{4}\cdot\frac{2}{5}\cdot\frac{4}{9}\), canceling common factors before multiplying. 3. The simplified product is \(\frac{2}{3}\).

\(\frac{2}{3}\)

- Interpret “of” as multiplication. - How do you multiply two fractions? - Rewrite the mixed number as an improper fraction. - Can you cancel common factors before multiplying?

1. For a), \(\frac{3}{5}\cdot\frac{10}{9}=\frac{2}{3}\). 2. For b), rewrite \(1 \frac{1}{7}=\frac{8}{7}\). Then \(\frac{7}{8}\cdot\frac{8}{7}=1\). 3. For c), \(\frac{5}{12}\cdot\frac{18}{25}=\frac{3}{10}\) after canceling common factors.

a) \(\frac{2}{3}\) b) \(1\) c) \(\frac{3}{10}\)

- Calculate the two products separately. - Simplify common factors before multiplying. - Compare the simplified results.

1. Value A is \(\frac{4}{9}\cdot\frac{27}{16}=\frac{3}{4}\) after simplifying common factors. 2. Value B is \(\frac{5}{6}\cdot\frac{9}{10}=\frac{3}{4}\). 3. Therefore the values are equal.

The values are equal; both are \(\frac{3}{4}\).

- Convert each decimal to a fraction before multiplying. - Simplify common factors before multiplying the numerators and denominators. - A fraction is in simplest form when its numerator and denominator have no common factor greater than \(1\).

1. For a), write \(0.35 = \frac{35}{100} = \frac{7}{20}\). Then \(\frac{4}{7} \cdot \frac{7}{20} = \frac{4}{20} = \frac{1}{5}\). 2. For b), write \(1.2 = \frac{12}{10} = \frac{6}{5}\). Then \(\frac{6}{5} \cdot \frac{5}{9} = \frac{6}{9} = \frac{2}{3}\). 3. For c), write \(0.16 = \frac{16}{100} = \frac{4}{25}\). Then \(\frac{3}{8} \cdot \frac{4}{25} = \frac{3}{50}\).

a) \(\frac{1}{5}\) b) \(\frac{2}{3}\) c) \(\frac{3}{50}\)

- What operation does “of” represent when working with fractions? - How do you multiply three fractions? - Think about whether repeatedly taking a fraction of a fraction should make the value larger or smaller.

1. Write each “one third” as \(\frac{1}{3}\). 2. The phrase “one third of one third of one third” means \(\frac{1}{3}\cdot\frac{1}{3}\cdot\frac{1}{3}\). 3. Multiply: \(\frac{1\cdot1\cdot1}{3\cdot3\cdot3}=\frac{1}{27}\). 4. Compare \(\frac{1}{27}\) with the claimed value \(\frac{1}{9}\). Since \(\frac{1}{27}\neq\frac{1}{9}\), Jordan's claim is false.

No. \(\frac{1}{3}\cdot\frac{1}{3}\cdot\frac{1}{3}=\frac{1}{27}\), not \(\frac{1}{9}\).

- Calculate the value of Julia's expression first. - Then calculate one half of one eighth. - Compare the two simplified fractions.

1. Julia's expression is \(\frac{1}{4}\cdot\frac{1}{4}=\frac{1}{16}\). 2. The comparison expression is \(\frac{1}{2}\cdot\frac{1}{8}=\frac{1}{16}\). 3. Since \(\frac{1}{16}=\frac{1}{16}\), Marcus is correct.

Marcus is correct. Both expressions equal \(\frac{1}{16}\).

- Convert each mixed number first. - Look for a numerator and denominator sharing a factor before multiplying. - Keep the sign separate while simplifying magnitudes.

1. a) \(\frac{8}{3}\cdot\frac{9}{8}\to\frac{9}{3}=3\). 2. b) \(-\frac{9}{2}\cdot\frac{4}{9}\to-\frac{4}{2}=-2\). 3. c) \(\frac{10}{7}\cdot\left(-\frac{21}{10}\right)\to-\frac{21}{7}=-3\).

a) \(\frac{8}{3}\cdot\frac{9}{8}\to3\) b) \(-\frac{9}{2}\cdot\frac{4}{9}\to-2\) c) \(\frac{10}{7}\cdot\left(-\frac{21}{10}\right)\to-3\)

- Divide by a fraction by multiplying by its reciprocal. - Keep track of the signs in each quotient. - Rewrite mixed numbers as improper fractions first. - Cancel common factors before multiplying. - Convert an improper fraction to a mixed number at the end when appropriate.

1. For a), \(\frac{15}{16}\div\frac{5}{8}=\frac{15}{16}\cdot\frac{8}{5}=\frac{3}{2}=1 \frac{1}{2}\). 2. For b), rewrite the mixed numbers: \(-\frac{8}{3}\div\frac{10}{9}=-\frac{8}{3}\cdot\frac{9}{10}=-\frac{12}{5}=-2 \frac{2}{5}\). 3. For c), \(10\div\left(-\frac{2}{3}\right)=10\cdot\left(-\frac{3}{2}\right)=-15\). 4. For d), two negative numbers give a positive quotient: \(\frac{7}{12}\cdot\frac{9}{14}=\frac{3}{8}\).

a) \(1 \frac{1}{2}\) b) \(-2 \frac{2}{5}\) c) \(-15\) d) \(\frac{3}{8}\)

- For a missing factor, think about which inverse operation will help. - Recall how multiplying a decimal by \(10\), \(100\), or \(1000\) changes its place value. - Pay attention to the sign when a product is negative.

1. For a), use division: \(3.2\div0.4=8\). 2. For b), \(0.07\cdot8=0.56\). 3. For c), use division: \(-4.5\div1.5=-3\). 4. For d), \(0.003\cdot300=0.9\).

a) \(8\) b) \(0.56\) c) \(-3\) d) \(0.9\)

- Use place value carefully when dividing decimals. - Decide the sign of a quotient before dividing absolute values. - If the dividend is smaller in magnitude than the divisor, expect a quotient between \(-1\) and \(1\).

1. \(15.6\div6=2.6\). 2. \(8.4\div12=0.7\). 3. \(-13.5\div5=-2.7\). 4. \(0.64\div16=0.04\).

a) \(2.6\) b) \(0.7\) c) \(-2.7\) d) \(0.04\)

- Relate each decimal quotient to a familiar whole-number fact. - Scaling both dividend and divisor by the same power of \(10\) preserves a quotient. - State the sign separately when needed.

1. a) \(48\div6=8\), so \(0.48\div6=0.08\). 2. b) \(15\div3=5\), and the quotient is negative, so \(-1.5\div3=-0.5\). 3. c) Scale both numbers by \(10\): \(1.2\div0.4=12\div4=3\).

a) \(48\div6=8\); result \(0.08\) b) \(15\div3=5\); sign negative; result \(-0.5\) c) \(12\div4=3\); result \(3\)

- Convert the decimal to a fraction. - To divide by a fraction, multiply by its reciprocal. - Look for common factors before multiplying.

1. For a), simplify common factors before multiplying: \(\frac{9}{14} \cdot \frac{7}{12} = \frac{3}{8}\). 2. For b), write \(2.5 = \frac{5}{2}\). 3. Divide by multiplying by the reciprocal: \(\frac{5}{2} \cdot \frac{8}{5}\). 4. Simplify the common factors to get \(4\).

a) \(\frac{3}{8}\) b) \(4\)

- Record the benchmark dividend and divisor you choose. - Keep the estimate and exact calculation as separate lines. - Use the estimate to check sign and approximate size, not to replace exact division.

1. a) Using \(6\div0.1\) gives estimate \(60\). Exactly, \(560\div8=70\), which is reasonably close. 2. b) Using \(0.4\div2\) gives estimate \(0.2\). Exactly, \(4.41\div21=0.21\), consistent with the estimate. 3. c) Using \((-2)\div(-0.2)\) gives estimate \(10\). Exactly, \(225\div15=15\), with positive sign; the estimate has the correct sign and order of magnitude.

a) Benchmarks \(6\) and \(0.1\): estimate \(60\); exact \(70\); reasonable b) Benchmarks \(0.4\) and \(2\): estimate \(0.2\); exact \(0.21\); reasonable c) Benchmarks \(-2\) and \(-0.2\): estimate \(10\); exact \(15\); same sign and order of magnitude

- For Expression A, evaluate the difference inside the parentheses first. - For Expression B, evaluate the two products first. - Which property relates multiplying a difference to subtracting two products?

1. For Expression A, subtract inside the parentheses: \(\frac{1}{4} - \frac{1}{3} = \frac{3}{12} - \frac{4}{12} = -\frac{1}{12}\). Then \(12\left(-\frac{1}{12}\right) = -1\). 2. For Expression B, evaluate each product: \(12 \cdot \frac{1}{4} = 3\) and \(12 \cdot \frac{1}{3} = 4\). Then \(3 - 4 = -1\). 3. Both expressions equal \(-1\). 4. The two equivalent forms illustrate the distributive property.

a) Expression A: \(-1\) Expression B: \(-1\) b) The results are equal. This illustrates the distributive property.

- Determine the sign of each correct result first. - What rule do you use when dividing by a fraction? - What sign does the product of two negative numbers have?

1. For a), the product of two negative numbers must be positive. The magnitude \(0.2\cdot0.3=0.06\) is correct, so the correct result is \(0.06\). 2. For b), dividing by a fraction means multiplying by its reciprocal. The correct calculation is \(\frac{3}{4}\cdot\left(-\frac{2}{1}\right)=-\frac{6}{4}=-\frac{3}{2}\).

a) The sign is wrong. The correct result is \(0.06\). b) The divisor was not replaced by its reciprocal. The correct result is \(-\frac{3}{2}\), or \(-1.5\).

- Choose the form that makes each operation easiest. - Subtracting a negative number is equivalent to adding its opposite. - Determine the sign before multiplying or dividing. - To divide by a fraction, multiply by its reciprocal.

1. For a), write \(0.5 = \frac{1}{2}\). Then \(-\frac{3}{4} + \frac{1}{2} = -\frac{1}{4}\). 2. For b), write \(1\frac{1}{5} = 1.2\). Then \(-2.2 - (-1.2) = -2.2 + 1.2 = -1\). 3. For c), write \(1\frac{1}{2} = 1.5\). Then \(0.8 \cdot (-1.5) = -1.2\). 4. For d), divide by multiplying by the reciprocal: \(-2 \cdot \frac{5}{4} = -\frac{5}{2} = -2.5\).

a) \(-\frac{1}{4}\), or \(-0.25\) b) \(-1\) c) \(-1.2\), or \(-1\frac{1}{5}\) d) \(-2.5\), or \(-2\frac{1}{2}\)

- Determine the sign before multiplying or dividing. - Simplify common factors before multiplying fractions. - To divide by a fraction, multiply by its reciprocal.

1. For a), write \(-0.9 = -\frac{9}{10}\). Then \(\frac{2}{3} \cdot \left(-\frac{9}{10}\right) = -\frac{3}{5} = -0.6\). 2. For b), write \(-0.75 = -\frac{3}{4}\). Then \(-\frac{3}{4} \div \frac{3}{4} = -1\). 3. For c), write \(1.25 = \frac{5}{4}\). Then \(\frac{5}{4} \cdot \left(-\frac{4}{5}\right) = -1\).

a) \(-0.6\), or \(-\frac{3}{5}\) b) \(-1\) c) \(-1\)

- Determine the sign of a product from the number of negative factors. - For “exactly two negative,” consider every possible sign of the remaining factor. - Recall that integers are rational numbers. - Any factor of \(0\) makes the entire product \(0\).

1. For a), possible: \((-1)\cdot(-1)\cdot(-1)=-1\). 2. For b), not possible. Two negative factors have a positive product. If the third factor is positive, the product is positive; if the third factor is \(0\), the product is \(0\). Neither case gives \(-1\). 3. For c), possible: \((-1)\cdot1\cdot1=-1\). 4. For d), not possible. Any product with a factor of \(0\) equals \(0\).

a) Yes; for example, \((-1)\cdot(-1)\cdot(-1)=-1\). b) No; the third factor is nonnegative, so the product is positive or \(0\), not \(-1\). c) Yes; for example, \((-1)\cdot1\cdot1=-1\). d) No; the product would be \(0\).

- First evaluate the expression inside the parentheses as a check. - When distributing, multiply the outside factor by each term, including its sign. - Recall the sign of a product of two negative numbers.

1. Evaluate inside the parentheses: \(1.2 - 4 = -2.8\). Then \(-5(-2.8) = 14\), so Sophie is correct. 2. When distributing, the correct expression is \(-5 \cdot 1.2 - (-5 \cdot 4)\). 3. Therefore, \(-6 - (-20) = -6 + 20 = 14\). Lucas failed to account for the product of two negative factors.

1. Sophie is correct; the value is \(14\). 2. Lucas mishandled the signs. The second product contributes \(+20\), not \(-20\).

- For Method A, consider an equivalent fraction whose denominator is \(10\). - For Method B, compare the two fraction factors after converting the decimal. - Compare the amount of multiplication and simplification required by the two methods.

1. Method A: Write \(\frac{4}{5} = 0.8\). Then \(1.25 \cdot 0.8 = 1\). 2. Method B: Write \(1.25 = \frac{5}{4}\). Then \(\frac{5}{4} \cdot \frac{4}{5} = 1\). 3. Method B makes the reciprocal factors visible, so the product simplifies immediately. Method A is also valid.

Method A: \(\frac{4}{5}=0.8\), so \(1.25\cdot0.8=1\). Method B: \(1.25=\frac{5}{4}\), so \(\frac{5}{4}\cdot\frac{4}{5}=1\). Both methods are valid. For example, Method B is especially short because the two fraction factors are reciprocals and simplify immediately. A learner may reasonably prefer Method A instead if the explanation compares the actual work in the two methods.

- Multiply \(20\) by each term inside the parentheses. - Evaluate \(20 \cdot \frac{1}{4}\) and \(20 \cdot 0.2\). - Compare the resulting expression with Expression B.

1. Distribute \(20\) in Expression A: \(20 \cdot \frac{1}{4} - 20 \cdot 0.2\). 2. Evaluate the first product: \(20 \cdot \frac{1}{4} = 5\). 3. Evaluate the second product: \(20 \cdot 0.2 = 4\). 4. The expanded form is \(5 - 4\), which is exactly Expression B. Both expressions equal \(1\).

Distributing \(20\) changes Expression A to \(5 - 4\), so the expressions are equivalent and both have value \(1\).

- Determine the sign before calculating the magnitude. - First multiply or divide the absolute values. - The product of two negative numbers is positive.

1. Determine each sign first: unlike signs give a negative result, and like signs give a positive result. 2. For a), \(145\cdot14=2030\), so the result is \(-2030\). 3. For b), \(2352\div12=196\), so the result is \(-196\). 4. For c), \(26\cdot302=7852\), and two negative factors give a positive result. 5. For d), \(4464\div18=248\), so the result is \(-248\).

a) \(-2030\) b) \(-196\) c) \(7852\) d) \(-248\)

- Determine the sign before doing the division. - Use long division for the absolute values if needed. - What sign results when a positive number is divided by a negative number?

1. A positive number divided by a negative number gives a negative result. 2. Divide the absolute values: \(18{,}432\div16=1152\). 3. Apply the sign: \(-1152\).

\(-1152\)

- A negative product requires factors with opposite signs. - Start from familiar factor pairs of \(30\). - Scale one factor up and the other down to create decimal or fractional factors.

1. For a), one example is \(3\cdot(-10)=-30\). 2. For b), one example is \((-1.5)\cdot20=-30\). 3. For c), one example is \(\frac{3}{4}\cdot(-40)=-30\).

a) For example, \(3\cdot(-10)\) b) For example, \((-1.5)\cdot20\) c) For example, \(\frac{3}{4}\cdot(-40)\)

- A factor that produces \(1\) is the reciprocal of the other factor. - Use division to find a missing factor. - Recall the zero-product property.

1. For a), \(x=1\div(-2)=-\frac{1}{2}\). 2. For b), \(y=1\div0.1=10\). 3. A fraction and its reciprocal have product \(1\), such as \(\frac{2}{5}\cdot\frac{5}{2}=1\). 4. Yes. A product is \(0\) when at least one factor is \(0\), such as \(0\cdot5=0\).

a) \(x=-\frac{1}{2}\) b) \(y=10\) c) For example, \(\frac{2}{5}\) and \(\frac{5}{2}\) d) Yes; for example, \(0\cdot5=0\).

- Use place value to position the decimal point in each product. - Decide the sign of each product before multiplying. - Multiply the absolute values first, then apply the sign.

1. \(4.8\cdot1.5=7.2\). 2. \((-2.2)\cdot3.4=-7.48\). 3. \(0.05\cdot(-0.14)=-0.007\). 4. \((-1.25)\cdot(-0.8)=1\).

a) \(7.2\) b) \(-7.48\) c) \(-0.007\) d) \(1\)

- Notice how the same digit product can appear with different place values. - Count the decimal places in the factors. - Determine the sign of each product separately from its magnitude.

1. The base digit product is \(6\cdot7=42\). 2. For a), \(0.6\cdot0.7=0.42\). 3. For b), \(0.06\cdot0.7=0.042\). 4. For c), the product is positive: \((-0.6)\cdot(-0.07)=0.042\). 5. For d), the product is negative: \((-0.06)\cdot0.07=-0.0042\).

a) \(0.42\) b) \(0.042\) c) \(0.042\) d) \(-0.0042\)

- For a), consider a positive fraction between \(0\) and \(1\). - For b), recall the sign of a product of two negative numbers. - For c), identify the number that cannot be used as a divisor.

1. For a), use \(\frac{1}{2}\). Its reciprocal is \(2\), and \(2 > \frac{1}{2}\). 2. For b), use \(-4\). Its reciprocal is \(-\frac{1}{4}\), and \((-4)\left(-\frac{1}{4}\right) = 1\), which is positive. 3. For c), use \(0\). Zero has no reciprocal because division by \(0\) is undefined.

a) The reciprocal of \(\frac{1}{2}\) is \(2\), which is greater than \(\frac{1}{2}\). b) \((-4)\left(-\frac{1}{4}\right) = 1\), which is positive. c) \(0\) has no reciprocal.

- For a), test a product with exactly one zero factor. - For b), test a negative number. - For c), consider a positive number between \(0\) and \(1\).

1. For a), \(0 \cdot 5 = 0\), but only one factor is \(0\). 2. For b), \((-3)(-1) = 3\), and \(3 > -3\). 3. For c), \((0.5)^2 = 0.25\), and \(0.25 < 0.5\).

a) \(0 \cdot 5 = 0\), although \(5 \ne 0\). b) \((-3)(-1) = 3 > -3\). c) \((0.5)^2 = 0.25 < 0.5\).

- Determine the sign before calculating the magnitude. - What sign results from multiplying numbers with like or unlike signs? - Simplify common factors before multiplying fractions when possible. - Converting between fractions and decimals can make a product easier.

1. For a), unlike signs give a negative product: \((-12)\cdot11=-132\). 2. For b), two negative factors give a positive product: \((-0.5)\cdot(-0.08)=0.04\). 3. For c), \(\frac{3}{7}\cdot\left(-\frac{14}{9}\right)=-\frac{42}{63}=-\frac{2}{3}\). 4. For d), \(-2.5=-\frac{5}{2}\), so \(-\frac{5}{2}\cdot\frac{2}{5}=-1\).

a) \(-132\) b) \(0.04\) c) \(-\frac{2}{3}\) d) \(-1\)

- First determine the sign of each missing number. - Use the inverse operation to find a missing factor or dividend. - For the last part, think about factor pairs of \(18\).

1. For a), divide the product by the known factor: \(-56\div7=-8\). 2. For b), multiply the quotient by the divisor: \(4\cdot(-9)=-36\). 3. For c), divide the product by the known factor: \(10\div(-0.5)=-20\). 4. For d), choose two integer factors of \(18\) with opposite signs. One example is \(2\cdot(-9)=-18\).

a) \(-8\) b) \(-36\) c) \(-20\) d) For example, \(2\) and \(-9\)

- In a), evaluate the difference in parentheses before multiplying. - Decide the product's sign in each part. - In b), look for two factors with a product of \(1\).

1. For a), the parentheses give \(4.2-5.2=-1\). Then \((-15.8)(-1)=15.8\). 2. For b), regroup the factors: \((25\cdot0.04)(-17)=1(-17)=-17\).

a) \(15.8\) b) \(-17\)

- Substitute each given value for \(a\). - Which operation must happen before the final subtraction? - Could a fraction picture help you check the division in part a)? - For part b), estimate the sign before calculating exactly.

1. For a), divide first: \(\frac{1}{8}\div\frac{1}{2}=\frac{1}{4}\). 2. Then \(\frac{3}{4}-\frac{1}{4}=\frac{1}{2}\). 3. For b), divide first: \(-1.5\div\frac{1}{2}=-3\). 4. Then \(\frac{3}{4}-(-3)=3\frac{3}{4}\).

a) \(\frac{1}{2}\) b) \(3\frac{3}{4}\)

- Determine the sign before calculating each product or quotient. - How do you multiply fractions? - How do you divide by a fraction? - A whole number can be written as a fraction when helpful.

1. For a), product: \(-2.5\cdot0.4=-1\). Quotient: \(-2.5\div0.4=-6.25\). 2. For b), product: \(\frac{3}{10}\cdot\left(-\frac{6}{5}\right)=-\frac{9}{25}\). Quotient: \(\frac{3}{10}\div\left(-\frac{6}{5}\right)=\frac{3}{10}\cdot\left(-\frac{5}{6}\right)=-\frac{1}{4}\). 3. For c), product: \((-12)\cdot\left(-\frac{4}{3}\right)=16\). Quotient: \((-12)\div\left(-\frac{4}{3}\right)=(-12)\cdot\left(-\frac{3}{4}\right)=9\).

a) Product: \(-1\); quotient: \(-6.25\) b) Product: \(-\frac{9}{25}\); quotient: \(-\frac{1}{4}\) c) Product: \(16\); quotient: \(9\)

- Determine the sign before calculating. - Simplify common factors before multiplying when possible. - Dividing by a whole number is equivalent to multiplying by its reciprocal.

1. For a), a negative times a positive is negative: \((-15)\cdot4=-60\). 2. For b), dividing two negative numbers gives a positive result: \((-20)\div(-5)=4\). 3. For c), \(\frac{4}{7}\cdot14=8\). 4. For d), \(\frac{3}{10}\div3=\frac{3}{10}\cdot\frac{1}{3}=\frac{1}{10}\). 5. For e), two negative factors give a positive product: \(\left(-\frac{2}{5}\right)\cdot\left(-\frac{1}{2}\right)=\frac{1}{5}\).

a) \(-60\) b) \(4\) c) \(8\) d) \(\frac{1}{10}\), or \(0.1\) e) \(\frac{1}{5}\), or \(0.2\)

- Use an inverse operation to find each missing number. - A number multiplied by its reciprocal equals \(1\). - Look for factors you can cancel before multiplying fractions.

1. For a), multiply by the reciprocal of \(-\frac{5}{6}\): \(1\cdot\left(-\frac{6}{5}\right)=-\frac{6}{5}\). 2. For b), multiply the quotient by the divisor: \(\frac{8}{9}\cdot\left(-\frac{3}{4}\right)=-\frac{24}{36}=-\frac{2}{3}\).

a) \(-\frac{6}{5}\) b) \(-\frac{2}{3}\)

- Identify the inverse operations in each expression. - Determine which operations undo each other. - Compare the starting values after simplifying the structure.

1. In expression \(A\), multiplying by \(77\) and then dividing by \(77\) undo each other, so \(A = 333\). 2. In expression \(B\), adding \(77\) and then subtracting \(77\) undo each other, so \(B = 330\). 3. Since \(333 > 330\), expression \(A\) has the greater value.

Expression \(A\) is greater because \(A = 333\) and \(B = 330\).

- Look for pairs whose product is a convenient benchmark number. - Keep signs attached to factors when rearranging. - Show the grouping that makes each product short.

1. \((25\cdot4)(-13)=100(-13)=-1300\). 2. \((-125)(-8)\cdot7=1000\cdot7=7000\). 3. \(\left(\frac{4}{9}\cdot\frac{9}{4}\right)(-17)=1(-17)=-17\). 4. \((0.2\cdot5)(-47)=1(-47)=-47\).

1) \((25\cdot4)(-13)=-1300\) 2) \((-125)(-8)\cdot7=7000\) 3) \(\left(\frac{4}{9}\cdot\frac{9}{4}\right)(-17)=-17\) 4) \((0.2\cdot5)(-47)=-47\)

- Which partial products are easier for mental calculation? - One property changes factor order; another changes grouping. - Show the convenient grouping in b) before evaluating.

1. Marie reordered factors using the commutative property and regrouped them using the associative property. Both methods give \(170\), but Marie's intermediate products are easier. 2. In b), regroup as \((8\cdot125)((-1.5)(-2))\). 3. The two partial products are \(1000\) and \(3\), so the value is \(3000\).

a) Marie used the commutative and associative properties to get easier partial products; both methods equal \(170\). b) \((8\cdot125)((-1.5)(-2))=1000\cdot3=3000\)

- Follow the grouping symbols in each expression. - Determine the sign of each result before calculating. - Compare all three final values.

1. \((-96 \cdot 16) \div 8 = -1536 \div 8 = -192\). 2. \((-96 \div 8) \cdot 16 = -12 \cdot 16 = -192\). 3. \(-96(16 \div 8) = -96 \cdot 2 = -192\). 4. All three expressions have the same value.

All three expressions equal \(-192\).

- Think about the sign of each quotient. - In the second method, divide by the two factors in sequence. - Compare the results after showing both calculations.

1. Multiply first: \((-6)\cdot4=-24\), then \(144\div(-24)=-6\). 2. Divide successively: \(144\div(-6)=-24\), then \(-24\div4=-6\). 3. Both methods agree.

1. \(144\div(-24)=-6\) 2. \((144\div(-6))\div4=-6\)

- Compute each product before comparing. - Track the sign and magnitude separately. - A product of numbers with the same sign is positive. - Use the signed multiplication rule explicitly in part c).

1. a) \(9\cdot3=27\) and \(4\cdot2=8\), so \(27>8\). 2. b) \(6\cdot4=24\) and \((-3)\cdot1=-3\), so \(24>-3\). 3. c) \((-5)(-4)=20\) and \(2\cdot3=6\), so \(20>6\). 4. In part c), the product of two negative numbers is positive.

a) \(27>8\) b) \(24>-3\) c) \(20>6\) The key rule in part c) is that a negative times a negative is positive.

- Compare the number of columns covered by the shaded rectangle with the total number of columns. - Compare the number of rows covered by the shaded rectangle with the total number of rows. - After multiplying the two dimension fractions, compare the result with the fraction of individual cells that are shaded.

1. The grid has \(4\) columns, and the shaded rectangle spans \(3\) of them, so the shaded width is \(\frac{3}{4}\). 2. The grid has \(3\) rows, and the shaded rectangle spans \(2\) of them, so the shaded height is \(\frac{2}{3}\). 3. Multiply: \(\frac{3}{4}\cdot\frac{2}{3}=\frac{6}{12}=\frac{1}{2}\). 4. There are \(12\) equal cells and \(6\) are shaded, so \(\frac{6}{12}=\frac{1}{2}\), which checks the product.

a) \(\frac{3}{4}\) b) \(\frac{2}{3}\) c) \(\frac{1}{2}\) d) \(6\) of \(12\) cells are shaded, so \(\frac{6}{12}=\frac{1}{2}\).

- Identify the quantity given per minute and the amount of time that passes. - Choose a relationship whose units reduce to meters. - Interpret a negative signed vertical change after calculating.

1. The quantity relationship is signed rate times elapsed time: \((-2.5)\cdot1.6\). 2. \((-2.5)\cdot1.6=-4\). 3. The signed vertical change is \(-4\,\text{m}\), meaning \(4\,\text{m}\) downward.

\((-2.5\,\text{m/min})\cdot(1.6\,\text{min})=-4\,\text{m}\). The negative sign means a \(4\,\text{m}\) downward change.

- Identify the original signed quantity and the fractional scale applied to it. - Write a single expression that gives the scaled signed change. - Interpret the sign only after finding the scaled value.

1. Trail B's change is the fraction \(\frac{5}{8}\) of \(-12.8\,\text{m}\), so model it as \(\frac{5}{8}\cdot(-12.8)\). 2. \(12.8\div8=1.6\) and \(1.6\cdot5=8\), so the result is \(-8\,\text{m}\). 3. The negative sign represents a net descent.

\(\frac{5}{8}\cdot(-12.8\,\text{m})=-8\,\text{m}\), representing a net descent of \(8\,\text{m}\).

- Let \(x\) be the signed change from one transaction and relate six copies of \(x\) to the total. - Solve that relationship for one transaction's change. - Interpret a negative transaction change in the account context.

1. Split the total signed change equally among six transactions: \((-39.00)\div6\). 2. \((-39.00)\div6=-6.50\). 3. Each transaction decreases the balance by \(\$6.50\).

\((-39.00)\div6=-6.50\). Each transaction decreases the balance by \(\$6.50\).

- Let \(n\) be the number of equal steps and write a relationship between \(n\), the change per step, and the total change. - Solve for the count \(n\). - Check that the sign of a count makes sense in context.

1. The number of equal steps is modeled by \((-12)\div(-1.5)\). 2. \((-12)\div(-1.5)=8\). 3. The result is positive because it is a count of \(8\) steps; the quotient of the two negative signed changes is also positive.

\((-12)\div(-1.5)=8\). The answer is positive because it counts \(8\) lowering steps, and the quotient of two negative values is positive.

- The requested quantity is “per second,” so relate the total change to the elapsed seconds. - Choose an expression whose units become \(\text{m/s}\). - Interpret the sign of the resulting rate in terms of altitude.

1. A per-second quantity is the total signed change distributed over \(2.4\) seconds: \((-7.2)\div2.4\). 2. \((-7.2)\div2.4=-3\). 3. The rate is \(-3\,\text{m/s}\), meaning altitude decreases by \(3\,\text{m}\) per second on average.

\((-7.2\,\text{m})\div(2.4\,\text{s})=-3\,\text{m/s}\). The negative sign indicates an average downward change.

- Let \(x\) represent the full signed change. - Write an equation saying that three fifths of \(x\) equals the observed change. - Solve that equation while keeping the signed meaning of \(x\).

1. If \(x\) is the full signed change, then \(\frac{3}{5}x=-9\). 2. Solving gives \(x=(-9)\div\frac{3}{5}=(-9)\cdot\frac{5}{3}=-15\). 3. The full signed change is \(-15\,\text{m}\), a \(15\,\text{m}\) descent.

\(\frac{3}{5}x=-9\), so \(x=(-9)\div\frac{3}{5}=-15\,\text{m}\).

- Read the signed jump size and count the jumps. - How can equal repeated changes be written as multiplication? - Use the marked starting point to calculate each landing in order.

1. Each jump is \(-0.75\), so the total change is \(3\cdot(-0.75)=-2.25\). 2. The landings after the displayed start are \(0.75\), \(0\), and \(-0.75\).

a) \(3\cdot(-0.75)\) b) \(-2.25\) c) \(0.75\), \(0\), \(-0.75\)

- Check whether every landing is farther left than the one before it. - Which unit fraction has the same size as a decimal part of \(0.25\)? - Check the last landing against the product's sign and size.

1. Each \(-0.25\) jump is \(-\frac{1}{4}\). 2. The missing landings are \(-0.25\), \(-0.50\), and \(-0.75\). 3. The total change is \(3\cdot\left(-\frac{1}{4}\right)=-\frac{3}{4}\), the final position.

Landing values: \(-0.25\), \(-0.50\), \(-0.75\). Jump fraction: \(-\frac{1}{4}\). Equation and final position: \(3\cdot\left(-\frac{1}{4}\right)=-\frac{3}{4}\).

- Decide the sign before multiplying the fractional magnitudes. - Look for a common factor between one numerator and the other denominator. - Simplify the fraction after multiplication.

1. A negative times a positive is negative. 2. Multiply and simplify: \(-\frac{2\cdot3}{3\cdot5}=-\frac{2}{5}\).

\(-\frac{2}{5}\); negative because the factors have opposite signs.

- Is a drop a positive or negative change in water level? - What calculation gives the change for one day from a three-day total? - Check whether three copies of your daily change match the total.

1. The three-day change is \(-2.4\,\text{in.}\). 2. Divide by three days: \(-2.4\div3=-0.8\,\text{in.}\) each day.

\(-2.4\div3=-0.8\,\text{in.}\) per day.

- A quotient times its divisor should return the dividend. - Dividing by one-half asks how many halves fit into the amount. - Keep track of the sign of the quotient.

1. Dividing by \(\frac{1}{2}\) means multiplying by \(2\), so \(-\frac{3}{4}\div\frac{1}{2}=-\frac{3}{2}\). 2. Check: \(-\frac{3}{2}\cdot\frac{1}{2}=-\frac{3}{4}\).

\(-\frac{3}{2}\). Check: \(-\frac{3}{2}\cdot\frac{1}{2}=-\frac{3}{4}\).

- A loss corresponds to a negative signed rate. - Which operation gives the number of equal hourly changes? - Check your result against the total change.

1. The signed rate is \(-\frac{3}{8}\) inch per hour. 2. Time equals total signed change divided by rate: \(-\frac{9}{8}\div\left(-\frac{3}{8}\right)=3\). 3. Check: \(3\cdot\left(-\frac{3}{8}\right)=-\frac{9}{8}\).

\(-\frac{9}{8}\div\left(-\frac{3}{8}\right)=3\) hours

- Work out the subtraction above the long bar first. - Is the bottom of the fraction positive or negative? - Rewrite the long bar as division before multiplying by a reciprocal.

1. Simplify the top: \(\frac{3}{4}-\frac{1}{8}=\frac{6}{8}-\frac{1}{8}=\frac{5}{8}\). 2. Divide by the entire bottom number: \(\frac{5}{8}\div\left(-\frac{5}{2}\right)=\frac{5}{8}\cdot\left(-\frac{2}{5}\right)\). 3. Cancel common factors to get \(-\frac{1}{4}\).

After simplifying the top to \(\frac{5}{8}\), divide \(\frac{5}{8}\div\left(-\frac{5}{2}\right)=-\frac{1}{4}\).

- Evaluate each expression separately before comparing them. - Subtracting a negative number is equivalent to adding its opposite. - Find the difference between \(X\) and \(Y\).

1. Evaluate \(X\): \((-12)\cdot(-3)=36\), so \(X=36+(-16)=20\). 2. Evaluate \(Y\): \((-60)\div4=-15\), so \(Y=-15-(-5)=-10\). 3. Solve \(Y+z=X\): \(-10+z=20\), so \(z=30\).

\(X=20\), \(Y=-10\), and \(z=30\)

- Dividing by a number is equivalent to multiplying by its reciprocal. - Rewrite mixed numbers as improper fractions before dividing. - It may help to rewrite decimals as fractions. - Keep track of the sign when dividing positive and negative rational numbers.

1. For a), multiply by the reciprocal: \(\frac{5}{6}\cdot\frac{1}{10}=\frac{1}{12}\). 2. For b), rewrite \(3 \frac{1}{3}=\frac{10}{3}\). Then \(\frac{10}{3}\cdot\frac{9}{5}=6\). 3. For c), rewrite \(-2.5=-\frac{5}{2}\). Then \(-\frac{5}{2}\cdot\frac{8}{5}=-4\). 4. For d), rewrite \(-0.6=-\frac{3}{5}\). Then \(\frac{12}{25}\cdot\left(-\frac{5}{3}\right)=-\frac{4}{5}\).

a) \(\frac{1}{12}\) b) \(6\) c) \(-4\) d) \(-\frac{4}{5}\)

- Choose a representation that exposes cancellation or a reciprocal relationship. - Keep the chosen rewrite visible in the response. - Determine the sign independently of the magnitude calculation.

1. a) \(\frac{27}{8}\cdot\frac{4}{9}=\frac{3}{2}\). 2. b) Rewrite \(0.8=\frac{4}{5}\): \(\frac{4}{5}\div\frac{4}{15}=\frac{4}{5}\cdot\frac{15}{4}=3\). 3. c) Rewrite \(2.25=\frac{9}{4}\): \(\frac{9}{4}\cdot\left(-\frac{2}{3}\right)=-\frac{3}{2}\).

a) \(\left(\frac{3}{2}\right)^3=\frac{27}{8}\); result \(\frac{3}{2}\) b) \(0.8=\frac{4}{5}\); result \(3\) c) \(2.25=\frac{9}{4}\); result \(-\frac{3}{2}\)

- Evaluate the expression inside the parentheses first. - Write the decimal and mixed number as fractions before multiplying. - Look for common factors in numerators and denominators. - Count the negative factors to determine the sign of the product.

1. Convert the mixed number and add inside the parentheses: \(\frac{2}{3} + 1\frac{1}{6} = \frac{4}{6} + \frac{7}{6} = \frac{11}{6}\). 2. Write \(-1.5 = -\frac{3}{2}\). 3. Multiply: \(\left(-\frac{3}{2}\right) \cdot \frac{11}{6} \cdot \left(-\frac{4}{11}\right)\). 4. Two negative factors give a positive product. Cancel the common factor \(11\), then simplify: \(\frac{3 \cdot 4}{2 \cdot 6} = \frac{12}{12} = 1\). 5. Fractions keep the calculation exact and make common factors visible before multiplication.

The value is \(1\). Writing all the numbers as fractions keeps the calculation exact and makes it easy to cancel common factors.

- State the benchmark factors used in each estimate. - Keep the estimate separate from the exact computation. - Finish by comparing the size and sign of the exact value with the estimate.

1. a) Using \(4\) and \(2\) gives estimate \(8\). Exactly, \(\frac{7}{2}\cdot\frac{12}{5}=\frac{42}{5}=8\frac{2}{5}\), consistent with the estimate. 2. b) Using \(-5\) and \(1.5\) gives estimate \(-7.5\). Exactly, \(-\frac{14}{3}\cdot\frac{3}{2}=-7\), consistent with the estimate. 3. c) Using \(0.6\) and \(3\) gives estimate \(1.8\), about \(2\). Exactly, \(\frac{6}{10}\cdot\frac{10}{3}=2\), consistent with the estimate.

a) Benchmarks \(4\) and \(2\): estimate \(8\); exact \(8\frac{2}{5}\); consistent b) Benchmarks \(-5\) and \(1.5\): estimate \(-7.5\); exact \(-7\); consistent c) Benchmarks \(0.6\) and \(3\): estimate \(1.8\), about \(2\); exact \(2\); consistent

- Convert mixed numbers before looking for factors that cancel. - Show the reduced factors before carrying out the final multiplication. - Determine the product sign from the signs of the factors.

1. a) \(\frac{5}{8}\cdot\frac{6}{5}\cdot4\) cancels to \(\frac{1}{2}\cdot6=3\). 2. b) \(\frac{9}{4}\cdot\frac{4}{3}\cdot\frac{1}{2}\) cancels to \(3\cdot\frac{1}{2}=\frac{3}{2}\).

a) \(\frac{5}{8}\cdot\frac{6}{5}\cdot4\to\frac{1}{2}\cdot6=3\) b) \(\frac{9}{4}\cdot\frac{4}{3}\cdot\frac{1}{2}\to3\cdot\frac{1}{2}=\frac{3}{2}\)

- Record the rounded factors used in each estimate. - Make the comparison from the estimates before presenting exact products. - Then rewrite the mixed numbers as improper fractions for the exact check.

1. Estimate \(A\approx4.5\cdot1.3\approx5.9\) and \(B\approx3.3\cdot1.5\approx5.0\), so predict \(A>B\). 2. Exactly, \(A=\frac{9}{2}\cdot\frac{4}{3}=6\). 3. Exactly, \(B=\frac{10}{3}\cdot\frac{3}{2}=5\). 4. The exact comparison confirms the prediction.

Estimate: \(A\approx4.5\cdot1.3\approx5.9\), \(B\approx3.3\cdot1.5\approx5.0\), so predict \(A>B\). Exact: \(A=6\), \(B=5\). The prediction is confirmed.

- With several factors, you can look for common factors across all numerators and denominators before multiplying. - Count the negative factors first to determine the sign of the product. - Rewrite mixed numbers as improper fractions.

1. For a), rewrite the mixed numbers: \(\frac{6}{5}\cdot\frac{5}{3}\cdot\frac{5}{6}\). Cancel common factors to get \(\frac{5}{3}=1 \frac{2}{3}\). 2. For b), rewrite the mixed numbers: \(-\frac{12}{5}\cdot\frac{15}{8}\cdot\left(-\frac{1}{3}\right)\). Two negative factors give a positive product. Cancel common factors to get \(\frac{3}{2}=1 \frac{1}{2}\).

a) \(1 \frac{2}{3}\) b) \(1 \frac{1}{2}\)

- Calculate each expression separately. - Look for reciprocal factors that simplify to \(1\). - After finding the values, order the letters that represent them.

1. \(A=\frac{7}{4}\cdot\frac{8}{7}=2\). 2. \(B=\frac{8}{3}\cdot\frac{3}{8}=1\). 3. \(C=-\frac{6}{5}\cdot\left(-\frac{5}{2}\right)=3\). 4. Since \(1<2<3\), the order is \(B<A<C\).

\(B<A<C\), with values \(1<2<3\).

- Rewrite mixed numbers as improper fractions before looking for cancellation. - In products, a factor in a numerator may cancel with a factor in any denominator. - For the comparison, put both simplified products in comparable exact forms.

1. a) \(\frac{8}{5}\cdot\left(-\frac{5}{2}\right)\to-\frac{8}{2}=-4\). 2. b) \(-\frac{15}{4}\cdot\left(-\frac{8}{15}\right)\to\frac{8}{4}=2\). 3. c) \(\frac{7}{9}\cdot\frac{9}{7}\cdot(-3)=-3\). 4. d) The products simplify to \(\frac{3}{2}\) and \(\frac{5}{4}\), so the first is greater.

a) \(\frac{8}{5}\cdot\left(-\frac{5}{2}\right)\to-4\) b) \(-\frac{15}{4}\cdot\left(-\frac{8}{15}\right)\to2\) c) \(\frac{7}{9}\cdot\frac{9}{7}\cdot(-3)=-3\) d) \(\frac{3}{2}>\frac{5}{4}\), so \(2\frac{1}{4}\cdot\frac{2}{3}\) is greater.

- Rewrite each division as a fraction before evaluating. - Dividing by a fraction is the same as multiplying by its reciprocal. - For repeated division, work from left to right unless parentheses say otherwise. - Cancel common factors before multiplying. - Keep track of the negative sign in part b).

1. For a), \(7\div10=\frac{7}{10}\), so the complex fraction is \(\frac{\frac{14}{25}}{\frac{7}{10}}\). Then \(\frac{14}{25}\cdot\frac{10}{7}=\frac{4}{5}\). 2. For b), \(-18\div35=-\frac{18}{35}\), so the complex fraction is \(\frac{-\frac{18}{35}}{\frac{9}{14}}\). Then \(-\frac{18}{35}\cdot\frac{14}{9}=-\frac{4}{5}\). 3. For c), division is evaluated from left to right. The nested complex fraction is \(\frac{\frac{\frac{21}{16}}{\frac{7}{8}}}{3}\). First, \(\frac{21}{16}\div\frac{7}{8}=\frac{3}{2}\). Then \(\frac{3}{2}\div3=\frac{1}{2}\).

a) \(\frac{\frac{14}{25}}{\frac{7}{10}}=\frac{4}{5}\) b) \(\frac{-\frac{18}{35}}{\frac{9}{14}}=-\frac{4}{5}\) c) \(\frac{\frac{\frac{21}{16}}{\frac{7}{8}}}{3}=\frac{1}{2}\)

- Convert mixed numbers to improper fractions. - Identify whether \(x\) is a factor, dividend, or divisor. - Evaluate powers and complex fractions before solving.

1. For a), \(x=-\frac{10}{9}\div\frac{8}{3}=-\frac{10}{9}\cdot\frac{3}{8}=-\frac{5}{12}\). 2. For b), \(x=\frac{9}{2}\div\frac{3}{4}=6\). 3. For c), \(\left(-\frac{1}{2}\right)^3=-\frac{1}{8}\). Thus \(x=(-16)\cdot\left(-\frac{1}{8}\right)=2\). 4. For d), \(\frac{2}{5}\div\frac{4}{15}=\frac{3}{2}\). Then \(\frac{3}{2}\div x=\frac{5}{2}\), so \(x=\frac{3}{2}\div\frac{5}{2}=\frac{3}{5}\).

a) \(x=-\frac{5}{12}\) b) \(x=6\) c) \(x=2\) d) \(x=\frac{3}{5}\)

- Simplify the side that does not contain \(x\) first. - Treat the complex fraction as division. - Use the inverse operation to isolate \(x\).

1. Simplify the right side: \(\left(\frac{1}{2}\right)^2=\frac{1}{4}\), and \(\frac{1}{4}\div\frac{5}{8}=\frac{1}{4}\cdot\frac{8}{5}=\frac{2}{5}\). 2. Multiply both sides by \(-2\frac{1}{2}=-\frac{5}{2}\): \(x=\frac{2}{5}\cdot\left(-\frac{5}{2}\right)=-1\).

\(x=-1\)

- Treat the two sides of the subtraction as separate blocks. - Divide by a fraction by multiplying by its reciprocal. - Keep track of signs when subtracting rational numbers.

1. First parentheses: \(\frac{5}{8}-\frac{3}{2}=-\frac{7}{8}\). 2. Divide: \(-\frac{7}{8}\div\frac{7}{4}=-\frac{7}{8}\cdot\frac{4}{7}=-\frac{1}{2}\). 3. Second parentheses: \(2\div\frac{4}{3}=2\cdot\frac{3}{4}=\frac{3}{2}\). 4. Subtract: \(-\frac{1}{2}-\frac{3}{2}=-2\).

\(-2\)

- Focus on the signs of the factors and addends first. - Is the value inside the parentheses in \(A\) positive or negative? - In \(B\), which addend inside the parentheses has the greater absolute value? - Dividing by \(\frac{1}{2}\) is the same as multiplying by \(2\).

1. For \(A\), the quantity \(\frac{1}{2}-\frac{3}{4}\) is negative. A negative number multiplied by a negative number is positive, so \(A\) is positive. 2. For \(B\), \(-2 \frac{2}{3}+1 \frac{1}{6}\) is negative because the negative addend has the greater absolute value. Dividing by the positive number \(\frac{1}{2}\) keeps the result negative, so \(B\) is negative. 3. Evaluate \(A\): \(-\frac{25}{8}\cdot\left(-\frac{1}{4}\right)=\frac{25}{32}\). 4. Evaluate \(B\): \(\left(-\frac{8}{3}+\frac{7}{6}\right)\div\frac{1}{2}=-\frac{3}{2}\cdot2=-3\).

a) \(B\) has a negative value. b) \(A=\frac{25}{32}\); \(B=-3\)

- In a quotient, how can you change both numbers without changing the result? - For b), split one factor into two easier numbers and add the products. - For d), decide the sign first, then use the corresponding whole-number fact.

1. Multiply both numbers in a) by \(10\): \(196\div14=14\). 2. For b), \(1.8(2+0.2)=3.6+0.36=3.96\). 3. Multiply both numbers in c) by \(100\): \(-1001\div7=-143\). 4. For d), calculate \(13\cdot13=169\), then place two decimal digits and give the product a negative sign: \(-1.69\).

a) \(14\) b) \(3.96\) c) \(-143\) d) \(-1.69\)

- Use estimation or multiplication as an inverse operation to check a quotient. - Pay close attention to decimal place value. - Decide the sign before checking a quotient involving a negative number.

1. For a), \(4.5\div9=0.5\), so the equation is correct. 2. For b), \(1.44\div12=0.12\), so \(1.2\) is incorrect. 3. For c), \(-3.6\div4=-0.9\), so the equation is correct. 4. For d), \(0.07\div2=0.035\), so \(0.35\) is incorrect.

a) Correct b) Incorrect; \(0.12\) c) Correct d) Incorrect; \(0.035\)

- Choose an exact representation that creates common factors across the product. - Keep the conversion visible in the response. - Simplify before multiplying when possible.

1. a) \(0.25=\frac{1}{4}\), so \(\frac{1}{4}\cdot\frac{4}{7}=\frac{1}{7}\). 2. b) \(0.8=\frac{4}{5}\), so \(\frac{4}{5}\cdot\frac{3}{4}=\frac{3}{5}\). 3. c) \(1.25=\frac{5}{4}\) and \(0.4=\frac{2}{5}\), so the product is \(\frac{1}{2}\). 4. d) \(0.9=\frac{9}{10}\), so \(\frac{2}{3}\cdot\frac{9}{10}=\frac{3}{5}\).

a) \(0.25=\frac{1}{4}\); result \(\frac{1}{7}\) b) \(0.8=\frac{4}{5}\); result \(\frac{3}{5}\) c) \(1.25=\frac{5}{4}\), \(0.4=\frac{2}{5}\); result \(\frac{1}{2}\) d) \(0.9=\frac{9}{10}\); result \(\frac{3}{5}\)

- Decide whether estimation is enough before calculating exactly. - What happens when a positive number less than \(1\) is multiplied by another positive number less than \(1\)? - For negative numbers, which value lies farther to the right on a number line?

1. \(2.5\div5=0.5\) and \(2.5\cdot0.2=0.5\), so the values are equal. 2. \(-3.6\div6=-0.6\) and \(-3.6\div4=-0.9\). Since \(-0.6>-0.9\), use \(>\). 3. \(0.8\cdot0.8=0.64\), and \(0.64<0.8\), so use \(<\).

a) \(=\) b) \(>\) c) \(<\)

- Convert the decimals and percent to fractions. - Use a common denominator for the subtraction. - To divide by a fraction, multiply by its reciprocal. - Simplify common factors before multiplying.

1. For a), write \(0.5 = \frac{1}{2}\). Then \(\frac{5}{6} - \frac{1}{2} = \frac{5}{6} - \frac{3}{6} = \frac{1}{3}\). 2. For b), write \(1.4 = \frac{7}{5}\). Then \(\frac{7}{5} \cdot \frac{5}{7} = 1\). 3. For c), write \(25\% = \frac{1}{4}\) and \(3\frac{1}{5} = \frac{16}{5}\). Then \(\frac{1}{4} \cdot \frac{16}{5} = \frac{4}{5}\). 4. For d), write \(0.3 = \frac{3}{10}\). Then \(\frac{9}{10} \div \frac{3}{10} = \frac{9}{10} \cdot \frac{10}{3} = 3\).

a) \(\frac{1}{3}\) b) \(1\) c) \(\frac{4}{5}\), or \(0.8\) d) \(3\)

- Look for exact rewrites that expose cancellation or familiar decimal facts. - Show the chosen rewrite before carrying out the operation. - Keep sign reasoning separate from magnitude simplification.

1. a) Write \(0.5=\frac{1}{2}\) and regroup to simplify: \(\left(\frac{12}{25}\cdot\frac{5}{3}\right)\cdot\frac{1}{2}=\frac{2}{5}\). 2. b) Write \(3.6=\frac{18}{5}\), then \(\frac{18}{5}\cdot\frac{10}{9}=4\). 3. c) Cancel common factors in \(-\frac{3}{4}\cdot\frac{16}{21}\) to get \(-\frac{4}{7}\). 4. d) Write \(80\%=0.8\), then \(0.8\div0.4=2\).

a) \(0.5=\frac{1}{2}\); \(\left(\frac{12}{25}\cdot\frac{5}{3}\right)\cdot\frac{1}{2}=\frac{2}{5}\) b) \(3.6=\frac{18}{5}\); result \(4\) c) Cancel in \(-\frac{3}{4}\cdot\frac{16}{21}\); result \(-\frac{4}{7}\) d) \(80\%=0.8\); result \(2\)

- Compare the amount of work created by converting each value to the other representation. - A conversion that creates a repeating decimal may be less convenient than keeping an exact fraction. - For division, consider whether an equivalent quotient can make the divisor easier to use.

1. For a), fractions are efficient because \(\frac{5}{6}\) is a repeating decimal. Write \(0.4 = \frac{2}{5}\). Then \(\frac{2}{5} \cdot \frac{5}{6} = \frac{1}{3}\). 2. For b), fractions are efficient because \(\frac{1}{3}\) is a repeating decimal. Write \(0.9 = \frac{9}{10}\). Then \(\frac{1}{3} \cdot \frac{9}{10} = \frac{3}{10}\). 3. For c), decimals are efficient because both numbers are terminating decimals. Multiply both numbers by \(10\): \(1.25 \div 0.5 = 12.5 \div 5 = 2.5\).

a) \(\frac{1}{3}\). Fractions are especially efficient because converting \(\frac{5}{6}\) to a decimal would produce a repeating decimal; \(0.4=\frac{2}{5}\) then simplifies cleanly with \(\frac{5}{6}\). b) \(\frac{3}{10}=0.3\). Fractions are especially efficient because \(\frac{1}{3}\) is repeating as a decimal, while \(0.9=\frac{9}{10}\) multiplies cleanly as a fraction. c) \(2.5=\frac{5}{2}\). Decimals are efficient because scaling both terminating decimals by \(10\) gives \(12.5\div5\). Fractions are also efficient here: \(\frac{5}{4}\div\frac{1}{2}=\frac{5}{2}\). Other exact choices are acceptable when the explanation accurately compares the work created by the chosen representations.

- Evaluate the expression inside the parentheses first. - A fraction form may be more useful when a fraction has a repeating decimal representation. - Break the calculation into separate multiplication and division steps.

1. Evaluate the expression inside the parentheses first. Write \(0.6 = \frac{3}{5}\). 2. Multiply: \(\frac{3}{5} \cdot \frac{5}{9} = \frac{1}{3}\). 3. Write \(1.2 = \frac{6}{5}\). 4. Divide by multiplying by the reciprocal: \(\frac{1}{3} \div \frac{6}{5} = \frac{1}{3} \cdot \frac{5}{6} = \frac{5}{18}\).

\(\frac{5}{18}\)

- Focus on how the value of a factor changes when one numerator or denominator changes. - Express that change as a multiplier applied to \(P\). - Combine the two factor changes in part c) before applying them to \(P\).

1. a) Doubling a numerator doubles that factor, so the new product is \(2P=\frac{3}{5}\). 2. b) Doubling a denominator halves that factor, so the new product is \(\frac{1}{2}P=\frac{3}{20}\). 3. c) Doubling the numerator multiplies the factor by \(2\), and halving its denominator multiplies it by another \(2\). The new product is \(4P=\frac{6}{5}\).

a) Factor change \(\times2\); new product \(2P=\frac{3}{5}\) b) Factor change \(\times\frac{1}{2}\); new product \(\frac{1}{2}P=\frac{3}{20}\) c) Factor change \(\times4\); new product \(4P=\frac{6}{5}\)

- Evaluate the two expressions separately. - It may help to write fractions and decimals in the same form. - How do you divide by a fraction?

1. Expression A: \(\frac{3}{4}=0.75\), so \(0.75-1.25=-0.5\). Then \((-0.5)\cdot(-4)=2\). 2. Expression B: \(2.5\div\frac{1}{2}=5\), so \(5-6=-1\). 3. Since \(2>-1\), Expression A has the greater value.

Expression A is greater because \(2>-1\).

- Use the order of operations before adding or subtracting. - Determine the sign of each product or quotient before calculating. - For powers, think about how many negative factors are being multiplied. - Converting the simple fraction to a decimal may help in e).

1. For a), the negative product has magnitude \(0.48\), which is slightly less than \(0.5\), so predict a positive result. Then \((-0.4)\cdot1.2=-0.48\), and \(-0.48+0.5=0.02\). 2. For b), the quotient is positive and less than \(0.8\), so predict a negative result. Then \((-2.5)\div(-5)=0.5\), and \(0.5-0.8=-0.3\). 3. For c), \(0.2\cdot3\) is greater than \(0.12\), so predict a negative result. Then \(0.2\cdot3=0.6\), and \(0.12-0.6=-0.48\). 4. For d), \((-3)^3\) is negative with magnitude \(27\), which is less than \(30\), so predict a positive result. Then \((-3)^3=-27\), and \(-27+30=3\). 5. For e), \(0.3\div0.5=0.6\), which is greater than \(\frac{1}{2}\), so predict a positive result. Then \(-\frac{1}{2}+0.6=-0.5+0.6=0.1\).

a) Positive; \(0.02\) b) Negative; \(-0.3\) c) Negative; \(-0.48\) d) Positive; \(3\) e) Positive; \(0.1\)

- Sometimes the sign alone is enough to compare two results. - Think about which result lies farther to the right on a number line. - What happens to absolute value when you divide by a positive number between \(0\) and \(1\)? - Sign rules can often settle a comparison without exact arithmetic.

1. For a), \((-0.5)^2\) is positive and \((-0.5)^3\) is negative, so the left side is greater. 2. For b), adding \(2.1\) moves \(-5.5\) to the right on the number line, while subtracting \(2.1\) moves it to the left. Therefore, the left side is greater. 3. For c), the product of three negative factors is negative, so it is less than \(0\). 4. For d), dividing by \(0.5\) doubles the absolute value, while multiplying by \(0.5\) halves it. Both results are negative, so the division result is smaller.

a) \(>\) b) \(>\) c) \(<\) d) \(<\)

- Write the power as repeated multiplication. - How many times is the base used as a factor? - Track the decimal places when multiplying decimals. - What sign should an odd power of a negative number have?

1. Expand the power: \((-0.5)^3=(-0.5)\cdot(-0.5)\cdot(-0.5)\). 2. Multiply: \((-0.5)\cdot(-0.5)=0.25\), and \(0.25\cdot(-0.5)=-0.125\). Therefore, B is correct. 3. A results from multiplying the base by the exponent instead of using repeated multiplication. 4. C has the wrong sign; an odd power of a negative number is negative. D has a decimal place-value error.

B is correct. A: The exponent was treated as a factor. C: The sign is incorrect. D: The decimal place value is incorrect.

- Convert between fractions and decimals when useful. - Write an equation for each missing divisor or factor. - Apply the sign rules before calculating.

1. For a), \(-\frac{2}{5}\cdot0.4=-0.4\cdot0.4=-0.16\). 2. For b), solve \(-0.16\div x=-1\). Thus \(x=0.16=\frac{4}{25}\). 3. For c), \(-0.16+0.2=0.04=\frac{1}{25}\). Its reciprocal is \(25\), so \(0.04\cdot25=1\).

a) \(-0.16\), or \(-\frac{4}{25}\) b) \(0.16\), or \(\frac{4}{25}\) c) \(25\)

- Numbers whose product is \(1\) are reciprocals. - Determine the sign of the sum of two negative numbers. - Use the sign rule for division.

1. Write \(-2\frac{1}{2}=-\frac{5}{2}\). The other factor must be its reciprocal, \(-\frac{2}{5}\). 2. Their sum is \(-2.5+(-0.4)=-2.9\), which is negative. 3. Both numbers are negative, so their quotient is positive.

a) \(-\frac{2}{5}\), or \(-0.4\) b) \(-2.9\); negative c) Positive, because both numbers are negative.

- Determine the sign of each result first. - Think about where the results would lie on a number line. - Comparing absolute values can sometimes determine the correct symbol without exact arithmetic.

1. Multiplying \(-1.2\) by \(5\) increases its absolute value, while dividing by \(5\) decreases its absolute value. Both results are negative, so the multiplication result is smaller. Thus, use \(<\). 2. \(-3\cdot(-4)\) is positive, while \(-3+(-4)\) is negative. Therefore, use \(>\). 3. \(-\frac{1}{2}+\frac{1}{3}=-\frac{1}{6}\), and \(-\frac{1}{2}\cdot\frac{1}{3}=-\frac{1}{6}\). Therefore, use \(=\).

a) \(<\) b) \(>\) c) \(=\)

- If both factors are written as sums, identify every partial product required by the distributive property. - Convert the mixed number and decimal to fractions. - Simplify common factors before multiplying.

1. The factors were multiplied part by part, which omits the cross products required when multiplying two sums. The work also loses the negative sign on the fractional part of \(-2\frac{2}{3}\). Converting both factors to fractions is more efficient here. 2. Write \(-2\frac{2}{3} = -\frac{8}{3}\) and \(1.5 = \frac{3}{2}\). 3. Multiply and simplify: \(-\frac{8}{3} \cdot \frac{3}{2} = -4\).

The calculation omits the cross products required by the distributive property and loses the negative sign on the fractional part of the mixed number. The correct result is \(-2\frac{2}{3} \cdot 1.5 = -\frac{8}{3} \cdot \frac{3}{2} = -4\).

- Look for an exact fraction form that shares factors with the other rational number. - A repeating decimal should remain exact rather than being truncated. - Show the conversion you use before the operation.

1. a) \(-0.12=-\frac{3}{25}\), giving \(-\frac{1}{10}\). 2. b) \(0.375=\frac{3}{8}\), giving \(\frac{3}{8}\cdot\frac{4}{3}=\frac{1}{2}\). 3. c) \(-1.2=-\frac{6}{5}\), giving \(-\frac{2}{3}\). 4. d) \(0.\overline{6}=\frac{2}{3}\), giving \(4\cdot\frac{3}{2}=6\).

a) \(-0.12=-\frac{3}{25}\); result \(-\frac{1}{10}\) b) \(0.375=\frac{3}{8}\); result \(\frac{1}{2}\) c) \(-1.2=-\frac{6}{5}\); result \(-\frac{2}{3}\) d) \(0.\overline{6}=\frac{2}{3}\); result \(6\)

- Apply multiplication and division before subtraction. - Convert the mixed number to a decimal or improper fraction. - Determine the sign of each product or quotient. - Subtracting a negative number is equivalent to adding its opposite.

1. Convert the mixed number: \(1\frac{1}{5} = 1.2\). 2. Multiply: \(1.2 \cdot (-0.5) = -0.6\). 3. Divide: \(2.4 \div (-3) = -0.8\). 4. Subtract: \(-0.6 - (-0.8) = -0.6 + 0.8 = 0.2\).

\(0.2\)

- Recall the fraction equivalent of \(0.375\). - Complete both methods before comparing them. - Compare \(0.375\) with \(0.75\).

1. Method 1: Write \(\frac{3}{4} = 0.75\) and \(\frac{1}{2} = 0.5\). Then \(0.375 \div 0.75 = 0.5\), and \(0.5 + 0.5 = 1\). 2. Method 2: Write \(0.375 = \frac{3}{8}\). Then \(\frac{3}{8} \div \frac{3}{4} = \frac{3}{8} \cdot \frac{4}{3} = \frac{1}{2}\), and \(\frac{1}{2} + \frac{1}{2} = 1\). 3. Both methods are valid. The fraction method may feel easier because common factors simplify immediately; the decimal method may feel easier if you recognize that \(0.375\) is half of \(0.75\).

Both methods give \(1\). A valid explanation should compare the simplification in the fraction method with the recognizable decimal relationship in the decimal method.

- Convert a repeating decimal to a fraction if useful. - Consider what kind of number results from multiplying integers. - Determine the sign from the number of negative factors.

1. For a), possible. For example, \(0.\overline{3}\cdot1.8\cdot1=0.6\). 2. For b), not possible. A product of integers is an integer, but \(0.6\) is not an integer. 3. For c), not possible. Exactly one negative factor would make the product negative.

a) Yes; for example, \(0.\overline{3}\cdot1.8\cdot1=0.6\). b) No; a product of integers must be an integer. c) No; the product would be negative.

- In a), find the product of the two known factors before finding the missing one. - In b), test whether the proposed factor signs agree with the stated product. - How many separate factors are doubled in c)?

1. The known factors give \((-0.5)(-0.5)=0.25\). So \(c=-0.125\div0.25=-0.5\). 2. Two negative factors and one positive factor give a positive product, not \(-0.125\). This is impossible. 3. Doubling each factor multiplies the product by \(2\cdot2\cdot2=8\). The new product is \(8(-0.125)=-1\).

a) \(c=-0.5\) b) No. Two negative factors and one positive factor have a positive product. c) \(-1\)

- Distribute the factor in \(A\) and compare the resulting terms with \(B\). - Pay close attention to the product of two negative numbers. - Find the value of the first two terms in \(B\), then determine its additive inverse.

1. Distributing \(-\frac{3}{4}\) in \(A\) gives \(-\frac{3}{4}\cdot2.4-\left(-\frac{3}{4}\cdot6.4\right)+1.5\), which simplifies to \(B\). 2. Evaluate \(A\): \(2.4-6.4=-4\), so \(-\frac{3}{4}(-4)+1.5=3+1.5=4.5\). 3. The part of \(B\) before the final number equals \(3\). Solve \(3+x=0\), giving \(x=-3\).

1. The distributive property shows that the expressions are equivalent. 2. \(A=4.5\) 3. Replace \(1.5\) with \(-3\).

- Search for pairs whose product is \(1\). - Keep track of which property permits reordering and which permits regrouping. - Show the grouped product in the answer.

1. Regroup as \(\left(\frac{5}{9}\cdot1.8\right)\left((-0.4)(-2.5)\right)\). 2. Each group equals \(1\), so the product is \(1\). 3. The commutative property changes order and the associative property changes grouping.

\(\left(\frac{5}{9}\cdot1.8\right)\left((-0.4)(-2.5)\right)=1\cdot1=1\). Properties: commutative and associative properties of multiplication.

- Convert the mixed number to a decimal for the first method. - Determine the sign of the value inside the parentheses. - The product of two negative numbers is positive. - Compare the number of conversions and simplifications in each method.

1. Decimal method: Write \(1\frac{1}{5} = 1.2\). Then \(0.8 - 1.2 = -0.4\), and \(-0.4 \cdot (-2.5) = 1\). 2. Fraction method: Write \(0.8 = \frac{4}{5}\), \(1\frac{1}{5} = \frac{6}{5}\), and \(-2.5 = -\frac{5}{2}\). Then \(\frac{4}{5} - \frac{6}{5} = -\frac{2}{5}\), and \(-\frac{2}{5} \cdot \left(-\frac{5}{2}\right) = 1\). 3. Both methods are efficient because every number has a simple terminating decimal and fraction form.

Both methods give \(1\). A valid explanation may favor either method if it refers to the simple conversions or simplifications used.

- Determine whether the fractional parts have terminating decimal representations. - Add fractions with unlike denominators using a common denominator. - To divide by a fraction, multiply by its reciprocal.

1. Fractions are more useful because \(\frac{1}{3}\) and \(\frac{1}{6}\) have repeating decimal representations. 2. Write \(1.75 = \frac{7}{4}\), \(2\frac{1}{3} = \frac{7}{3}\), and \(-1\frac{1}{6} = -\frac{7}{6}\). 3. Add inside the parentheses: \(\frac{7}{4} + \frac{7}{3} = \frac{21}{12} + \frac{28}{12} = \frac{49}{12}\). 4. Divide by multiplying by the reciprocal: \(\frac{49}{12} \cdot \left(-\frac{6}{7}\right) = -\frac{7}{2} = -3.5\).

\(-\frac{7}{2}\), or \(-3.5\). Fractions avoid rounded repeating decimals in the calculation.

- Apply powers before multiplication or division, and multiplication or division before addition or subtraction. - How do the parentheses in b) change which operation is performed last? - Pay attention to the sign when squaring a negative number.

1. For a), the last operation is subtraction, so the expression is a difference. Compute \((-0.4)^2=0.16\) and \(0.16\div0.4=0.4\). Then \(0.16-0.4=-0.24\). 2. For b), the last operation is division, so the expression is a quotient. Inside the parentheses, \((-0.4)^2-0.16=0.16-0.16=0\). Then \(0\div0.4=0\).

a) Structure: difference; value: \(-0.24\) b) Structure: quotient; value: \(0\)

- Look for reciprocal factors or factors that make a power of \(10\). - You may change order and grouping but not the factors themselves. - Include the reordered/regrouped form in each answer.

1. a) \(\left(\frac{2}{9}\cdot\frac{9}{2}\right)\left(\frac{5}{11}\cdot22\right)=1\cdot10=10\). 2. b) \((12.5\cdot8)\cdot0.7=100\cdot0.7=70\).

a) \(\left(\frac{2}{9}\cdot\frac{9}{2}\right)\left(\frac{5}{11}\cdot22\right)=10\) b) \((12.5\cdot8)\cdot0.7=70\)

- Apply the order of operations inside each set of parentheses. - How do you divide by a fraction? - Simplify the final fraction or write it as a mixed number.

1. Evaluate the first parentheses: \(-4.5=-\frac{9}{2}\), so \(-\frac{9}{2}\div\frac{9}{10}=-\frac{9}{2}\cdot\frac{10}{9}=-5\). 2. Evaluate the second parentheses: \(1.2=\frac{6}{5}\), so \(\frac{2}{3}-\frac{6}{5}=\frac{10}{15}-\frac{18}{15}=-\frac{8}{15}\). 3. Multiply: \(-5\cdot\left(-\frac{8}{15}\right)=\frac{8}{3}=2 \frac{2}{3}\).

\(\frac{8}{3}\), or \(2 \frac{2}{3}\)

- A fraction bar acts like division and groups the numerator and denominator. - Which operation is performed last in the entire expression? - Evaluate the numerator and denominator separately. - Use sign rules when dividing rational numbers.

1. The overall expression is a quotient. Its numerator is the sum of \(-3.6\) and \(1.2\), and its denominator is the quotient of \(-3.6\) and \(1.2\). 2. Numerator: \(-3.6+1.2=-2.4\). 3. Denominator: \(-3.6\div1.2=-3\). 4. Final division: \(-2.4\div(-3)=0.8\).

a) The expression is the quotient of the sum \(-3.6+1.2\) and the quotient \(-3.6\div1.2\). b) \(T=0.8\)

- Start with the innermost parentheses. - What sign results when two negative numbers are multiplied? - How do you divide by a fraction? - Decide whether a fraction or decimal form is more useful for the final result.

1. Evaluate the inner parentheses: \(\frac{1}{5}-0.6=0.2-0.6=-0.4\). 2. Multiply inside the brackets: \((-2.5)\cdot(-0.4)=1\). 3. Divide: \(1\div\left(-\frac{1}{2}\right)=-2\). 4. Subtract: \(\frac{5}{8}-(-2)=\frac{21}{8}=2 \frac{5}{8}=2.625\).

\(2 \frac{5}{8}\), or \(2.625\)

- What happens when \(-1\) is raised to an odd power? - Does an even or odd exponent change the sign? - Does changing the order of factors change a product?

1. Since \(3\) is odd, \((-1)^3=-1\). Therefore, Expression A is \(-1\cdot4.2=-4.2\). 2. Since \(7\) is odd, \((-1)^7=-1\). Therefore, Expression B is \(4.2\cdot(-1)=-4.2\). 3. Both expressions have the same value.

Yes. Both expressions equal \(-4.2\) because both odd powers of \(-1\) equal \(-1\), and multiplication can be written in either factor order.

- In a), rearrange the factors to pair the two decimals. - Convert the decimals to fractions to check whether their product is \(1\). - In b), compare dividing by twice a number with dividing by the number and then doubling.

1. For a), rearrange and regroup the factors: \((0.8 \cdot 1.25)\frac{3}{7}\). Since \(0.8 = \frac{4}{5}\) and \(1.25 = \frac{5}{4}\), their product is \(1\). The equation is true. 2. For b), dividing by a product is equivalent to dividing successively by both factors: \(a \div (bc) = a \div b \div c\). The right side divides by \(\frac{1}{3}\) and then multiplies by \(2\), so the equation is false.

a) True, because \(0.8 \cdot 1.25 = 1\). b) False, because dividing by \(\frac{1}{3} \cdot 2\) is not the same as dividing by \(\frac{1}{3}\) and then multiplying by \(2\).

- For multiplication only, you can regroup factors to make the arithmetic easier. - Determine the sign as you work through each expression. - When multiplication and division are mixed, work from left to right.

1. For a), regroup the factors: \((-4)\cdot(-8)=32\), then \(32\cdot125=4000\). 2. For b), work from left to right: \((-3600)\div(-15)=240\), then \(240\div(-4)=-60\). 3. For c), \(12\cdot(-15)=-180\), then \(-180\div(-9)=20\), and \(20\cdot(-2)=-40\).

a) \(4000\) b) \(-60\) c) \(-40\)

- Evaluate the parentheses first. - Use a common denominator for the subtraction. - Simplify a fraction before the next step when possible.

1. Subtract inside the parentheses: \(\frac{5}{6}-\frac{1}{2}=\frac{5}{6}-\frac{3}{6}=\frac{2}{6}=\frac{1}{3}\). 2. Multiply: \(\frac{3}{4}\cdot\frac{1}{3}=\frac{1}{4}\).

\(\frac{1}{4}\)

- Would fractions or decimals be easier for this expression? - Use one form consistently before calculating. - Determine the sign of the product before multiplying.

1. Write the quantities in a common form: \(1 \frac{1}{5}=1.2\) and \(\frac{7}{10}=0.7\). 2. Evaluate the parentheses: \(0.2-0.7=-0.5\). 3. Multiply: \(1.2\cdot(-0.5)=-0.6=-\frac{3}{5}\).

\(-0.6\), or \(-\frac{3}{5}\)

- Evaluate each set of parentheses separately. - How do you divide by a fraction? - What sign results when a negative number is divided by a negative number?

1. First parentheses: \(\frac{5}{6}-\frac{4}{3}=\frac{5}{6}-\frac{8}{6}=-\frac{1}{2}\). 2. Second parentheses: \(0.75-1=-0.25=-\frac{1}{4}\). 3. Divide: \(-\frac{1}{2}\div\left(-\frac{1}{4}\right)=-\frac{1}{2}\cdot(-4)=2\).

\(2\)

- Track only the signs first for part a). - Subtracting a negative number is equivalent to adding its opposite. - How do you divide by a fraction?

1. For a), \(\frac{1}{4}\cdot(-3)\) is negative, so subtracting it makes the bracket value larger. In fact, its magnitude is \(\frac{3}{4}>\frac{5}{12}\), so the bracket value is positive. A positive number divided by a negative number is negative. 2. For b), \(\frac{1}{4}\cdot(-3)=-\frac{3}{4}\). 3. Then \(-\frac{5}{12}-\left(-\frac{3}{4}\right)=-\frac{5}{12}+\frac{9}{12}=\frac{1}{3}\). 4. Divide: \(\frac{1}{3}\div\left(-\frac{1}{6}\right)=\frac{1}{3}\cdot(-6)=-2\).

a) The final result is negative. b) \(-2\)

- For an estimate, round to numbers that are easy to use mentally. - Follow the order of operations for the exact calculation. - Convert a mixed number to a decimal if that makes the arithmetic easier.

1. For a), estimate \(4.2\approx4\) and \(3 \frac{4}{5}=3.8\approx4\), so \((4+4)\cdot0.5\approx4\). 2. Exactly, \(4.2+3.8=8\), and \(8\cdot0.5=4\). 3. For b), estimate \(10.5\approx11\) and \(2.5\approx3\), so \(11-3\cdot3\approx2\). 4. Exactly, multiply before subtracting: \(10.5-2.5\cdot3=10.5-7.5=3\).

a) Estimate: about \(4\); exact value: \(4\) b) Estimate: about \(2\); exact value: \(3\)

- What order of operations applies when there are no parentheses? - How do parentheses change which quantity is the divisor? - Evaluate each expression step by step.

1. For \(A\), evaluate the parentheses first: \(1.2+0.9=2.1\). Then \(8.4\div2.1=4\). 2. For \(B\), division comes before addition: \(8.4\div1.2=7\), then \(7+0.9=7.9\). 3. The parentheses in \(A\) make the entire sum the divisor, while in \(B\) only \(1.2\) is the divisor.

\(A=4\); \(B=7.9\). The parentheses change the structure and order of operations.

- Converting the fraction to a decimal may simplify the first subtraction. - Pay attention to the sign when dividing. - Evaluate both sets of parentheses first.

1. First parentheses: \(1.2-\frac{3}{4}=1.2-0.75=0.45\). 2. Second parentheses: \(0.2-0.5=-0.3\). 3. Divide: \(0.45\div(-0.3)=-1.5=-\frac{3}{2}\).

\(-1.5\), or \(-\frac{3}{2}\)

- Treat the numerator and denominator as separate expressions first. - Convert between fractions and decimals when that makes a step easier. - A fraction bar means divide the completed numerator by the completed denominator.

1. Numerator: \(\frac{1}{5}-0.6=0.2-0.6=-0.4\), so \(2.5\cdot(-0.4)=-1\). 2. Denominator: \(0.25\cdot8=2\), then \(2-1.5=0.5\). 3. Divide numerator by denominator: \(-1\div0.5=-2\).

\(-2\)

- Follow the grouping and work from left to right. - Determine the sign at each division step. - What sign results when dividing two negative numbers?

1. First divide: \(-2160\div18=-120\). 2. Then divide again: \(-120\div(-3)=40\).

\(40\)

- Convert fractions and decimals to the same form when useful. - Pay attention to signs in division and subtraction. - Apply multiplication and division before addition or subtraction. - Among negative numbers, the value closer to \(0\) is greater.

1. For a), \(1.2\div(-3)=-0.4\), so \(-0.5-0.4=-0.9\). 2. For b), \(0.4\cdot(-2)=-0.8\), and \(-0.8+0.6=-0.2\). 3. For c), \(-\frac{1}{4}-(-0.15)=-0.25+0.15=-0.1\). 4. Since \(-0.1>-0.2>-0.9\), expression c) has the greatest value.

Expression c) has the greatest value, \(-0.1\).

- Review the sign rules for multiplication and division. - Adding two negative numbers gives a negative sum. - Subtracting a negative number is equivalent to adding.

1. The product of two negative numbers is positive: \((-5)\cdot(-2)=10\). 2. The sum of two negative numbers is negative: \(-5+(-2)=-7\). 3. Subtracting a negative number gives \(y-x=-2-(-5)=-2+5=3\), which is positive. 4. The quotient of two negative numbers is positive: \(\frac{-5}{-2}=\frac{5}{2}=2.5\).

a) Positive; \(10\) b) Negative; \(-7\) c) Positive; \(3\) d) Positive; \(2.5\)

- Count negative factors to determine the sign of a product. - You may regroup factors in a product to make the arithmetic easier. - What sign results when a negative number in parentheses is squared?

1. For a), four negative factors give a positive product: \(1\cdot2\cdot3\cdot4=24\). 2. For b), \(-\frac{1}{2}\cdot8=-4\), then \((-4)\cdot(-0.25)=1\). 3. For c), \((-0.1)^2=0.01\), then \(0.01\cdot(-1000)=-10\). 4. For d), \((-5)\cdot\left(-\frac{2}{5}\right)=2\), then \(2\cdot(-3)=-6\).

a) \(24\) b) \(1\) c) \(-10\) d) \(-6\)

- Determine the sign before calculating the magnitude. - Simplify fractions with whole-number factors when possible. - Expand a power such as \((-1)^4\) as repeated multiplication if needed. - Converting a simple decimal to a fraction may help.

1. For a), \((-12)\cdot\frac{3}{4}=-9\), then \((-9)\cdot(-2)=18\). 2. For b), \(-\frac{2}{5}\cdot(-10)=4\), then \(4\cdot(-0.5)=-2\). 3. For c), \((-1)^4=1\), so \(1\cdot7\cdot(-3)=-21\). 4. For d), \(\frac{4}{9}\cdot(-18)=-8\), then \((-8)\cdot\left(-\frac{1}{2}\right)=4\).

a) \(18\) b) \(-2\) c) \(-21\) d) \(4\)

- Evaluate each expression separately. - What sign results when a negative number is squared? - Among negative numbers, which one lies farther to the right on a number line?

1. For \(X\), \((-4)^2=16\), so \(X=16\cdot(-0.5)=-8\). 2. For \(Y\), \((-2)\cdot(-2)=4\), then \(4\cdot(-2.5)=-10\). 3. Since \(-8>-10\), \(X\) is greater.

\(X\) is greater because \(-8>-10\).

- Work from the bottom row upward. - Each missing block depends only on the two blocks directly below it. - Pay attention to the sign of each product.

1. For the left middle block, multiply the two blocks below it: \((-1.2)\cdot(-5)=6\). 2. For the right middle block, \((-5)\cdot0.25=-1.25\). 3. For the top block, multiply the two middle values: \(6\cdot(-1.25)=-7.5\).

Middle row: \(6\) and \(-1.25\) Top block: \(-7.5\)

- Order the numbers before identifying the least and greatest. - Look for a decimal whose magnitude is the reciprocal of an integer in the set. - A positive product comes from factors with the same sign.

1. The least number is \(-8\), and the greatest is \(4\). Their product is \(-32\). 2. Since \(0.125=\frac{1}{8}\), \((-8)\cdot0.125=-1\). 3. The possible positive products from equal signs are \((-8)\cdot(-0.5)=4\) and \(0.125\cdot4=0.5\). The greater product is \(4\).

a) \(-32\) b) \(-8\) and \(0.125\) c) \(-8\) and \(-0.5\); the product is \(4\).

- Check whether each side uses addition or multiplication. - Determine the sign of a product before comparing values. - Evaluate both sides before choosing the comparison symbol.

1. For a), \(0.2\cdot0.3=0.06\) and \(0.2+0.3=0.5\), so \(0.06<0.5\). 2. For b), \(1.2\cdot0.5=0.6\) and \(1.2\cdot1.5=1.8\), so \(0.6<1.8\). 3. For c), \((-0.4)\cdot0.5=-0.2\) and \((-0.4)\cdot(-0.5)=0.2\), so \(-0.2<0.2\). 4. For d), \(0.25\cdot4=1\) and \(0.5\cdot2=1\), so the values are equal.

a) \(<\) b) \(<\) c) \(<\) d) \(=\)

- The sign depends on whether the number of negative factors is even or odd. - In c), represent the number of negative factors with a variable. - Use the total of \(15\) factors to determine how many are negative.

1. A product with an odd number of negative factors is negative; a product with an even number of negative factors is positive. 2. For a), \(7\) is odd, so the product is negative. 3. For b), all \(15\) factors are negative. Since \(15\) is odd, the product is negative. 4. For c), let \(n\) be the number of negative factors. Then there are \(2n\) positive factors, so \(n + 2n = 15\). Thus, \(n = 5\). Since \(5\) is odd, the product is negative.

a) Negative, because \(7\) is odd. b) Negative, because \(15\) is odd. c) Negative, because there are \(5\) negative factors and \(10\) positive factors.

- Count the negative factors and decide whether that count is even or odd. - In b), subtract the number of positive factors from \(12\). - In c), determine how changing one sign affects the number of negative factors.

1. For a), yes. With \(12\) negative factors, the number of negative factors is even, so the product is positive. 2. For b), the other \(12 - 3 = 9\) factors are negative. Since \(9\) is odd, the product is negative. 3. For c), changing the sign of one factor changes the number of negative factors by \(1\). Its parity switches from even to odd or from odd to even, so the sign of the product always reverses. Jonah is correct.

a) Yes; the product would be positive. b) Negative, because there are \(9\) negative factors. c) Jonah is correct. Reversing one factor’s sign changes the parity of the number of negative factors, so the product’s sign reverses.

- Count the negative factors in each product. - Use the sign rule for multiplying a positive number by a negative number. - A product of negative factors depends on whether their count is even or odd.

1. Product \(A\) has \(10\) negative factors. Since \(10\) is even, \(A\) is positive. Product \(B\) has \(11\) negative factors, so \(B\) is negative. 2. A positive number times a negative number is negative, so \(A \cdot B\) is negative. 3. Equivalently, \(A \cdot B\) contains \(10 + 11 = 21\) negative factors, and \(21\) is odd. 4. A product of \(n\) negative numbers is negative exactly when \(n\) is odd.

a) \(A\) is positive. b) \(A \cdot B\) is negative. c) The product is negative when \(n\) is odd.

- Evaluate parentheses first. - How do you divide by a fraction? - Multiplication and division come before subtraction. - What sign results when multiplying two negative numbers?

1. For a), \(\frac{2}{5}-1=-\frac{3}{5}\) and \(-2 \frac{1}{2}=-\frac{5}{2}\). Then \(-\frac{3}{5}\cdot\left(-\frac{5}{2}\right)=\frac{3}{2}=1.5\). 2. For b), \(\frac{3}{8}\div\left(-\frac{1}{4}\right)=\frac{3}{8}\cdot(-4)=-1.5\). Also, \(0.5\cdot3=1.5\). Therefore, \(-1.5-1.5=-3\).

a) \(1.5\), or \(1 \frac{1}{2}\) b) \(-3\)

- Recognize common fraction-decimal equivalents. - In part b), look for terms that become additive inverses after conversion. - Convert the decimal to a fraction before dividing in part c).

1. For a), write \(0.25=\frac{1}{4}\). Then \(\frac{1}{4}\cdot\frac{4}{7}=\frac{1}{7}=0.\overline{142857}\). 2. For b), write \(0.125=\frac{1}{8}\). Then \(\frac{3}{4}+\frac{1}{8}-\frac{1}{8}=\frac{3}{4}=0.75\). 3. For c), write \(0.6=\frac{3}{5}\). Then \(\frac{3}{5}\div\frac{2}{3}=\frac{3}{5}\cdot\frac{3}{2}=\frac{9}{10}=0.9\).

a) \(\frac{1}{7}=0.\overline{142857}\) b) \(\frac{3}{4}=0.75\) c) \(\frac{9}{10}=0.9\)

- In a), how could you rewrite the division with a whole-number divisor? - In b), look for factors that can be canceled before multiplying. - In c), group factors that make easy products, but keep track of their signs.

1. For a), divide first: \((-15.3)\div0.3=-153\div3=-51\). Then \((-51)\cdot(-0.5)=25.5\). 2. For b), cancel common factors: \(\frac{7}{11}\cdot\frac{22}{14}=\frac{1}{2}\cdot2=1\). Then \(1-1=0\). 3. For c), group the first two and last two factors: \(((-8)\cdot1.25)((-3)\cdot(-2))=(-10)\cdot6=-60\).

a) \((-15.3)\div0.3=-51\), then \((-51)\cdot(-0.5)=25.5\) b) \(\frac{7}{11}\cdot\frac{22}{14}=1\), then \(1-1=0\) c) \(((-8)\cdot1.25)((-3)\cdot(-2))=-60\)

- Use inverse operations to find each missing number. - For division, ask which divisor produces the given quotient. - Pay close attention to the signs when rearranging each equation.

1. For a), divide by the known factor: \(1.5\div(-0.5)=-3\). 2. For b), let the missing number be \(x\). From \(\frac{2}{7}\div x=-\frac{4}{21}\), compute \(x=\frac{2}{7}\div\left(-\frac{4}{21}\right)=\frac{2}{7}\cdot\left(-\frac{21}{4}\right)=-\frac{3}{2}\). 3. For c), multiply the quotient by the divisor: \(\frac{9}{4}\cdot\left(-\frac{2}{3}\right)=-\frac{3}{2}\).

a) \(-3\) b) \(-\frac{3}{2}\) c) \(-\frac{3}{2}\)

- Find the reciprocal by switching the numerator and denominator. - Use the sign rules for multiplying and dividing rational numbers. - Compare the division in b) with multiplication by the reciprocal in c).

1. For a), \(-\frac{3}{4}\cdot\frac{2}{3}=-\frac{6}{12}=-\frac{1}{2}\). 2. For b), \(-\frac{3}{4}\div\frac{2}{3}=-\frac{3}{4}\cdot\frac{3}{2}=-\frac{9}{8}\). 3. For c), the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). Then \(-\frac{3}{4}\cdot\frac{3}{2}=-\frac{9}{8}\). 4. The results in b) and c) are equal, confirming that dividing by a nonzero fraction is equivalent to multiplying by its reciprocal.

a) \(-\frac{1}{2}\) b) \(-\frac{9}{8}\) c) \(-\frac{9}{8}\); it matches the result from part b).

- Multiplication and division come before subtraction. - What sign results when a positive number is divided by a negative number? - Pay attention to both the subtraction sign and the negative product.

1. Divide: \(18\div(-3)=-6\). 2. Multiply: \(4\cdot(-2.5)=-10\). 3. Subtract: \(-6-(-10)=4\).

\(4\)

- How do the parentheses change the order of operations? - Compare the structures of the two expressions carefully. - Predict the sign of each result before calculating.

1. For \(A\), evaluate the parentheses first: \(2.4-5.4=-3\). Then \(-3\cdot(-3)=9\). 2. For \(B\), multiply first: \(-3\cdot2.4=-7.2\). Then \(-7.2-5.4=-12.6\). 3. Since \(9>-12.6\), expression \(A\) has the greater value.

Expression \(A\) has the greater value because \(9>-12.6\).

- Rewrite division by the fraction as multiplication by its reciprocal. - Think about how a positive factor less than \(1\) affects the magnitude of a negative product. - Compare the two negative results by their positions on a number line.

1. \(x=(-10)\cdot\frac{2}{5}=-4\). 2. \(y=(-10)\div\frac{2}{5}=(-10)\cdot\frac{5}{2}=-25\). 3. Since \(-4>-25\), \(x\) lies farther to the right.

\(x=-4\) and \(y=-25\). The value \(x\) lies farther to the right because \(-4>-25\).

- Convert the mixed number and decimal to fractions. - Evaluate parentheses first. - How do you divide two fractions? - Track signs carefully.

1. First parentheses: \(\frac{4}{3}-\frac{1}{2}=\frac{8}{6}-\frac{3}{6}=\frac{5}{6}\). 2. Second parentheses: \(\frac{1}{6}-1=-\frac{5}{6}\). 3. Divide: \(\frac{5}{6}\div\left(-\frac{5}{6}\right)=-1\). 4. Multiply: \(-1\cdot\frac{3}{5}=-\frac{3}{5}\).

\(-\frac{3}{5}\), or \(-0.6\)

- What is the difference between \(-x^2\) and \((-x)^2\)? - Work from inside parentheses outward when powers are involved. - Track signs carefully in multiplication.

1. For a), \(-2^4=-16\) because the exponent applies before the leading negative. Also, \((-3)^2=9\), so \(9\cdot\frac{1}{9}=1\). Thus, \(-16+1=-15\). 2. For b), \(\frac{1}{2}-\frac{2}{3}=-\frac{1}{6}\). Squaring gives \(\frac{1}{36}\), so \(\frac{5}{6}-\frac{1}{36}=\frac{29}{36}\). 3. For c), \(10-10^2=10-100=-90\). Then \(-0.1\cdot(-90)=9\).

a) \(-15\) b) \(\frac{29}{36}\) c) \(9\)

- Replace division by a fraction with multiplication by its reciprocal. - Evaluate each expression separately for the given values. - For part b), rewrite \(T_1\) symbolically and compare its numerator and denominator with \(T_2\).

1. For a), \(T_1=\frac{2}{3}\div\frac{4}{5}=\frac{2}{3}\cdot\frac{5}{4}=\frac{5}{6}\). 2. Also, \(T_2=\frac{2\cdot5}{3\cdot4}=\frac{5}{6}\). 3. For b), dividing by \(\frac{c}{d}\) is equivalent to multiplying by its reciprocal \(\frac{d}{c}\): \(\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}=\frac{ad}{bc}\). 4. This is exactly \(T_2\), so the expressions are equivalent whenever the stated nonzero conditions hold.

a) \(T_1=T_2=\frac{5}{6}\) b) \(\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}=\frac{ad}{bc}\), so the expressions are equivalent.

- Find the scale factor caused by each change. - Multiply the scale factors when both factors change. - Test with simple factors if needed.

1. Multiplying one factor by \(4\) multiplies the product by \(4\). 2. The combined scale factor is \(10 \div 2 = 5\), so the product is multiplied by \(5\). 3. The combined scale factor is \(3 \cdot 3 = 9\), so the product is multiplied by \(9\).

a) The product is multiplied by \(4\). b) The product is multiplied by \(5\). c) The product is multiplied by \(9\).

- Compare both factors with those in the known product. - Check separately how the sign and the size of each factor change. - When both factors change, think about whether their effects cancel.

1. Doubling the negative first factor doubles the magnitude: \((-2.4)\cdot1.5=-3.6\). 2. Multiplying the positive second factor by \(-3\) multiplies the product by \(-3\): \((-1.2)\cdot(-4.5)=5.4\). 3. Halving one factor and doubling the other leave the product unchanged: \((-0.6)\cdot3=-1.8\). 4. Dividing the first factor by \(3\) divides the product by \(3\): \((-0.4)\cdot1.5=-0.6\).

a) \(-3.6\); doubling the negative factor doubles the negative product. b) \(5.4\); the second factor is multiplied by \(-3\), reversing the product's sign and tripling its size. c) \(-1.8\); half of one factor and twice the other leave the product unchanged. d) \(-0.6\); the first factor and product are both divided by \(3\).

- Generate several examples with different positive divisors. - For a chosen divisor, determine the dividend that gives quotient \(10\). - Decide whether there is a greatest positive integer.

1. Choose any positive integer \(n\) as the divisor. 2. Use \(10n\) as the dividend. Then \(10n \div n = 10\). 3. Because there are infinitely many positive integers \(n\), there are infinitely many such division equations.

There are infinitely many. For every positive integer \(n\), \(10n \div n = 10\).

- Change the sign to find an opposite. - Rewrite decimals and mixed numbers as fractions. - Interchange the numerator and denominator to find a reciprocal. - The sign of a reciprocal stays the same as the original number.

1. For \(5 = \frac{5}{1}\), the opposite is \(-5\), and the reciprocal is \(\frac{1}{5}\). Check: \(5 \cdot \frac{1}{5} = 1\). 2. For \(-0.75 = -\frac{3}{4}\), the opposite is \(0.75\), and the reciprocal is \(-\frac{4}{3}\). Check: \(-\frac{3}{4} \cdot -\frac{4}{3} = 1\). 3. For \(\frac{4}{9}\), the opposite is \(-\frac{4}{9}\), and the reciprocal is \(\frac{9}{4}\). Check: \(\frac{4}{9} \cdot \frac{9}{4} = 1\). 4. For \(-3\frac{1}{5} = -\frac{16}{5}\), the opposite is \(3\frac{1}{5}\), and the reciprocal is \(-\frac{5}{16}\). Check: \(-\frac{16}{5} \cdot -\frac{5}{16} = 1\).

Opposites: \(-5\), \(0.75\), \(-\frac{4}{9}\), \(3\frac{1}{5}\) Reciprocals: \(\frac{1}{5}\), \(-\frac{4}{3}\), \(\frac{9}{4}\), \(-\frac{5}{16}\) Each number multiplied by its reciprocal equals \(1\).

- Test a fraction times its flipped form. - Reverse the sign to undo an opposite. - Write \(2.5\) as a fraction before taking its reciprocal. - Recall the rule for division by zero.

1. For a), if \(a \ne 0\), then \(a \cdot \frac{1}{a} = \frac{a}{a} = 1\). 2. For b), reverse the sign. The original number is \(-4.5\). 3. For c), \(2.5 = \frac{5}{2}\). Taking the reciprocal again gives \(\frac{2}{5} = 0.4\). 4. For d), the reciprocal of \(0\) would be \(\frac{1}{0}\), but division by zero is undefined.

a) \(1\) b) \(-4.5\) c) \(0.4\) or \(\frac{2}{5}\) d) \(0\) has no reciprocal because \(\frac{1}{0}\) is undefined.

- Follow the connections in each tree to identify which operation happens first. - In Tree 1, evaluate the exponent before adding and multiplying. - Use grouping symbols when an operation must happen before another operation that would normally have higher precedence. - Review the sign rules for multiplying and dividing negative numbers. - Work from the upper branches toward the final box.

1. Tree 1 first evaluates \((-2)^3=-8\), then adds \(-8+7=-1\), and then multiplies by \(-4\). Its expression is \(((-2)^3+7)\cdot(-4)\), and its final value is \((-1)\cdot(-4)=4\). 2. Tree 2 first divides \(60\div(-5)=-12\), then adds that result to \(-100\). Its expression is \(-100+(60\div(-5))\), and its final value is \(-100+(-12)=-112\).

a) Tree 1: \(((-2)^3+7)\cdot(-4)\); Tree 2: \(-100+(60\div(-5))\) b) Tree 1: \(4\); Tree 2: \(-112\)

- Evaluate the exponent before adding on the left branch. - Work through the tree from the branch operations toward the final operation. - Subtracting a negative number is the same as adding its opposite. - Use grouping symbols to preserve the operations shown in each branch.

1. The tree represents \((-20+6^2)+(10-(-15))\). 2. Evaluate the exponent and the left branch: \(6^2=36\), so \(-20+36=16\). 3. Evaluate the right branch: \(10-(-15)=10+15=25\). 4. Add the branch results: \(16+25=41\).

The expression is \((-20+6^2)+(10-(-15))\), and its value is \(41\).

- Evaluate the exponent before the addition. - Start with the box connected directly to the two values on the right branch. - Review the sign rule for division involving a negative number. - Use grouping symbols so the addition is performed before the division.

1. Evaluate the exponent: \(3^2=9\). 2. Evaluate the branch operation: \(9+3=12\). 3. Use that result in the final division: \(-60\div12=-5\). 4. Because the addition must happen before the division, the expression is \(-60\div(3^2+3)\).

The intermediate value is \(12\), the final value is \(-5\), and the expression is \(-60\div(3^2+3)\).

- Evaluate the exponent before multiplying. - Start with the values at the ends of the upper branches and follow the tree toward the bottom. - Each operation combines the values directly connected to it. - Track the signs carefully when multiplying, adding, and dividing decimals.

1. Evaluate the exponent: \(2^2=4\). 2. Evaluate the multiplication branch: \(-0.4\cdot4=-1.6\). 3. Add that result to the other branch value: \(0.6+(-1.6)=-1.0\). 4. Divide by the final value: \(-1.0\div0.2=-5\). 5. The expression is \((0.6+(-0.4)\cdot2^2)\div0.2\).

The expression is \((0.6+(-0.4)\cdot2^2)\div0.2\), and its value is \(-5\).

- Count the total columns and the shaded columns to determine the width fraction. - Count the total rows and the shaded rows to determine the height fraction. - Convert each fraction to a decimal, multiply, and then write the product as a simplified fraction.

1. The shaded rectangle spans \(4\) of the \(5\) columns, so it covers \(\frac{4}{5}=0.8\) of the wall’s width. It spans \(3\) of the \(4\) rows, so it covers \(\frac{3}{4}=0.75\) of the wall’s height. 2. Multiply the decimal forms: \(0.8\cdot0.75=0.6\). 3. Convert the product to a fraction: \(0.6=\frac{6}{10}=\frac{3}{5}\).

Width: \(\frac{4}{5}=0.8\) Height: \(\frac{3}{4}=0.75\) Decimal product: \(0.6\) Fraction of the wall: \(\frac{3}{5}\)

- Identify which of the three quantities in the given relationship is unknown in each part. - Substitute the known quantities into the relationship before deciding how to isolate the unknown. - Use units as a check on the quantity you solve for.

1. a) The unknown is total change, so substitute into the relationship: \((-0.8)\cdot7.5=-6\). Multiplication combines rate with elapsed time. 2. b) The unknown is rate, so \(5.4=r\cdot1.8\). Solving gives \(r=5.4\div1.8=3\). Division isolates the per-second rate.

a) Unknown: total change. \((-0.8\,\text{m/s})(7.5\,\text{s})=-6\,\text{m}\). Multiplication is appropriate because rate and time are known. b) Unknown: rate. \(5.4=r(1.8)\), so \(r=5.4\div1.8=3\,\text{m/s}\). Division is appropriate because the total and time are known and the per-unit rate is unknown.

- In each part, identify whether the unknown is a total signed change or a count of equal changes. - Write the relationship among change per command, number of commands, and total change. - Interpret the kind of quantity requested before deciding what its sign should mean.

1. a) \((-1.25)\cdot6=-7.5\), so Robot A's total signed change is \(-7.5\,\text{m}\). 2. b) \((-7.5)\div(-1.5)=5\), so Robot B received \(5\) commands. 3. Part a) asks for a signed downward change; part b) asks for a count of equal changes, which is nonnegative.

a) \((-1.25)\cdot6=-7.5\,\text{m}\) b) \((-7.5)\div(-1.5)=5\) commands c) Part a) is a signed downward change, while part b) is a count of equal negative changes.

- Which terms sit entirely above the long fraction bar? - Add the top terms before dividing by the bottom. - Would Sofia's expression carry out the same addition first?

1. The long bar groups all of \(\frac{1}{2}+\frac{1}{4}\) as the numerator, so Sofia's expression does not match it. 2. Add the numerator: \(\frac{1}{2}+\frac{1}{4}=\frac{3}{4}\). 3. Divide by the denominator: \(\frac{3}{4}\div\left(-\frac{3}{4}\right)=-1\). 4. Sofia's expression instead gives \(\frac{1}{2}-\frac{1}{3}=\frac{1}{6}\), confirming that it means something different.

No. The long bar groups the whole sum on top, so the original fraction is \(-1\), not \(\frac{1}{6}\).

- Convert the percent to a decimal before dividing. - Converting \(0.75\) to a fraction can simplify the first product. - Evaluate the power and multiplication or division before adding and subtracting.

1. Write \(0.75=\frac{3}{4}\): \(\frac{2}{3}\cdot\frac{3}{4}=\frac{1}{2}=0.5\). 2. Write \(1 \frac{1}{2}=1.5\) and \(75\%=0.75\): \(1.5\div0.75=2\). 3. Evaluate the power: \(\left(\frac{1}{2}\right)^3=\frac{1}{8}=0.125\). 4. Combine: \(0.5+2-0.125=2.375\).

\(2.375\)

- Rewrite the fraction with a denominator of \(10\). - Make the divisor a whole number before dividing decimals. - Check a quotient by multiplying it by the divisor.

1. The fraction was converted incorrectly. Since \(\frac{2}{5} = \frac{4}{10}\), its decimal form is \(0.4\), not \(0.25\). 2. The displayed decimal division was also calculated incorrectly. In fact, \(1.5 \div 0.25 = 150 \div 25 = 6\), not \(60\). 3. For the original expression, \(1.5 \div 0.4 = 15 \div 4 = 3.75\). Equivalently, \(\frac{3}{2} \div \frac{2}{5} = \frac{3}{2} \cdot \frac{5}{2} = \frac{15}{4} = 3.75\).

The first error is that \(\frac{2}{5} = 0.4\), not \(0.25\). The second error is that \(1.5 \div 0.25 = 6\), not \(60\). The correct value is \(3.75\).

- Converting mixed numbers and decimals to fractions may make the arithmetic easier. - In a), determine the sign of the quantity in parentheses before multiplying. - After applying the order of operations, perform multiplication and division from left to right. - What sign results when a larger positive quantity is subtracted from a smaller one?

1. For a), \(2 \frac{1}{3}=\frac{7}{3}\) and \(4.5-6=-1.5=-\frac{3}{2}\). 2. Then \(\frac{7}{3}\cdot\left(-\frac{3}{2}\right)=-\frac{7}{2}\), and \(-\frac{7}{2}\div\frac{7}{4}=-\frac{7}{2}\cdot\frac{4}{7}=-2\). 3. For b), \(\frac{5}{12}\div\frac{25}{18}=\frac{5}{12}\cdot\frac{18}{25}=\frac{3}{10}\). 4. Also, \(0.4\cdot1 \frac{1}{2}=\frac{2}{5}\cdot\frac{3}{2}=\frac{3}{5}\). Therefore, \(\frac{3}{10}-\frac{3}{5}=-\frac{3}{10}\).

a) \(-2\) b) \(-\frac{3}{10}\), or \(-0.3\)

- Write the original product as a single fraction before considering the changes. - Think about the commutative property of multiplication for part b). - If one factor becomes three times as large, what change to another factor would keep the overall product unchanged?

1. The original product is \(P=\frac{2}{5}\cdot\frac{3}{7}=\frac{6}{35}\). 2. For a), the new product is \(\frac{4}{5}\cdot\frac{3}{14}=\frac{6}{35}\). The factor of \(2\) in the numerator is canceled by the factor of \(2\) in the denominator, so the product is unchanged. 3. For b), swapping the numerators gives \(\frac{3}{5}\cdot\frac{2}{7}=\frac{6}{35}\). The product is unchanged because \(2\cdot3=3\cdot2\). 4. For c), tripling the first factor must be balanced by multiplying the second factor by \(\frac{1}{3}\). The second factor becomes \(\frac{3}{7}\cdot\frac{1}{3}=\frac{1}{7}\), and \(\frac{6}{5}\cdot\frac{1}{7}=\frac{6}{35}\).

a) The product stays the same: \(\frac{6}{35}\). b) Yes. The product remains \(\frac{6}{35}\). c) Multiply the second factor by \(\frac{1}{3}\), so it becomes \(\frac{1}{7}\).

- Treat the numerator and denominator as separate expressions first. - Is there a common factor in both products in the numerator? - Simplify the denominator before doing the final division.

1. Factor the common factor in the numerator: \(12.5\cdot3.7+12.5\cdot4.3=12.5(3.7+4.3)=12.5\cdot8=100\). 2. Simplify the denominator: \(\frac{2}{3}+\frac{1}{6}=\frac{5}{6}\), so \(\frac{5}{6}\div\frac{1}{12}=\frac{5}{6}\cdot12=10\). 3. Divide: \(100\div10=10\).

\(10\)

- Convert the percent to a decimal. - Use the completed diagonal to find the common product. - Then use rows, columns, or diagonals with only one missing value.

1. Use the main diagonal to find the common product: \(0.25\cdot0.5\cdot1=0.125\). 2. Convert \(10\%=0.1\). 3. Top middle: \(0.25\cdot x\cdot0.1=0.125\), so \(x=5\). 4. Bottom left: \(0.1\cdot0.5\cdot x=0.125\), so \(x=2.5\). 5. Middle left: \(0.25\cdot x\cdot2.5=0.125\), so \(x=0.2\). 6. Middle right: \(0.2\cdot0.5\cdot x=0.125\), so \(x=1.25\). 7. Bottom middle: \(5\cdot0.5\cdot x=0.125\), so \(x=0.05\).

<table> <tr><td>\(0.25\)</td><td>\(5\)</td><td>\(10\%\)</td></tr> <tr><td>\(0.2\)</td><td>\(0.5\)</td><td>\(1.25\)</td></tr> <tr><td>\(2.5\)</td><td>\(0.05\)</td><td>\(1\)</td></tr> </table> The common product is \(0.125\), or \(\frac{1}{8}\).

- For estimation, replace complicated values with nearby easy numbers. - In c), determine the sign by counting the negative factors. - Convert between fractions and decimals when that makes the exact calculation easier.

1. For a), \((-1.5)^3\approx-3.4\) and \(0.375\approx0.4\), so the result is about \(-3\). Exactly, \((-1.5)^3+0.375=-3.375+0.375=-3\). 2. For b), the parentheses are about \(-1.5\), and dividing by \(-0.5\) gives about \(3\). Exactly, \((2.25-3.75)\div(-0.5)=3\). 3. For c), the parentheses are about \(-3.8\), so \(0.1\cdot(-3.8)\cdot(-3)\approx1.1\). Exactly, \(0.1\cdot\left(-\frac{23}{6}\right)\cdot(-3)=\frac{23}{20}=1.15\).

a) Estimate: about \(-3\); exact value: \(-3\) b) Estimate: about \(3\); exact value: \(3\) c) Estimate: about \(1.1\); exact value: \(1.15\)

- Treat the numerator and denominator as separate expressions first. - Apply powers before multiplication or subtraction. - A fraction bar groups the entire numerator and denominator. - Simplify common factors before multiplying fractions when possible.

1. Numerator: \(\left(\frac{1}{3}\right)^2=\frac{1}{9}\), so \(\frac{1}{2}-\frac{1}{9}=\frac{7}{18}\). 2. Denominator: \(1.25=\frac{5}{4}\), so \(\frac{5}{4}\cdot\frac{8}{3}=\frac{10}{3}\). Then \(\frac{10}{3}-4=-\frac{2}{3}\). 3. Divide: \(\frac{7}{18}\div\left(-\frac{2}{3}\right)=\frac{7}{18}\cdot\left(-\frac{3}{2}\right)=-\frac{7}{12}\).

\(-\frac{7}{12}\)

- Distinguish subtraction signs from negative signs. - Evaluate nested grouping symbols from the inside out. - Break each expression into smaller parts. - Check the sign of every product.

1. \(-8-\left(\frac{-8-(-4)}{-2}\right)\cdot(-6)=-8-2\cdot(-6)=4\). 2. \((-5-2)\cdot\left[-5-(-4)\cdot(-4)\right]+(-5+4)\cdot(-3)=(-7)\cdot(-21)+3=150\).

1) \(4\) 2) \(150\)