Free Math Worksheets

- 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) \(1\)

- First think about the product without the decimal point. - Use place value to position the decimal point in the product. - For a negative factor, remember the sign rule for multiplication.

1. \(0.8\cdot4=3.2\). 2. \(-0.5\cdot9=-4.5\). 3. \(1.2\cdot5=6\). 4. \(0.006\cdot7=0.042\).

a) \(3.2\) b) \(-4.5\) c) \(6\) d) \(0.042\)

- 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) \(-20\)

- Rewrite a factor as a sum or difference involving \(10\), \(100\), or \(1\). - For the division, split the dividend into parts that are each divisible by \(3\). - Use the sign rules when multiplying a negative number.

1. For a), decompose \(101\) as \(100 + 1\) and distribute: \(15 \cdot 100 + 15 \cdot 1 = 1500 + 15 = 1515\). 2. For b), decompose \(9\) as \(10 - 1\) and distribute: \(18 \cdot 10 - 18 \cdot 1 = 180 - 18 = 162\). 3. For c), decompose the dividend into divisible parts: \(315 \div 3 = (300 \div 3) + (15 \div 3) = 100 + 5 = 105\). 4. For d), decompose \(11\) as \(10 + 1\) and distribute: \(-22 \cdot 10 + (-22) \cdot 1 = -220 - 22 = -242\).

a) \(1515\) b) \(162\) c) \(105\) d) \(-242\)

- Look for a common factor between the whole number and the denominator before multiplying. - Working with smaller factors can simplify the arithmetic. - Keep track of the sign in the negative product.

1. For a), cancel a factor of \(5\) between \(15\) and \(25\): \(15 \cdot \frac{4}{25}=3 \cdot \frac{4}{5}=\frac{12}{5}\). 2. For b), cancel a factor of \(8\): \(24 \cdot \frac{5}{16}=3 \cdot \frac{5}{2}=\frac{15}{2}\). 3. For c), cancel a factor of \(6\): \(-18 \cdot \frac{7}{12}=-3 \cdot \frac{7}{2}=-\frac{21}{2}\). 4. For d), cancel a factor of \(7\): \(35 \cdot \frac{3}{14}=5 \cdot \frac{3}{2}=\frac{15}{2}\).

a) \(\frac{12}{5}\) b) \(\frac{15}{2}\) c) \(-\frac{21}{2}\) d) \(\frac{15}{2}\)

- 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) \(-\frac{4}{5}\)

- 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) \(1\)

- 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) \(\frac{3}{10}\)

- 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 \(\frac{1}{9}\).

- How do you rewrite a mixed number as an improper fraction? - Look for common factors that can be canceled before multiplying. - Keep track of the sign of each product.

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

a) \(3\) b) \(-2\) c) \(-3\)

- 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) \(\frac{3}{8}\)

- 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) \(0.04\)

- Use familiar whole-number facts when possible. - Decide the sign of the quotient before calculating. - For division by a decimal, create an equivalent division with a whole-number divisor.

1. \(0.48\div6=0.08\). 2. \(-1.5\div3=-0.5\). 3. \(1.2\div0.4=12\div4=3\).

a) \(0.08\) b) \(-0.5\) c) \(3\)

- 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) \(4\)

- Choose nearby numbers that make each estimate easy to compute. - Scale dividend and divisor by the same power of \(10\) for exact decimal division. - Apply the sign rules for division. - Use the estimate to check whether the exact quotient is reasonable.

1. For a), an estimate is \(6\div0.1=60\). Exactly, \(5.6\div0.08=560\div8=70\). 2. For b), an estimate is \(0.4\div2=0.2\). Exactly, \(0.441\div2.1=4.41\div21=0.21\). 3. For c), an estimate is \((-2)\div(-0.2)=10\). Exactly, the quotient is positive and \(-2.25\div(-0.15)=225\div15=15\).

a) Estimate: about \(60\); exact: \(70\) b) Estimate: about \(0.2\); exact: \(0.21\) c) Estimate: about \(10\); exact: \(15\)

- 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 \(-1.5\).

- Determine the sign of a product from the number of negative factors. - 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, and multiplying by a positive third factor keeps it positive. 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 product would be positive. c) Yes; for example, \((-1)\cdot1\cdot1=-1\). d) No; the product would be \(0\).

- Convert a fraction with denominator \(10\), \(100\), or \(1000\) to a decimal. - Notice what happens when two reciprocal fractions are multiplied. - Compare the amount of multiplication and simplification in 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.

Both methods give \(1\). A valid comparison may favor either method if the explanation refers to the work required.

- 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) \(1\)

- 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 \(-9\)

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

- Look for factors that combine to make \(1\), \(100\), or \(1000\). - Reciprocal factors have product \(1\). - Determine the sign before multiplying magnitudes.

1. Group \(25\) and \(4\): \((25 \cdot 4)(-13) = 100(-13) = -1300\). 2. Group \(-125\) and \(-8\): \((-125)(-8) \cdot 7 = 1000 \cdot 7 = 7000\). 3. Group the reciprocal fractions: \(\frac{4}{9} \cdot \frac{9}{4} = 1\), so the product is \(-17\). 4. Group \(0.2\) and \(5\): \((0.2 \cdot 5)(-47) = 1(-47) = -47\).

1) \(-1300\) 2) \(7000\) 3) \(-17\) 4) \(-47\)

- Look for factors that create simple partial products. - One property allows reordering; another allows regrouping. - Determine the sign of the final product before calculating its magnitude.

1. Marie used the commutative and associative properties to reorder and regroup factors so that the first product was \(10\). 2. For b), group \(8\) and \(125\): \(8 \cdot 125 = 1000\). 3. Group the remaining factors: \((-1.5)(-2) = 3\). 4. Multiply: \(1000 \cdot 3 = 3000\).

a) Marie used the commutative and associative properties of multiplication. b) \(3000\)

- 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 \(-192\).

- Determine the sign of a positive number divided by a negative number. - For the second method, divide by the factors one at a time in their given order. - Compare the two final results.

1. Multiply first: \((-6) \cdot 4 = -24\), then \(144 \div (-24) = -6\). 2. Divide successively: \(144 \div (-6) = -24\), then \(-24 \div 4 = -6\). 3. Both methods give the same result.

Both methods give \(-6\).

- Can you pair or regroup factors so that the decimal product becomes a whole number? - Rewrite \(-101\) using \(-100\) and \(-1\). - Split \(728\) into two numbers that are both divisible by \(7\).

1. For a), decompose \(12\) as \(10 + 2\): \(2.5 \cdot 10 + 2.5 \cdot 2 = 25 + 5 = 30\). Another efficient method is \((2.5 \cdot 4) \cdot 3 = 10 \cdot 3 = 30\). 2. For b), decompose \(-101\) as \(-100 - 1\): \(14(-100) + 14(-1) = -1400 - 14 = -1414\). 3. For c), decompose the dividend as \(700 + 28\): \(700 \div 7 + 28 \div 7 = 100 + 4 = 104\).

a) \(30\) b) \(-1414\) c) \(104\)

- 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) \(\frac{1}{20}\)

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

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

- 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 \(\frac{3}{4}\).

- Decide which expressions are easier to evaluate using fractions. - To divide by a fraction, multiply by its reciprocal. - Determine the sign of the product in part c). - Recall that raising a fraction to the third power means multiplying three equal factors.

1. For a), \(\left(\frac{3}{2}\right)^3 = \frac{27}{8}\). Then \(\frac{27}{8} \cdot \frac{4}{9} = \frac{3}{2}\). 2. For b), write \(0.8 = \frac{4}{5}\). Divide by multiplying by the reciprocal: \(\frac{4}{5} \cdot \frac{15}{4} = 3\). 3. For c), write \(2.25 = \frac{9}{4}\). Then \(\frac{9}{4} \cdot \left(-\frac{2}{3}\right) = -\frac{3}{2}\).

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

- 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 \(\frac{1}{16}\).

- How can you rewrite a mixed number as an improper fraction? - Look for common factors you can cancel before multiplying. - What sign will a product have when one factor is negative and the other is positive? - How can you write a decimal as a fraction?

1. For a), an estimate is \(4\cdot2=8\). Rewrite the mixed numbers: \(\frac{7}{2}\cdot\frac{12}{5}=\frac{42}{5}=8 \frac{2}{5}\). 2. For b), an estimate is \(-5\cdot1.5=-7.5\). Rewrite the mixed numbers: \(-\frac{14}{3}\cdot\frac{3}{2}=-7\). 3. For c), \(0.6\cdot3=1.8\), so an estimate is about \(2\). For the exact value, \(0.6=\frac{6}{10}\) and \(3 \frac{1}{3}=\frac{10}{3}\), so \(\frac{6}{10}\cdot\frac{10}{3}=2\).

a) Estimate: about \(8\); exact: \(8 \frac{2}{5}\) b) Estimate: about \(-7.5\); exact: \(-7\) c) Estimate: about \(2\); exact: \(2\)

- Rewrite each mixed number as an improper fraction before multiplying. - What happens to the sign when two negative factors are multiplied? - Look for common factors in numerators and denominators that can be canceled first.

1. For a), rewrite the mixed number: \(\frac{5}{8}\cdot\frac{6}{5}\cdot4\). Cancel the factors \(5\), then simplify \(\frac{4}{8}=\frac{1}{2}\). The product is \(\frac{1}{2}\cdot6=3\). 2. For b), two negative factors give a positive product. Rewrite the mixed numbers: \(\frac{9}{4}\cdot\frac{4}{3}\cdot\frac{1}{2}\). Cancel the factors \(4\), then simplify \(\frac{9}{3}=3\). The product is \(3\cdot\frac{1}{2}=\frac{3}{2}=1 \frac{1}{2}\).

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

- Compare the sizes of the factors in the two products. - Use nearby decimal values to estimate each product. - Rewrite the mixed numbers as improper fractions for the exact calculation.

1. Estimate: \(A\approx4.5\cdot1.3\approx5.9\), while \(B\approx3.3\cdot1.5\approx5.0\). So \(A\) is expected to be greater. 2. For \(A\), rewrite the mixed numbers: \(\frac{9}{2}\cdot\frac{4}{3}=6\). 3. For \(B\), rewrite the mixed numbers: \(\frac{10}{3}\cdot\frac{3}{2}=5\). 4. Since \(6>5\), the estimate was correct and \(A\) is greater.

\(A\) is greater. \(A=6\) and \(B=5\).

- 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. - Use the sign rules for products of positive and negative rational numbers. - Cancel common factors before multiplying. - For the comparison, simplify both products before deciding which is greater.

1. For a), \(1 \frac{3}{5}=\frac{8}{5}\) and \(-2 \frac{1}{2}=-\frac{5}{2}\). Then \(\frac{8}{5}\cdot\left(-\frac{5}{2}\right)=-4\). 2. For b), \(-3 \frac{3}{4}=-\frac{15}{4}\). Then \(-\frac{15}{4}\cdot\left(-\frac{8}{15}\right)=2\). 3. For c), \(1 \frac{2}{7}=\frac{9}{7}\). Then \(\frac{7}{9}\cdot\frac{9}{7}\cdot(-3)=-3\). 4. For d), \(2 \frac{1}{4}\cdot\frac{2}{3}=\frac{9}{4}\cdot\frac{2}{3}=\frac{3}{2}\), while \(1 \frac{1}{2}\cdot\frac{5}{6}=\frac{3}{2}\cdot\frac{5}{6}=\frac{5}{4}\). Since \(\frac{3}{2}>\frac{5}{4}\), the first product is greater.

a) \(-4\) b) \(2\) c) \(-3\) d) \(2 \frac{1}{4}\cdot\frac{2}{3}\) is greater; the products are \(\frac{3}{2}\) and \(\frac{5}{4}\).

- 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) \(\frac{\frac{\frac{21}{16}}{\frac{7}{8}}}{3}=\frac{1}{2}\)

- Look for factors that combine to make \(10\) or \(100\). - A division by a composite number can sometimes be carried out as two successive divisions by its factors. - Determine the sign of each answer before calculating its magnitude.

1. For a), write \(28 = 4 \cdot 7\) and group \(25\) with \(4\): \((25 \cdot 4) \cdot 7 = 100 \cdot 7 = 700\). 2. For b), use \(18 = 9 \cdot 2\): \(126 \div 9 = 14\), then \(14 \div 2 = 7\). 3. For c), determine that the product is negative and decompose \(12\) as \(10 + 2\): \(-15(10 + 2) = -150 - 30 = -180\). 4. For d), determine that the quotient is negative and decompose \(105\) as \(70 + 35\): \(-\left(70 \div 7 + 35 \div 7\right) = -(10 + 5) = -15\).

a) \(700\) b) \(7\) c) \(-180\) d) \(-15\)

- 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 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; \(0.035\)

- Decide whether a fraction or decimal form makes each product easier. - Convert familiar decimals such as \(0.25\) and \(1.25\) to simple fractions. - Simplify common factors before multiplying.

1. For a), write \(0.25 = \frac{1}{4}\). Then \(\frac{1}{4} \cdot \frac{4}{7} = \frac{1}{7}\). 2. For b), write \(0.8 = \frac{4}{5}\). Then \(\frac{4}{5} \cdot \frac{3}{4} = \frac{3}{5} = 0.6\). 3. For c), write \(1.25 = \frac{5}{4}\) and \(0.4 = \frac{2}{5}\). Then \(\frac{5}{4} \cdot \frac{2}{5} = \frac{1}{2} = 0.5\). 4. For d), write \(0.9 = \frac{9}{10}\). Then \(\frac{2}{3} \cdot \frac{9}{10} = \frac{3}{5} = 0.6\).

a) \(\frac{1}{7}\) b) \(\frac{3}{5}\), or \(0.6\) c) \(\frac{1}{2}\), or \(0.5\) d) \(\frac{3}{5}\), or \(0.6\)

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

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

- Reorder multiplication factors when doing so makes common factors easier to simplify. - Determine the sign before multiplying a negative number by a positive number. - Choose fractions or decimals based on which form makes the operation easier. - Convert the percent to a decimal or fraction before dividing.

1. For a), write \(0.5 = \frac{1}{2}\), then reorder the factors to simplify: \(\left(\frac{12}{25} \cdot \frac{5}{3}\right) \cdot \frac{1}{2} = \frac{4}{5} \cdot \frac{1}{2} = \frac{2}{5}\). 2. For b), write \(3.6 = \frac{18}{5}\). Then \(\frac{18}{5} \div \frac{9}{10} = \frac{18}{5} \cdot \frac{10}{9} = 4\). 3. For c), the product is negative. Simplify before multiplying: \(\left(-\frac{3}{4}\right) \cdot \frac{16}{21} = -\frac{4}{7}\). 4. For d), write \(80\% = 0.8\). Then \(0.8 \div 0.4 = 2\).

a) \(\frac{2}{5}\), or \(0.4\) b) \(4\) c) \(-\frac{4}{7}\) d) \(2\)

- Consider whether converting a fraction to a decimal would produce a repeating decimal. - Familiar terminating decimals can often be written as simple fractions. - When dividing decimals, multiply both numbers by the same power of \(10\) to make the divisor a whole number.

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 efficient. b) \(\frac{3}{10}\), or \(0.3\); fractions are efficient. c) \(2.5\), or \(\frac{5}{2}\); decimals are efficient.

- What happens to a fraction when only its numerator is doubled? - What happens to a fraction when only its denominator is doubled? - In c), consider the effect of each change to the first factor before finding the new product. - Compare each new product with the original product.

1. The original product is \(P=\frac{3}{8}\cdot\frac{4}{5}=\frac{3}{10}\). 2. For a), the first factor becomes \(\frac{6}{8}\). The new product is \(\frac{6}{8}\cdot\frac{4}{5}=\frac{3}{5}=2\cdot\frac{3}{10}\), so the product doubles. 3. For b), the second factor becomes \(\frac{4}{10}\). The new product is \(\frac{3}{8}\cdot\frac{4}{10}=\frac{3}{20}=\frac{1}{2}\cdot\frac{3}{10}\), so the product is halved. 4. For c), the first factor becomes \(\frac{6}{4}\). The new product is \(\frac{6}{4}\cdot\frac{4}{5}=\frac{6}{5}=4\cdot\frac{3}{10}\), so the product is four times as large.

The original product is \(\frac{3}{10}\). a) It doubles to \(\frac{3}{5}\). b) It is halved to \(\frac{3}{20}\). c) It becomes four times as large, \(\frac{6}{5}\).

- 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.

- 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.

- 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 \(-2\frac{2}{3} \cdot 1.5 = -\frac{8}{3} \cdot \frac{3}{2} = -4\).

- Simplify common factors before multiplying fractions. - Convert repeating decimals to fractions for exact calculations. - Recognize terminating decimals such as \(0.375\) as fractions with powers of \(10\) in the denominator.

1. For a), write \(-0.12 = -\frac{3}{25}\). Then \(\frac{5}{6} \cdot \left(-\frac{3}{25}\right) = -\frac{1}{10} = -0.1\). 2. For b), write \(0.375 = \frac{3}{8}\). Then \(\frac{3}{8} \div \frac{3}{4} = \frac{3}{8} \cdot \frac{4}{3} = \frac{1}{2}\). 3. For c), write \(-1.2 = -\frac{6}{5}\). Then \(-\frac{6}{5} \cdot \frac{5}{9} = -\frac{2}{3}\). 4. For d), write \(0.\overline{6} = \frac{2}{3}\). Then \(4 \div \frac{2}{3} = 4 \cdot \frac{3}{2} = 6\).

a) \(-0.1\) b) \(0.5\), or \(\frac{1}{2}\) c) \(-\frac{2}{3}\) d) \(6\)

- 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) \(-1\)

- 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.

- Look for pairs whose products are \(1\) or another simple number. - Determine the sign of the product of the two negative factors. - Convert a decimal to a fraction when that makes cancellation visible.

1. Use the commutative and associative properties of multiplication to form convenient pairs: \(\left(\frac{5}{9} \cdot 1.8\right)\left((-0.4)(-2.5)\right)\). 2. Evaluate the first pair: \(\frac{5}{9} \cdot \frac{18}{10} = 1\). 3. Evaluate the second pair: \((-0.4)(-2.5) = 1\). 4. Multiply the partial products: \(1 \cdot 1 = 1\).

The product is \(1\). The commutative and associative properties of multiplication were used.

- In a), look for reciprocal factors and factors that cancel. - In b), find two factors whose product is \(100\). - You may change the order and grouping of factors in a product.

1. For a), rearrange and regroup reciprocal or compatible factors: \(\left(\frac{2}{9} \cdot \frac{9}{2}\right)\left(\frac{5}{11} \cdot 22\right)\). 2. The first group equals \(1\), and the second equals \(5 \cdot 2 = 10\). Therefore, the product is \(10\). 3. For b), regroup to make \(100\): \((12.5 \cdot 8) \cdot 0.7 = 100 \cdot 0.7 = 70\).

a) \(10\) b) \(70\)

- 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) \(-40\)

- 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: \(-120\div(-3)=40\).

\(40\)

- 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. - Pay attention to sign rules when multiplying negative numbers. - Converting \(\frac{1}{4}\) to a decimal may simplify the calculation.

1. Left block in Row 2: \((-1.2)\cdot(-5)=6\). 2. Right block in Row 2: \((-5)\cdot\frac{1}{4}=-\frac{5}{4}=-1.25\). 3. Top block: \(6\cdot(-1.25)=-7.5\).

Row 2: \(6\) and \(-1.25\) Row 3: \(-7.5\)

- 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) \(=\)

- 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) \(-0.0042\)

- 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.

- Substitute each value for \(a\). - Divide before subtracting. - Dividing by \(\frac{1}{2}\) is equivalent to multiplying by \(2\). - Track the negative sign carefully in part b).

1. For \(a=\frac{1}{8}\), \(T(a)=\frac{3}{4}-\frac{1}{8}\div\frac{1}{2}\). 2. \(\frac{1}{8}\div\frac{1}{2}=\frac{1}{8}\cdot 2=\frac{1}{4}\), so \(T(a)=\frac{3}{4}-\frac{1}{4}=\frac{1}{2}\). 3. For \(a=-1.5\), \(T(a)=\frac{3}{4}-(-1.5)\div\frac{1}{2}\). 4. \(-1.5\div 0.5=-3\), so \(T(a)=\frac{3}{4}-(-3)=3\frac{3}{4}\).

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

- 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).

- Rewrite division by \(\frac{2}{5}\) as multiplication by its reciprocal. - Which is greater, \(-4\) or \(-25\)? - On a number line, greater numbers lie farther to the right.

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

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

- 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.

- 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 \(3.75\).

- 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 \(\frac{1}{7}\).

- Find a number whose cube is \(-0.125\). - Use the sign rule for a product with two negative factors. - Track the factor contributed by doubling each of three factors.

1. For a), \((-0.5)^3=-0.125\), so \(a=b=c=-0.5\). 2. For b), no. Two negative factors and one positive factor produce a positive product. 3. For c), \((2a)(2b)(2c)=8abc\). Thus the new product is \(8(-0.125)=-1\).

a) \(a=b=c=-0.5\) b) No; the product would be positive. c) \(-1\)