Add and subtract fractions and decimals

Learning goals

Adds and subtracts signed fractions using a common denominator.
Adds and subtracts signed decimals using place value.
Chooses fraction or decimal forms for a calculation.
Evaluates rational sums and differences containing brackets.
Uses signed fractional or decimal changes to find an initial or final value.

Difficulty: 1 – 5

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Add and subtract fractions and decimals 1

Evaluate each expression and simplify the result. a) $45-72$ b) $-18+(-12)$ c) $\frac{5}{6}-\frac{1}{6}$ d) $-\frac{3}{8}-\frac{1}{8}$

Hints

- Determine the sign before calculating. - When adding two negative numbers, add their absolute values and keep the negative sign. - With like denominators, combine the numerators. - Simplify fraction results.

Solution

1. For a), $45-72=-27$. 2. For b), $-18+(-12)=-30$. 3. For c), $\frac{5}{6}-\frac{1}{6}=\frac{4}{6}=\frac{2}{3}$. 4. For d), $-\frac{3}{8}-\frac{1}{8}=-\frac{4}{8}=-\frac{1}{2}$.

Answer

a) $-27$ b) $-30$ c) $\frac{2}{3}$ d) $-\frac{1}{2}$

Add and subtract fractions and decimals 2

Point $A$ is your starting position. Move $\frac{1}{2}$ unit right. What fraction do you land on? Write an addition equation.

Hints

- Find $A$ by counting the equal tick marks from zero. - How many quarter-unit steps make one half? - Check where the move ends relative to zero.

Solution

1. Point $A$ is one quarter to the left of zero, so $A=-\frac{1}{4}$. 2. Moving right half a unit means adding $\frac{1}{2}$. 3. $-\frac{1}{4}+\frac{1}{2}=-\frac{1}{4}+\frac{2}{4}=\frac{1}{4}$.

Answer

$-\frac{1}{4}+\frac{1}{2}=\frac{1}{4}$

Add and subtract fractions and decimals 3

Find $\frac{3}{8}+\left(-\frac{5}{8}\right)$. Is the result positive or negative?

Hints

- Adding a negative means the total decreases. - Compare the sizes of the positive and negative parts. - Keep the sign when simplifying the fraction.

Solution

1. The parts have the same denominator: $\frac{3}{8}-\frac{5}{8}=\frac{-2}{8}$. 2. Simplify: $-\frac{1}{4}$. The result is negative because the negative part has greater magnitude.

Answer

$-\frac{1}{4}$; negative

Add and subtract fractions and decimals 4

Calculate $-\frac{1}{2}-\left(-\frac{1}{4}\right)$. Explain whether the result is to the left or right of $-\frac{1}{2}$ on a number line.

Hints

- How does subtracting a negative change the direction of movement? - Write both fractional parts in fourths. - A move right does not always end at a positive number.

Solution

1. Subtracting $-\frac{1}{4}$ means adding $\frac{1}{4}$. 2. $-\frac{1}{2}+\frac{1}{4}=-\frac{2}{4}+\frac{1}{4}=-\frac{1}{4}$. 3. Adding a positive amount moves right, so the result is to the right of $-\frac{1}{2}$.

Answer

$-\frac{1}{4}$, to the right of $-\frac{1}{2}$

Add and subtract fractions and decimals 5

a) Read the coordinates of $A$ and $B$. b) Write their sum. Explain your answer using the number line.

Hints

- Use the spacing between the labeled values and the smaller ticks. - Compare the two positions relative to zero. - Think about what happens when a number and its opposite are added.

Solution

1. The points are $A=-\frac{3}{4}$ and $B=\frac{3}{4}$. 2. Their sum is $-\frac{3}{4}+\frac{3}{4}=0$, since equal distances on opposite sides of zero cancel.

Answer

a) $A=-\frac{3}{4}$, $B=\frac{3}{4}$ b) $-\frac{3}{4}+\frac{3}{4}=0$; they are opposites.

Add and subtract fractions and decimals 6

Complete the equation: $-\frac{2}{5}+\square=0$.

Hints

- What number would cancel the negative fraction? - Think of moving back to zero from a negative position. - Check by adding your value to the given fraction.

Solution

1. A number and its opposite add to zero. 2. The opposite of $-\frac{2}{5}$ is $\frac{2}{5}$.

Answer

$\frac{2}{5}$

Add and subtract fractions and decimals 7

Which expression is another way to write $-\frac{4}{9}-\left(-\frac{1}{9}\right)$? a) $-\frac{4}{9}+\left(-\frac{1}{9}\right)$ b) $-\frac{4}{9}+\frac{1}{9}$ c) $\frac{4}{9}-\frac{1}{9}$ Explain why your chosen expression has the same value.

Hints

- Change only the operation involving the second number. - Does the first fraction change when you rewrite the subtraction? - Check whether the proposed expression gives the same result.

Solution

1. Subtracting a negative number means adding its positive opposite. 2. Keep the first number negative and change $-(-\frac{1}{9})$ to $+\frac{1}{9}$.

Answer

b) $-\frac{4}{9}+\frac{1}{9}$; subtracting a negative becomes adding its opposite.

Add and subtract fractions and decimals 8

At dawn, an outdoor thermometer reads $-\frac{1}{4}\,^{\circ}\text{C}$. The temperature drops by $\frac{1}{2}\,^{\circ}\text{C}$. What is the new temperature? Write the calculation.

Hints

- What operation represents a drop in temperature? - Keep the sign of the starting temperature. - Write both fractions in the same-sized parts.

Solution

1. A temperature drop is subtraction: $-\frac{1}{4}-\frac{1}{2}$. 2. Convert to fourths: $-\frac{1}{4}-\frac{2}{4}=-\frac{3}{4}\,^{\circ}\text{C}$.

Answer

$-\frac{1}{4}-\frac{1}{2}=-\frac{3}{4}\,^{\circ}\text{C}$

Add and subtract fractions and decimals 9

Calculate $-\frac{2}{7}+\left(-\frac{3}{7}\right)$. Explain why the answer is not positive.

Hints

- Imagine both terms as movements on a number line. - What do their directions tell you about the final position? - Keep the denominator when adding seventh parts.

Solution

1. Both terms are negative, so their sum is negative. 2. Add the magnitudes: $\frac{2}{7}+\frac{3}{7}=\frac{5}{7}$. 3. Hence the result is $-\frac{5}{7}$.

Answer

$-\frac{5}{7}$; adding two negative values moves farther below zero.

Add and subtract fractions and decimals 10

Evaluate the expression. Show the value of the quantity in parentheses before finding the final value. $7 \frac{5}{12} - \left(3 \frac{5}{6} + 2 \frac{1}{2}\right)$

Hints

- Evaluate the grouped expression before doing the outside subtraction. - Use a common denominator when adding the fractional parts. - Keep the exact fractional value of the grouped expression for the final subtraction.

Solution

1. Evaluate the parentheses: $3\frac{5}{6}+2\frac{1}{2}=3\frac{5}{6}+2\frac{3}{6}=6\frac{1}{3}$. 2. Rewrite $6\frac{1}{3}=6\frac{4}{12}$. 3. Subtract: $7\frac{5}{12}-6\frac{4}{12}=1\frac{1}{12}$.

Answer

Quantity in parentheses: $6\frac{1}{3}$ Final value: $1\frac{1}{12}$

Add and subtract fractions and decimals 11

Which number is exactly halfway between $-2.4$ and $3.6$? a) $0.6$ b) $0.8$ c) $1.0$ d) $1.2$

Hints

- The midpoint is the average of the two endpoints. - Add the two values first. - Divide the sum by $2$.

Solution

1. Find the midpoint by adding the two numbers and dividing by $2$: $\frac{-2.4+3.6}{2}$. 2. The sum is $-2.4+3.6=1.2$. 3. Divide by $2$: $1.2\div2=0.6$.

Answer

a) $0.6$

Add and subtract fractions and decimals 12

Calculate the value of the expression and write the result in simplest form: $\frac{5}{12} - \left(\frac{1}{4} + \frac{1}{3}\right)$

Hints

- Which part of the expression should you evaluate first? - How can you rewrite fractions with different denominators so they have a common denominator? - Check whether the final fraction can be simplified.

Solution

1. Evaluate the parentheses first: $\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$. 2. Subtract: $\frac{5}{12} - \frac{7}{12} = -\frac{2}{12}$. 3. Simplify: $-\frac{2}{12} = -\frac{1}{6}$.

Answer

$-\frac{1}{6}$

Add and subtract fractions and decimals 13

Calculate: $\left(\frac{3}{8} - \frac{5}{6}\right) + \frac{11}{12}$

Hints

- First rewrite the fractions inside the parentheses with a common denominator. - What sign should the result have when the larger fraction is subtracted from the smaller fraction? - How do you add a negative fraction and a positive fraction?

Solution

1. Evaluate the parentheses using a common denominator of $24$: $\frac{3}{8} - \frac{5}{6} = \frac{9}{24} - \frac{20}{24} = -\frac{11}{24}$. 2. Rewrite $\frac{11}{12}$ as $\frac{22}{24}$. 3. Add: $-\frac{11}{24} + \frac{22}{24} = \frac{11}{24}$.

Answer

$\frac{11}{24}$

Add and subtract fractions and decimals 14

Calculate each expression. Write each answer as a fraction in simplest form or as a mixed number. a) $3 \frac{5}{12}+1 \frac{1}{4}$ b) $6 \frac{2}{5}-4 \frac{7}{10}$ c) $2 \frac{3}{4}+0.125$

Hints

- Rewrite fractional parts with common denominators before adding or subtracting. - For the decimal, consider an equivalent fraction. - If the fractional part of a mixed number is too small to subtract, regroup one whole.

Solution

1. For a), rewrite $1 \frac{1}{4}$ as $1 \frac{3}{12}$. Then $3 \frac{5}{12}+1 \frac{3}{12}=4 \frac{8}{12}=4 \frac{2}{3}$. 2. For b), rewrite $6 \frac{2}{5}$ as $6 \frac{4}{10}$. Regroup one whole: $6 \frac{4}{10}=5 \frac{14}{10}$. Then $5 \frac{14}{10}-4 \frac{7}{10}=1 \frac{7}{10}$. 3. For c), rewrite $0.125$ as $\frac{1}{8}$ and $2 \frac{3}{4}$ as $2 \frac{6}{8}$. Then $2 \frac{6}{8}+\frac{1}{8}=2 \frac{7}{8}$.

Answer

a) $4 \frac{2}{3}$ b) $1 \frac{7}{10}$ c) $2 \frac{7}{8}$

Add and subtract fractions and decimals 15

Calculate each expression. Write each answer as a mixed number in simplest form. a) $3 \frac{2}{5}+4 \frac{5}{6}$ b) $8 \frac{1}{4}-5 \frac{5}{8}$ c) $10-\left(2 \frac{1}{3}+4 \frac{3}{5}\right)$

Hints

- Rewrite fractions with unlike denominators using a common denominator. - In a mixed-number subtraction, you may need to regroup one whole. - Evaluate expressions inside parentheses before the rest of the calculation. - A whole number can be rewritten as a fraction or mixed number when that makes the subtraction easier.

Solution

1. For a), use denominator $30$: $3 \frac{12}{30}+4 \frac{25}{30}=7 \frac{37}{30}=8 \frac{7}{30}$. 2. For b), rewrite $8 \frac{1}{4}$ as $8 \frac{2}{8}$, then regroup: $7 \frac{10}{8}-5 \frac{5}{8}=2 \frac{5}{8}$. 3. For c), evaluate the parentheses: $2 \frac{1}{3}+4 \frac{3}{5}=2 \frac{5}{15}+4 \frac{9}{15}=6 \frac{14}{15}$. Then $10-6 \frac{14}{15}=3 \frac{1}{15}$.

Answer

a) $8 \frac{7}{30}$ b) $2 \frac{5}{8}$ c) $3 \frac{1}{15}$

Add and subtract fractions and decimals 16

Rearrange and group the terms to make the calculation efficient. Write the equivalent regrouped expression you use, then evaluate it. $5\frac{3}{8} + 4\frac{2}{5} - 2\frac{3}{8} + 1\frac{3}{5}$

Hints

- Keep each sign attached to its term when changing the order. - Look for terms whose fractional parts combine without introducing a new denominator. - Your response should show the regrouping, not only the final number.

Solution

1. Rewrite subtraction as addition of the opposite and regroup: $\left(5\frac{3}{8}-2\frac{3}{8}\right)+\left(4\frac{2}{5}+1\frac{3}{5}\right)$. 2. The groups are $3$ and $6$. 3. Therefore, the value is $9$.

Answer

One efficient regrouping is $\left(5\frac{3}{8}-2\frac{3}{8}\right)+\left(4\frac{2}{5}+1\frac{3}{5}\right)$, which equals $9$.

Add and subtract fractions and decimals 17

For each expression, rewrite any fraction as a terminating decimal before evaluating. If an expression has no fraction, state that no conversion is needed. a) $0.25 + \frac{1}{2}$ b) $0.4 - \frac{3}{5}$ c) $0.75 - 1$ d) $1.1 + \frac{9}{10}$

Hints

- Write each fraction with a denominator of $10$ or $100$ when possible. - Keep the converted decimal visible in your response before evaluating. - Check the sign of each result against the relative sizes of the terms.

Solution

1. a) $\frac{1}{2}=0.5$, so $0.25+0.5=0.75$. 2. b) $\frac{3}{5}=0.6$, so $0.4-0.6=-0.2$. 3. c) No conversion is needed: $0.75-1=-0.25$. 4. d) $\frac{9}{10}=0.9$, so $1.1+0.9=2$.

Answer

a) $\frac{1}{2}=0.5$; result $0.75$ b) $\frac{3}{5}=0.6$; result $-0.2$ c) No fraction is present, so no conversion is needed; result $-0.25$ d) $\frac{9}{10}=0.9$; result $2$

Add and subtract fractions and decimals 18

Evaluate $\frac{5}{6} + 0.4 + \frac{1}{6} + 1.1$. Decide whether to convert every value to fractions or decimals, or to combine compatible terms separately. Briefly justify your choice.

Hints

- Compare the work created by converting the sixths to decimals with the work created by keeping them as fractions. - Look for terms that can combine exactly without introducing a new representation. - Your justification should refer to the actual numbers in this expression.

Solution

1. Converting the sixths to decimals would create repeating decimals, so keep them as fractions. 2. Group the fractions: $\frac{5}{6} + \frac{1}{6} = 1$. 3. Group the decimals: $0.4 + 1.1 = 1.5$. 4. Add the partial sums: $1 + 1.5 = 2.5$.

Answer

One particularly efficient choice is to combine the fractions and decimals separately. Converting the sixths to decimals would introduce repeating decimals, while $\frac{5}{6}+\frac{1}{6}=1$ exactly and $0.4+1.1=1.5$, so the value is $2.5$. Another exact strategy is acceptable if it is shown correctly and the efficiency justification refers to the actual conversions and arithmetic required.

Add and subtract fractions and decimals 19

Evaluate the expression by converting the fractions to terminating decimals and then showing one regrouping that creates convenient decimal pairs. $\frac{2}{5} + 1.35 - \frac{3}{4} + 0.1$

Hints

- Convert each fraction exactly before doing the addition or subtraction. - Keep each sign attached to its term when regrouping. - Show the regrouped decimal expression in your answer.

Solution

1. Convert: $\frac{2}{5}=0.4$ and $\frac{3}{4}=0.75$. 2. Regroup: $(0.4+0.1)+(1.35-0.75)$. 3. Evaluate: $0.5+0.6=1.1$.

Answer

$\frac{2}{5}=0.4$, $\frac{3}{4}=0.75$, and $(0.4+0.1)+(1.35-0.75)=1.1$.

Add and subtract fractions and decimals 20

Evaluate the expression by switching strategically between fractions and decimals. $4.5 - \left(1.25 + \frac{1}{3}\right) + \frac{1}{12}$

Hints

- Distribute the subtraction sign before regrouping. - Combine the decimal terms separately from the fractional terms. - Convert the resulting fraction to a decimal only at the final step.

Solution

1. Remove the parentheses: $4.5 - 1.25 - \frac{1}{3} + \frac{1}{12}$. 2. Combine the decimals: $4.5 - 1.25 = 3.25$. 3. Combine the fractions: $-\frac{1}{3} + \frac{1}{12} = -\frac{4}{12} + \frac{1}{12} = -\frac{1}{4} = -0.25$. 4. Add: $3.25 - 0.25 = 3$.

Answer

$3$

Add and subtract fractions and decimals 21

Estimate the value first, then calculate the exact value. $18.75 - \left(4\frac{1}{2} + 3.2\right) + 1\frac{1}{4}$

Hints

- Round to convenient whole numbers for the estimate. - Convert the mixed numbers to terminating decimals for the exact calculation. - Follow the grouping symbols before adding the final term.

Solution

1. Estimate by rounding to convenient whole numbers: $19 - (5 + 3) + 1 = 12$. 2. Convert the mixed numbers: $4\frac{1}{2} = 4.5$ and $1\frac{1}{4} = 1.25$. 3. Evaluate the parentheses: $4.5 + 3.2 = 7.7$. 4. Calculate exactly: $18.75 - 7.7 + 1.25 = 11.05 + 1.25 = 12.3$.

Answer

Estimate: about $12$ Exact value: $12.3$

Add and subtract fractions and decimals 22

Evaluate the expression and write the result as a decimal: $\frac{7}{10}-\left(\frac{2}{5}+\frac{1}{2}\right)$

Hints

- Evaluate the parentheses first. - Fractions need a common denominator before you add them. - How do you write tenths as decimals?

Solution

1. Add inside the parentheses: $\frac{2}{5}+\frac{1}{2}=\frac{4}{10}+\frac{5}{10}=\frac{9}{10}$. 2. Subtract: $\frac{7}{10}-\frac{9}{10}=-\frac{2}{10}$. 3. Convert to a decimal: $-\frac{2}{10}=-0.2$.

Answer

$-0.2$

Add and subtract fractions and decimals 23

Find the error in the calculation and give the correct result. $7\frac{1}{4} - 2.5 = (7 - 2) + (0.25 + 0.5) = 5 + 0.75 = 5.75$

Hints

- The entire second number must be subtracted. - Track the subtraction sign when decomposing a number into whole-number and decimal parts. - Rewrite both numbers as decimals or both as fractions.

Solution

1. The error is that $0.5$, which is part of the number being subtracted, was added instead of subtracted. 2. Convert the mixed number: $7\frac{1}{4} = 7.25$. 3. Subtract correctly: $7.25 - 2.5 = 4.75$.

Answer

The $0.5$ was added instead of subtracted. The correct calculation is $7.25 - 2.5 = 4.75$.

Add and subtract fractions and decimals 24

For each expression, choose a fraction or decimal form that avoids unnecessary work. Write the conversion or rewrite you chose before evaluating. a) $\frac{3}{4} + 0.15$ b) $0.8 - \frac{1}{5}$ c) $-\frac{2}{3} + 0.5$ d) $1.125 - \frac{1}{8}$

Hints

- Ask whether the fraction has a short terminating decimal before converting it. - A repeating decimal is often less convenient than keeping a fraction exact. - Show the representation change you actually use.

Solution

1. a) $\frac{3}{4}=0.75$, so $0.75+0.15=0.9$. 2. b) $\frac{1}{5}=0.2$, so $0.8-0.2=0.6$. 3. c) $0.5=\frac{1}{2}$, so $-\frac{2}{3}+\frac{1}{2}=-\frac{1}{6}$. 4. d) $\frac{1}{8}=0.125$, so $1.125-0.125=1$.

Answer

a) $\frac{3}{4}=0.75$; result $0.9$ b) $\frac{1}{5}=0.2$; result $0.6$ c) $0.5=\frac{1}{2}$; result $-\frac{1}{6}$ d) $\frac{1}{8}=0.125$; result $1$

Add and subtract fractions and decimals 25

Evaluate each expression. For every part, first write the exact fraction-to-decimal conversion you use; then calculate. a) $0.45 + \frac{3}{20}$ b) $\frac{5}{8} - 0.125$ c) $-1.2 + \frac{1}{5}$

Hints

- Rewrite each fraction with denominator $10$, $100$, or $1000$ when possible. - Keep the conversion visible before combining the values. - Check the sign of the final sum or difference.

Solution

1. a) $\frac{3}{20}=0.15$, so $0.45+0.15=0.6$. 2. b) $\frac{5}{8}=0.625$, so $0.625-0.125=0.5$. 3. c) $\frac{1}{5}=0.2$, so $-1.2+0.2=-1$.

Answer

a) $\frac{3}{20}=0.15$; $0.45+0.15=0.6$ b) $\frac{5}{8}=0.625$; $0.625-0.125=0.5$ c) $\frac{1}{5}=0.2$; $-1.2+0.2=-1$

Add and subtract fractions and decimals 26

Before evaluating, show an equivalent rearrangement and regrouping that creates two whole-number partial sums. Then evaluate and name the properties that justify changing the order and grouping. $3.7 + \frac{2}{3} + 1.3 + 1\frac{1}{3}$

Hints

- Keep each addend unchanged when you reorder the sum. - Look for pairs that produce especially simple partial sums. - Name the properties that permit changing order and grouping.

Solution

1. Use the commutative and associative properties of addition to write $(3.7+1.3)+\left(\frac{2}{3}+1\frac{1}{3}\right)$. 2. The decimal group is $5$. 3. The fraction group is $\frac{2}{3}+\frac{4}{3}=2$. 4. Therefore, the sum is $7$.

Answer

A valid regrouping is $(3.7+1.3)+\left(\frac{2}{3}+1\frac{1}{3}\right)=5+2=7$. The commutative and associative properties of addition justify the changed order and grouping.

Add and subtract fractions and decimals 27

Evaluate each expression. a) $-4.8-2.35$ b) $-\frac{5}{6}+\frac{1}{3}$ c) $0.75+(-1.2)$ d) $-\left(\frac{1}{4}-\frac{5}{8}\right)$

Hints

- Pay attention to signs when removing parentheses. - Use common denominators before adding or subtracting fractions. - Before calculating with decimals, decide whether the result should be positive or negative.

Solution

1. For a), $-4.8-2.35=-7.15$. 2. For b), $-\frac{5}{6}+\frac{1}{3}=-\frac{5}{6}+\frac{2}{6}=-\frac{1}{2}$. 3. For c), $0.75+(-1.2)=-0.45$. 4. For d), $\frac{1}{4}-\frac{5}{8}=\frac{2}{8}-\frac{5}{8}=-\frac{3}{8}$. The leading negative changes the sign, giving $\frac{3}{8}$.

Answer

a) $-7.15$ b) $-\frac{1}{2}$ c) $-0.45$ d) $\frac{3}{8}$

Add and subtract fractions and decimals 28

Evaluate the expression: $(-4.2+1.7)-(0.8-2.3)$

Hints

- Evaluate both sets of parentheses first. - What happens when you subtract a negative number? - A number line can help you check the final sign.

Solution

1. First parentheses: $-4.2+1.7=-2.5$. 2. Second parentheses: $0.8-2.3=-1.5$. 3. Subtract: $-2.5-(-1.5)=-1$.

Answer

$-1$

Add and subtract fractions and decimals 29

Calculate the value of the expression and write the result as a mixed number: $\left(5 \frac{3}{4}-1 \frac{1}{2}\right)+2 \frac{5}{8}$

Hints

- Evaluate the expression inside the parentheses first. - Rewrite the fractional parts with common denominators before subtracting or adding. - Simplify the result inside the parentheses before continuing. - Convert the intermediate result to eighths before the final addition.

Solution

1. Evaluate the parentheses: $5 \frac{3}{4}-1 \frac{1}{2}=5 \frac{3}{4}-1 \frac{2}{4}=4 \frac{1}{4}$. 2. Rewrite $4 \frac{1}{4}$ as $4 \frac{2}{8}$. 3. Add: $4 \frac{2}{8}+2 \frac{5}{8}=6 \frac{7}{8}$.

Answer

$6 \frac{7}{8}$

Add and subtract fractions and decimals 30

A point starts at $-2.3$ on a number line. It moves $4.5$ units to the right and then $1.4$ units to the left. Where does the point end?

Hints

- A move to the right corresponds to addition. - A move to the left corresponds to subtraction. - Carry out the moves in order.

Solution

1. Moving right means add: $-2.3+4.5=2.2$. 2. Moving left means subtract: $2.2-1.4=0.8$. 3. The point ends at $0.8$.

Answer

$0.8$

Add and subtract fractions and decimals 31

Evaluate and write the result as a fraction in simplest form: $\frac{2}{5}+\frac{1}{2}-\frac{3}{10}$

Hints

- Find a common denominator for $5$, $2$, and $10$. - Rewrite all three fractions before adding and subtracting. - Simplify the final fraction.

Solution

1. Use denominator $10$: $\frac{2}{5}=\frac{4}{10}$ and $\frac{1}{2}=\frac{5}{10}$. 2. Calculate: $\frac{4}{10}+\frac{5}{10}-\frac{3}{10}=\frac{6}{10}=\frac{3}{5}$.

Answer

$\frac{3}{5}$

Add and subtract fractions and decimals 32

Evaluate each sum or difference of rational numbers. a) $-8.7+(-13.5)$ b) $4.2-9.8$ c) $-\frac{5}{9}+\frac{1}{6}$ d) $\frac{2}{3}-\frac{3}{4}+\left(-\frac{1}{2}\right)$

Hints

- What sign results when two negative numbers are added? - For unlike signs, compare absolute values. - Use a common denominator for fractions. - You can rewrite subtraction as addition of the opposite.

Solution

1. For a), $-8.7+(-13.5)=-22.2$. 2. For b), $4.2-9.8=-5.6$. 3. For c), use denominator $18$: $-\frac{10}{18}+\frac{3}{18}=-\frac{7}{18}$. 4. For d), use denominator $12$: $\frac{8}{12}-\frac{9}{12}-\frac{6}{12}=-\frac{7}{12}$.

Answer

a) $-22.2$ b) $-5.6$ c) $-\frac{7}{18}$ d) $-\frac{7}{12}$

Add and subtract fractions and decimals 33

Evaluate each expression. Write the result as either a decimal or a fraction. a) $\frac{3}{4}-1.25$ b) $-0.6-\left(-\frac{1}{5}\right)$ c) $2-\frac{1}{2}-0.75$ d) $-\frac{1}{8}-0.125$

Hints

- Choose fractions or decimals and convert values to the same form. - Recall common fraction-decimal equivalents such as $\frac{1}{2}$ and $\frac{1}{5}$. - Pay close attention to subtracting a negative number.

Solution

1. For a), $0.75-1.25=-0.5=-\frac{1}{2}$. 2. For b), $-0.6-(-0.2)=-0.4=-\frac{2}{5}$. 3. For c), $2-0.5-0.75=0.75=\frac{3}{4}$. 4. For d), $-0.125-0.125=-0.25=-\frac{1}{4}$.

Answer

a) $-0.5$, or $-\frac{1}{2}$ b) $-0.4$, or $-\frac{2}{5}$ c) $0.75$, or $\frac{3}{4}$ d) $-0.25$, or $-\frac{1}{4}$

Add and subtract fractions and decimals 34

The expressions are $A=x-5.4$ and $B=-3.2-y$. Evaluate them for $x=2.1$ and $y=-1.8$. Which value is greater?

Hints

- Substitute each given value into its expression. - Subtracting a negative number is equivalent to adding its opposite. - On a number line, the number farther to the right is greater.

Solution

1. $A=2.1-5.4=-3.3$. 2. $B=-3.2-(-1.8)=-3.2+1.8=-1.4$. 3. Since $-1.4>-3.3$, $B$ has the greater value.

Answer

$A=-3.3$ and $B=-1.4$. The value of $B$ is greater.

Add and subtract fractions and decimals 35

Ms. Weber's bank account balance is $-\$75.20$. She wants the balance to be exactly $\$120.00$. How much money must she deposit?

Hints

- A negative balance is below $0$. - First think about how much is needed to reach $0$, then how much more is needed to reach $120$. - Subtract the current signed balance from the target balance.

Solution

1. The current balance is $-\$75.20$, and the target balance is $\$120.00$. 2. The required deposit is the change from $-75.20$ to $120.00$: $120.00-(-75.20)=195.20$. 3. Therefore, she must deposit $\$195.20$.

Answer

$\$195.20$

Add and subtract fractions and decimals 36

Ms. Schneider's checking account balance is $-\$125.50$. In the morning, she deposits $\$350.00$. In the afternoon, she pays an $\$80.00$ bill from the account. Find the end-of-day balance using one expression with rational numbers.

Hints

- A deposit increases the account balance. - A bill payment decreases the account balance. - Write all three amounts in one signed-number expression before calculating.

Solution

1. Represent the transactions in one expression: $-125.50+350.00-80.00$. 2. Calculate: $-125.50+350.00=224.50$, then $224.50-80.00=144.50$. 3. The end-of-day balance is $\$144.50$.

Answer

$-125.50+350.00-80.00=144.50$. The end-of-day balance is $\$144.50$.

Add and subtract fractions and decimals 37

Evaluate $T(a,b)=a-b$ for each pair of rational numbers. Complete the table using fractions in simplest form or mixed numbers. <table> <tr><td>$a-b$</td><td>$-\frac{2}{3}$</td><td>$\frac{5}{6}$</td></tr> <tr><td>$-\frac{1}{2}$</td><td></td><td></td></tr> <tr><td>$\frac{3}{4}$</td><td></td><td></td></tr> </table>

Hints

- Use a common denominator before subtracting fractions. - Subtracting a negative number means adding. - Simplify each result or write it as a mixed number.

Solution

1. $-\frac{1}{2}-\left(-\frac{2}{3}\right)=-\frac{3}{6}+\frac{4}{6}=\frac{1}{6}$. 2. $-\frac{1}{2}-\frac{5}{6}=-\frac{3}{6}-\frac{5}{6}=-\frac{8}{6}=-\frac{4}{3}=-1\frac{1}{3}$. 3. $\frac{3}{4}-\left(-\frac{2}{3}\right)=\frac{9}{12}+\frac{8}{12}=\frac{17}{12}=1\frac{5}{12}$. 4. $\frac{3}{4}-\frac{5}{6}=\frac{9}{12}-\frac{10}{12}=-\frac{1}{12}$.

Answer

Add and subtract fractions and decimals 38

Determine whether each result is positive or negative without finding its exact value. For each part, give a sign rule or an absolute-value comparison that proves your choice; an exact numerical result alone is not a justification. a) $-12+(-15)$ b) $24-(-10)$ c) $-0.5+0.3$ d) $\frac{1}{2}-\frac{3}{4}$ e) $-8-(-5)$

Hints

- Base each decision on signs, opposites, or relative magnitudes rather than exact arithmetic. - For unlike signs, compare absolute values. - Rewrite subtraction of a negative before deciding its effect.

Solution

1. a) The sum of two negative numbers is negative. 2. b) Subtracting a negative is adding a positive, so the result is positive. 3. c) The addends have unlike signs and $0.5>0.3$, so the sign is negative. 4. d) Since $\frac{3}{4}>\frac{1}{2}$, subtracting the larger positive number gives a negative result. 5. e) Rewrite as $-8+5$. Since $8>5$, the result is negative.

Answer

a) negative — both addends are negative b) positive — subtracting a negative adds a positive c) negative — $|-0.5|>|0.3|$ d) negative — $\frac{3}{4}>\frac{1}{2}$ e) negative — $-8-(-5)=-8+5$ and $8>5$

Add and subtract fractions and decimals 39

Rewrite the expression as an equivalent sum that groups compatible fractions, then evaluate it. $\frac{7}{15} - \frac{3}{4} + \frac{8}{15} - \frac{1}{4}$

Hints

- First view every subtraction as addition of a signed term. - Which terms can be combined without creating a new denominator? - Include the regrouped expression in your response.

Solution

1. Rewrite and regroup: $\left(\frac{7}{15}+\frac{8}{15}\right)+\left(-\frac{3}{4}-\frac{1}{4}\right)$. 2. The groups are $1$ and $-1$. 3. Their sum is $0$.

Answer

$\left(\frac{7}{15}+\frac{8}{15}\right)+\left(-\frac{3}{4}-\frac{1}{4}\right)=0$

Add and subtract fractions and decimals 40

Evaluate without converting every number to the same form. Show an equivalent grouping that lets the decimals combine with decimals and the fractions combine with fractions. $2.7 + \frac{2}{9} - 1.7 + \frac{16}{9}$

Hints

- Keep each sign attached to its term when changing order. - Look for same-form terms that combine exactly as written. - Do not convert a repeating fraction to a decimal just to make every term look alike.

Solution

1. Regroup as $(2.7-1.7)+\left(\frac{2}{9}+\frac{16}{9}\right)$. 2. The groups equal $1$ and $2$. 3. Therefore, the value is $3$.

Answer

$(2.7-1.7)+\left(\frac{2}{9}+\frac{16}{9}\right)=1+2=3$

Add and subtract fractions and decimals 41

For each expression, write a regrouped equivalent expression that creates convenient pairs, then evaluate. a) $12.3 - 4.5 - 2.3 + 1.5$ b) $\frac{3}{8} - 1.4 + \frac{5}{8} - 0.6$

Hints

- Rewrite each subtraction as addition of a signed term before changing order. - Look for pairs whose sum or difference is a whole number. - Show the regrouping explicitly.

Solution

1. a) Regroup as $(12.3-2.3)+(-4.5+1.5)=10-3=7$. 2. b) Regroup as $\left(\frac{3}{8}+\frac{5}{8}\right)+(-1.4-0.6)=1-2=-1$.

Answer

a) $(12.3-2.3)+(-4.5+1.5)=7$ b) $\left(\frac{3}{8}+\frac{5}{8}\right)+(-1.4-0.6)=-1$

Add and subtract fractions and decimals 42

For each expression, rearrange and group terms to create at least one multiple of $10$ or $100$. Write the regrouped expression, then evaluate. a) $34 + 128 - 134$ b) $12.5 - 6.7 - 2.5 + 1.7$ c) $-48 + 56 - 52 + 44$

Hints

- Keep each sign attached to its term when rearranging. - Search for pairs whose combined value has a zero in the ones place. - Your response must display the convenient grouping.

Solution

1. a) $(34-134)+128=-100+128=28$. 2. b) $(12.5-2.5)+(-6.7+1.7)=10-5=5$. 3. c) $(-48-52)+(56+44)=-100+100=0$.

Answer

a) $(34-134)+128=28$ b) $(12.5-2.5)+(-6.7+1.7)=5$ c) $(-48-52)+(56+44)=0$

Add and subtract fractions and decimals 43

For each expression, show one regrouping that creates a whole-number partial sum or difference, then evaluate. a) $15.6 - 7.8 - 2.2$ b) $-4.9 + 12.5 - 5.1$ c) $24.3 + 8.7 - 14.3 + 1.3$

Hints

- Look for decimal pairs whose sum or difference is a whole number. - Preserve the sign of every term when changing grouping. - Include the regrouped expression in your answer.

Solution

1. a) $15.6-(7.8+2.2)=15.6-10=5.6$. 2. b) $(-4.9-5.1)+12.5=-10+12.5=2.5$. 3. c) $(24.3-14.3)+(8.7+1.3)=10+10=20$.

Answer

a) $15.6-(7.8+2.2)=5.6$ b) $(-4.9-5.1)+12.5=2.5$ c) $(24.3-14.3)+(8.7+1.3)=20$

Add and subtract fractions and decimals 44

Use properties of addition to evaluate efficiently. $7.25-4.8+\frac{11}{4}-1.2$

Hints

- Convert the fraction to a decimal. - Look for values whose decimal parts combine to whole numbers. - Rewrite the expression as a sum before reordering terms.

Solution

1. Write $\frac{11}{4}=2.75$. 2. Rewrite subtraction as addition and regroup: $(7.25+2.75)+(-4.8-1.2)$. 3. Evaluate the grouped sums: $10+(-6)=4$.

Answer

$4$

Add and subtract fractions and decimals 45

a) Start at $-1.25$ on a number line and move $0.75$ unit to the right. Where do you land? b) From your new position, move $1.25$ units left. Where do you land? Write an addition equation for each move.

Hints

- Think about which direction each move takes you. - Use the landing point from a) as the starting point for b). - A move to the left can be written as adding a negative number.

Solution

1. Moving right means adding a positive amount: $-1.25+0.75=-0.5$. 2. Moving left means adding a negative amount: $-0.5+(-1.25)=-1.75$.

Answer

a) $-1.25+0.75=-0.5$ b) $-0.5+(-1.25)=-1.75$

Add and subtract fractions and decimals 46

For each expression, write one equivalent regrouping that produces convenient whole-number partial results, then evaluate. a) $14.7 - 3.9 + 5.3 - 6.1$ b) $-\frac{3}{8} + 12.5 - 5.5 + 0.375$ c) $8.4 - (2.4 + 3.9) + 0.9$

Hints

- Search for pairs that make whole numbers or additive inverses. - A fraction-decimal conversion is useful only if it creates such a pair. - Keep signs attached to terms when regrouping.

Solution

1. a) $(14.7+5.3)+(-3.9-6.1)=20-10=10$. 2. b) Since $-\frac{3}{8}=-0.375$, $(-0.375+0.375)+(12.5-5.5)=7$. 3. c) Remove the parentheses and regroup: $(8.4-2.4)+(0.9-3.9)=6-3=3$.

Answer

a) $(14.7+5.3)+(-3.9-6.1)=10$ b) $(-0.375+0.375)+(12.5-5.5)=7$ c) $(8.4-2.4)+(0.9-3.9)=3$

Add and subtract fractions and decimals 47

Evaluate the expression: $\frac{2}{3}-\left(-\frac{1}{4}\right)+\frac{5}{6}-1$

Hints

- What happens when you subtract a negative number? - Use a common denominator for all fractions. - Write $1$ as a fraction with the same denominator.

Solution

1. Subtracting a negative gives addition: $\frac{2}{3}+\frac{1}{4}+\frac{5}{6}-1$. 2. Use denominator $12$: $\frac{8}{12}+\frac{3}{12}+\frac{10}{12}-\frac{12}{12}=\frac{9}{12}$. 3. Simplify: $\frac{9}{12}=\frac{3}{4}=0.75$.

Answer

$\frac{3}{4}$, or $0.75$

Add and subtract fractions and decimals 48

Evaluate each expression. Use common denominators where needed. a) $\frac{4}{5}+\left(-\frac{1}{2}\right)$ b) $-\frac{3}{4}-\frac{5}{6}$ c) $-1.5+\frac{1}{4}$ d) $\frac{7}{9}-\left(-\frac{2}{3}\right)$

Hints

- Subtracting a negative number is the same as adding its opposite. - Fractions need common denominators before addition or subtraction. - Convert between fractions and decimals when helpful. - Simplify or write improper fractions as mixed numbers if useful.

Solution

1. For a), $\frac{4}{5}-\frac{1}{2}=\frac{8}{10}-\frac{5}{10}=\frac{3}{10}$. 2. For b), $-\frac{3}{4}-\frac{5}{6}=-\frac{9}{12}-\frac{10}{12}=-\frac{19}{12}=-1 \frac{7}{12}$. 3. For c), $-1.5+0.25=-1.25=-\frac{5}{4}$. 4. For d), subtracting a negative gives addition: $\frac{7}{9}+\frac{2}{3}=\frac{7}{9}+\frac{6}{9}=\frac{13}{9}=1 \frac{4}{9}$.

Answer

a) $\frac{3}{10}$ b) $-\frac{19}{12}$, or $-1 \frac{7}{12}$ c) $-1.25$, or $-\frac{5}{4}$ d) $\frac{13}{9}$, or $1 \frac{4}{9}$

Add and subtract fractions and decimals 49

A mountain weather station tracks temperature changes during the day. In the morning, the temperature rises by $x\,^{\circ}\text{C}$. In the afternoon, it falls by $y\,^{\circ}\text{C}$. a) Write an expression for the net temperature change. b) Find the net change on each day. 1) Day 1: $x=5.4$, $y=3.1$ 2) Day 2: $x=2.8$, $y=4.5$ c) Explain what the sign of each result means.

Hints

- Think about the direction of each change on a thermometer. - Subtract the amount of the decrease from the amount of the increase. - Interpret a negative result as an overall drop.

Solution

1. The net change is the increase minus the decrease: $x-y$. 2. Day 1: $5.4-3.1=2.3$, so the temperature rises by $2.3\,^{\circ}\text{C}$ overall. 3. Day 2: $2.8-4.5=-1.7$, so the temperature falls by $1.7\,^{\circ}\text{C}$ overall. 4. A positive result indicates a net warming, and a negative result indicates a net cooling.

Answer

a) $x-y$ b) Day 1: $2.3\,^{\circ}\text{C}$; Day 2: $-1.7\,^{\circ}\text{C}$ c) A positive value means the day ends warmer than it began; a negative value means it ends cooler.

Add and subtract fractions and decimals 50

Evaluate by rearranging and grouping the addends. Show the reordered expression, the two grouped partial sums, and the final value. $17.4 + (-5.9) + 2.6 + (-4.1)$ Which property allows you to change the order of the addends?

Hints

- Look for decimal pairs whose sum is a whole number. - Separate changing the order from changing the grouping. - The answer must show both the reordered expression and the resulting groups.

Solution

1. Reorder: $17.4+2.6+(-5.9)+(-4.1)$. 2. Group: $(17.4+2.6)+((-5.9)+(-4.1))$. 3. The partial sums are $20$ and $-10$, so the final value is $10$. 4. The commutative property of addition allows the order to change.

Answer

Reordered expression: $17.4+2.6+(-5.9)+(-4.1)$ Grouped expression: $(17.4+2.6)+((-5.9)+(-4.1))$ Partial sums: $20$ and $-10$ Final value: $10$ Property: commutative property of addition

Add and subtract fractions and decimals 51

Find each sum of rational numbers. 1) $(-17)+(+25)+(-13)$ 2) $(+4.8)+(-6.3)+(-2.5)$ 3) $\left(-\frac{3}{10}\right)+0.75$ 4) $\left(-2 \frac{1}{4}\right)+\left(-1 \frac{5}{8}\right)$

Hints

- Pay close attention to signs when adding rational numbers. - You can combine several addends step by step. - When fractions and decimals appear together, convert them to a common form if that makes the calculation easier. - Fractions need a common denominator before they can be added.

Solution

1. For 1), $-17+25=8$, then $8+(-13)=-5$. 2. For 2), $4.8+(-6.3)=-1.5$, then $-1.5+(-2.5)=-4$. 3. For 3), $-\frac{3}{10}=-0.3$, so $-0.3+0.75=0.45$. Equivalently, $-\frac{6}{20}+\frac{15}{20}=\frac{9}{20}$. 4. For 4), write the mixed numbers with denominator $8$: $-2 \frac{2}{8}+\left(-1 \frac{5}{8}\right)=-3 \frac{7}{8}$.

Answer

1) $-5$ 2) $-4$ 3) $0.45$, or $\frac{9}{20}$ 4) $-3 \frac{7}{8}$

Add and subtract fractions and decimals 52

For each pair of values, find $c$ so that $a+b+c=0$. a) $a=-8.5$, $b=3\frac{1}{4}$ b) $a=\frac{2}{3}$, $b=-\frac{1}{6}$ c) $a=15.7$, $b=-20$

Hints

- First find $a+b$. - The total is zero when $c$ is the opposite of $a+b$. - Use a common denominator when adding or subtracting fractions.

Solution

1. For a), $a+b=-8.5+3.25=-5.25$. The opposite of $-5.25$ is $5.25$, so $c=5.25$. 2. For b), $a+b=\frac{2}{3}-\frac{1}{6}=\frac{4}{6}-\frac{1}{6}=\frac{1}{2}$. Its opposite is $-\frac{1}{2}$, so $c=-\frac{1}{2}$. 3. For c), $a+b=15.7-20=-4.3$. Its opposite is $4.3$, so $c=4.3$.

Answer

a) $c=5.25$ b) $c=-\frac{1}{2}$ c) $c=4.3$

Add and subtract fractions and decimals 53

Evaluate each expression. 1) $(-18)-(+12)-(-20)$ 2) $(-5.4)-(-2.6)-(+1.2)$ 3) $(+3.7)-(+8.7)-(-5)$

Hints

- What happens to the sign of a number in parentheses when a minus sign is directly before it? - Can you rewrite subtraction as addition of the opposite? - After rewriting, work from left to right. - Keep decimal place values aligned and track the signs at each step.

Solution

1. For 1), rewrite subtraction of a negative as addition: $-18-12+20=-30+20=-10$. 2. For 2), $-5.4-(-2.6)-1.2=-5.4+2.6-1.2=-2.8-1.2=-4$. 3. For 3), $3.7-8.7-(-5)=3.7-8.7+5=-5+5=0$.

Answer

1) $-10$ 2) $-4$ 3) $0$

Add and subtract fractions and decimals 54

For each sum, write the grouping that creates an additive inverse or a whole-number partial sum, then evaluate. 1) $(-34) + 57 + 34$ 2) $12.4 + (-6.8) + (-3.2)$ 3) $-2\frac{1}{5} + 8 - 1\frac{4}{5}$

Hints

- Search for opposites or terms that combine to an integer. - Keep each sign with its term when grouping. - Include the grouping that makes the arithmetic short.

Solution

1. $(-34+34)+57=57$. 2. $12.4+(-6.8-3.2)=12.4-10=2.4$. 3. $\left(-2\frac{1}{5}-1\frac{4}{5}\right)+8=-4+8=4$.

Answer

1) $(-34+34)+57=57$ 2) $12.4+(-6.8-3.2)=2.4$ 3) $\left(-2\frac{1}{5}-1\frac{4}{5}\right)+8=4$

Add and subtract fractions and decimals 55

Calculate $x-y$ for each pair. a) $x=-2.4$, $y=5.6$ b) $x=\frac{3}{5}$, $y=-\frac{7}{10}$ c) $x=-1\frac{1}{2}$, $y=-2\frac{1}{4}$ d) $x=-0.75$, $y=0.75$

Hints

- Subtracting a negative number is the same as adding its opposite. - Use a common denominator before subtracting fractions. - For decimal differences, think about the expected sign before calculating.

Solution

1. For a), $x-y=-2.4-5.6=-8.0$. 2. For b), $x-y=\frac{3}{5}-\left(-\frac{7}{10}\right)=\frac{6}{10}+\frac{7}{10}=\frac{13}{10}=1\frac{3}{10}$. 3. For c), $x-y=-1\frac{1}{2}-\left(-2\frac{1}{4}\right)=-\frac{6}{4}+\frac{9}{4}=\frac{3}{4}$. 4. For d), $x-y=-0.75-0.75=-1.5$.

Answer

a) $-8.0$ b) $1\frac{3}{10}$, or $1.3$ c) $\frac{3}{4}$, or $0.75$ d) $-1.5$

Add and subtract fractions and decimals 56

Point $A$ is a starting position and point $B$ is an ending position on the number line. a) Read $A$ and $B$ as fractions in simplest form. b) Write the signed change from $A$ to $B$ as $B-A$, and evaluate it. c) Does the sign of the change match the direction of movement? Explain.

Hints

- Determine the value of one small tick interval from the labeled integers. - Read both coordinates before forming the signed difference. - Relate the sign of $B-A$ to the direction from the starting point to the ending point.

Solution

1. The ticks divide each unit into eighths, so $A=-\frac{5}{8}$ and $B=\frac{1}{4}$. 2. $B-A=\frac{1}{4}-\left(-\frac{5}{8}\right)=\frac{2}{8}+\frac{5}{8}=\frac{7}{8}$. 3. The change is positive, matching movement to the right from $A$ to $B$.

Answer

a) $A=-\frac{5}{8}$, $B=\frac{1}{4}$ b) $B-A=\frac{7}{8}$ c) Yes. The positive change matches movement to the right.

Add and subtract fractions and decimals 57

Calculate $0.3-0.85$. Explain why your result is negative.

Hints

- Line up tenths and hundredths before subtracting. - Predict whether the result is above or below zero. - Check that your answer is less than the starting value.

Solution

1. Rewrite $0.3$ as $0.30$. 2. $0.30-0.85=-0.55$. 3. The amount being subtracted is greater than the starting number, so the result is below zero.

Answer

$-0.55$; $0.85$ is greater than $0.30$.

Add and subtract fractions and decimals 58

Find the missing addend in $-\frac{1}{4}+x=\frac{3}{8}$. Show how you check it.

Hints

- What could you add to undo the negative quarter? - Use eighths to compare the two fractions. - Put the found value back into the original equation.

Solution

1. Add $\frac{1}{4}$ to the right side: $x=\frac{3}{8}+\frac{1}{4}$. 2. $x=\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$. 3. Check: $-\frac{1}{4}+\frac{5}{8}=-\frac{2}{8}+\frac{5}{8}=\frac{3}{8}$.

Answer

$x=\frac{5}{8}$. Check: $-\frac{1}{4}+\frac{5}{8}=\frac{3}{8}$.

Add and subtract fractions and decimals 59

Malik writes $-\frac{1}{3}-\frac{2}{3}=1$. What mistake did he make? Give the correct value.

Hints

- Predict the sign before computing. - Subtracting a positive number makes the result smaller. - Combine thirds without changing the sign.

Solution

1. Subtracting a positive fraction from a negative fraction moves farther below zero; it cannot produce a positive result. 2. $-\frac{1}{3}-\frac{2}{3}=-\frac{3}{3}=-1$.

Answer

Malik lost the negative sign. The correct result is $-\frac{1}{3}-\frac{2}{3}=-1$.

Add and subtract fractions and decimals 60

Calculate $\frac{7}{8}-\left(-\frac{1}{4}\right)$. Explain why the result must be greater than $1$, then check it using addition.

Hints

- Think about what subtracting a negative does. - Compare your expected result with the whole-number benchmark. - Use the inverse of the original subtraction to check.

Solution

1. Subtracting a negative gives $\frac{7}{8}+\frac{1}{4}=\frac{7}{8}+\frac{2}{8}=\frac{9}{8}$. 2. Since $\frac{7}{8}$ is only $\frac{1}{8}$ below $1$, adding $\frac{2}{8}$ passes $1$. 3. Check by adding the subtracted number to the result: $\frac{9}{8}+\left(-\frac{1}{4}\right)=\frac{9}{8}-\frac{2}{8}=\frac{7}{8}$.

Answer

$\frac{9}{8}=1\frac{1}{8}$, which is above $1$. Check: $\frac{9}{8}+\left(-\frac{1}{4}\right)=\frac{7}{8}$.

Add and subtract fractions and decimals 61

Find the fraction exactly halfway between $\frac{2}{3}$ and $\frac{4}{5}$. Write your answer in simplest form.

Hints

- The number halfway between two values is their mean. - Use a common denominator before adding the fractions. - Divide the sum by $2$.

Solution

1. The midpoint is the mean: $\frac{\frac{2}{3}+\frac{4}{5}}{2}$. 2. Use a common denominator: $\frac{2}{3}=\frac{10}{15}$ and $\frac{4}{5}=\frac{12}{15}$. 3. Add: $\frac{10}{15}+\frac{12}{15}=\frac{22}{15}$. 4. Divide by $2$: $\frac{22}{15}\div2=\frac{22}{15}\cdot\frac{1}{2}=\frac{11}{15}$.

Answer

$\frac{11}{15}$

Add and subtract fractions and decimals 62

Point $P$ is at $-3.4$ on a number line, and point $Q$ is at $1.2$. a) Find the number $m$ exactly halfway between $P$ and $Q$. b) Is $m$ an integer? Is it a rational number? c) List all integers strictly between $-3.4$ and $1.2$.

Hints

- Find the midpoint by adding the endpoints and dividing by $2$. - A terminating decimal can be written as a fraction. - Move from left to right on the number line and record each integer between the endpoints.

Solution

1. Find the midpoint: $m=\frac{-3.4+1.2}{2}=\frac{-2.2}{2}=-1.1$. 2. The value $-1.1$ is not an integer. It is rational because $-1.1=-\frac{11}{10}$. 3. The integers satisfying $-3.4<z<1.2$ are $-3, -2, -1, 0, 1$.

Answer

a) $m=-1.1$ b) $m$ is not an integer, but it is rational. c) $-3, -2, -1, 0, 1$

Add and subtract fractions and decimals 63

Compare $\frac{19}{8}$ and $2.37$. a) Which number is greater? b) Find the decimal exactly halfway between the two numbers.

Hints

- Convert the fraction to a decimal. - Add a trailing zero when comparing $2.37$ with a number having three decimal places. - The number halfway between two values is their average.

Solution

1. Convert the fraction: $\frac{19}{8}=2.375$. 2. Compare $2.375$ and $2.370$. Therefore, $\frac{19}{8}>2.37$. 3. Find the midpoint: $\frac{2.375+2.37}{2}=\frac{4.745}{2}=2.3725$.

Answer

a) $\frac{19}{8}$ is greater. b) $2.3725$

Add and subtract fractions and decimals 64

Find the number exactly halfway between each pair. a) Between $\frac{5}{8}$ and $0.63$. Give the answer as a decimal. b) Between $0.7$ and $0.71$. Give the answer as a fraction in simplest form.

Hints

- The number exactly halfway between two values is their average. - Convert the fraction in part a) to a decimal. - In part b), use equivalent fractions with a denominator large enough to leave a numerator between the two endpoints.

Solution

1. For a), $\frac{5}{8}=0.625$. The midpoint is $\frac{0.625+0.63}{2}=\frac{1.255}{2}=0.6275$. 2. For b), write the endpoints as fractions: $0.7=\frac{140}{200}$ and $0.71=\frac{142}{200}$. 3. The fraction halfway between them is $\frac{141}{200}$, which is already in simplest form.

Answer

a) $0.6275$ b) $\frac{141}{200}$

Add and subtract fractions and decimals 65

Find the decimal exactly halfway between $\frac{1}{5}$ and $\frac{1}{4}$. Then give one decimal greater than $\frac{1}{5}$ but less than your midpoint.

Hints

- Convert both fractions to decimals. - Find the average of the two endpoint values. - Choose a value strictly between the lower endpoint and the midpoint.

Solution

1. Convert the fractions: $\frac{1}{5} = 0.2$ and $\frac{1}{4} = 0.25$. 2. Average the endpoints: $\frac{0.2 + 0.25}{2} = \frac{0.45}{2} = 0.225$. 3. Any decimal strictly between $0.2$ and $0.225$ works, such as $0.21$.

Answer

The midpoint is $0.225$. One possible additional decimal is $0.21$.

Add and subtract fractions and decimals 66

Let $x=\frac{5}{8}$ and $y=0.62\overline{5}$. 1. Compare $x$ and $y$ using $<$, $>$, or $=$. 2. Find the rational number $z$ exactly halfway between $0.62$ and $x$. Write $z$ as a decimal.

Hints

- Put $x$ and $y$ into forms that can be compared digit by digit. - For the halfway value, think about what it means for the two distances from $z$ to be equal. - Check that your proposed $z$ is the same distance from both endpoints.

Solution

1. Convert $x$: $\frac{5}{8}=0.625$. 2. Since $y=0.62555\ldots$, it follows that $x<y$. 3. Find the midpoint by averaging: $z=\frac{0.62+0.625}{2}$. 4. Compute $z=\frac{1.245}{2}=0.6225$.

Answer

1. $x<y$ 2. $z=0.6225$

Add and subtract fractions and decimals 67

Find the rational number $x$ exactly halfway between $-\frac{1}{2}$ and $0.3$. Then determine whether $x$ is greater than or less than $-\frac{1}{8}$. Justify your comparison.

Hints

- The number halfway between two values is their average. - Convert the fractions and decimals to a common form. - On a number line, the value farther right is greater.

Solution

1. Convert $-\frac{1}{2}$ to $-0.5$. 2. Find the midpoint: $x=\frac{-0.5+0.3}{2}=\frac{-0.2}{2}=-0.1$. 3. Convert the comparison value: $-\frac{1}{8}=-0.125$. 4. Since $-0.1>-0.125$, $x> -\frac{1}{8}$.

Answer

$x=-0.1$, and $x> -\frac{1}{8}$.

Add and subtract fractions and decimals 68

Calculate each expression. Write each answer as a fraction in simplest form or as a mixed number. a) $1 \frac{3}{8} + \frac{7}{8} - \frac{5}{8}$ b) $\frac{4}{15} - \left(\frac{7}{15} + \frac{12}{15}\right)$ c) $2 \frac{1}{12} - \frac{5}{12} - \frac{11}{12}$

Hints

- How can you rewrite a mixed number before calculating? - Remember to evaluate the expression inside parentheses first. - A subtraction involving fractions can have a negative result. - At the end, simplify the fraction or rewrite an improper fraction as a mixed number when appropriate.

Solution

1. For a), rewrite $1 \frac{3}{8}$ as $\frac{11}{8}$. Then $\frac{11+7-5}{8} = \frac{13}{8} = 1 \frac{5}{8}$. 2. For b), evaluate the parentheses first: $\frac{7}{15} + \frac{12}{15} = \frac{19}{15}$. Then $\frac{4}{15} - \frac{19}{15} = -\frac{15}{15} = -1$. 3. For c), rewrite $2 \frac{1}{12}$ as $\frac{25}{12}$. Then $\frac{25-5-11}{12} = \frac{9}{12} = \frac{3}{4}$.

Answer

a) $1 \frac{5}{8}$ b) $-1$ c) $\frac{3}{4}$

Add and subtract fractions and decimals 69

Calculate the value of the nested expression: $\frac{2}{5} - \left(\frac{1}{2} - \left(\frac{3}{10} + \frac{1}{4}\right)\right)$

Hints

- With nested parentheses, work from the innermost parentheses outward. - Subtracting a negative number is the same as adding its opposite. - Find a useful common denominator at each step.

Solution

1. Start with the innermost parentheses: $\frac{3}{10} + \frac{1}{4} = \frac{6}{20} + \frac{5}{20} = \frac{11}{20}$. 2. Evaluate the next parentheses: $\frac{1}{2} - \frac{11}{20} = \frac{10}{20} - \frac{11}{20} = -\frac{1}{20}$. 3. Subtract the negative fraction: $\frac{2}{5} - \left(-\frac{1}{20}\right) = \frac{2}{5} + \frac{1}{20}$. 4. Use denominator $20$: $\frac{8}{20} + \frac{1}{20} = \frac{9}{20}$.

Answer

$\frac{9}{20}$

Add and subtract fractions and decimals 70

Compare the results of calculations $A$ and $B$. Insert $<$, $>$, or $=$, and justify your answer with calculations. $A=5 \frac{1}{3}-2 \frac{1}{2}$ $B=1 \frac{3}{4}+1 \frac{1}{6}$

Hints

- Calculate $A$ and $B$ separately first. - To compare fractional parts, rewrite them with a common denominator. - When subtracting mixed numbers, check whether you need to regroup one whole.

Solution

1. Calculate $A$: $5 \frac{1}{3}-2 \frac{1}{2}=5 \frac{2}{6}-2 \frac{3}{6}=4 \frac{8}{6}-2 \frac{3}{6}=2 \frac{5}{6}$. 2. Calculate $B$: $1 \frac{3}{4}+1 \frac{1}{6}=1 \frac{9}{12}+1 \frac{2}{12}=2 \frac{11}{12}$. 3. Rewrite $A$ with denominator $12$: $2 \frac{5}{6}=2 \frac{10}{12}$. Since $2 \frac{10}{12}<2 \frac{11}{12}$, $A<B$.

Answer

$A<B$, because $A=2 \frac{5}{6}=2 \frac{10}{12}$ and $B=2 \frac{11}{12}$.

Add and subtract fractions and decimals 71

Evaluate the expression using the correct order of operations. Write the answer as a mixed number. $7 \frac{1}{5}-\left(2 \frac{1}{2}+1 \frac{3}{4}\right)+0.8$

Hints

- Evaluate the expression inside the parentheses first. - Before changing the order of terms, rewrite subtraction in a form that can be regrouped using addition properties. - You can rewrite the decimal as a fraction before calculating.

Solution

1. Evaluate the parentheses: $2 \frac{1}{2}+1 \frac{3}{4}=2 \frac{2}{4}+1 \frac{3}{4}=4 \frac{1}{4}$. 2. Rewrite $0.8$ as $\frac{4}{5}$, and rewrite subtraction as addition of the opposite: $7 \frac{1}{5}+\left(-4 \frac{1}{4}\right)+\frac{4}{5}$. 3. Use the commutative and associative properties of addition to group the fifths: $\left(7 \frac{1}{5}+\frac{4}{5}\right)+\left(-4 \frac{1}{4}\right)=8-4 \frac{1}{4}$. 4. Subtract: $8-4 \frac{1}{4}=3 \frac{3}{4}$.

Answer

$3 \frac{3}{4}$

Add and subtract fractions and decimals 72

Amani writes $6 \frac{1}{4}-2 \frac{2}{3}=(6-2)+\left(\frac{2}{3}-\frac{1}{4}\right)=4+\left(\frac{8}{12}-\frac{3}{12}\right)=4 \frac{5}{12}$. Explain the error, then calculate the correct result.

Hints

- Check whether the fractional subtraction in the written work keeps the same order as the original expression. - If the fractional part of the first mixed number is smaller, consider how an equivalent regrouping could help. - Verify your corrected result in the original subtraction.

Solution

1. Amani reversed the order of the fractional parts. In the original subtraction, the fractional difference is $\frac{1}{4}-\frac{2}{3}$, not $\frac{2}{3}-\frac{1}{4}$. 2. Rewrite with denominator $12$: $6 \frac{3}{12}-2 \frac{8}{12}$. 3. Regroup one whole: $5 \frac{15}{12}-2 \frac{8}{12}=3 \frac{7}{12}$.

Answer

Amani incorrectly reversed $\frac{1}{4}-\frac{2}{3}$ to $\frac{2}{3}-\frac{1}{4}$. The correct result is $3 \frac{7}{12}$.

Add and subtract fractions and decimals 73

Simplify and remove the parentheses, then write an equivalent expression whose grouping makes the final calculation efficient. Evaluate the expression. $\left(6\frac{4}{15} + \frac{18}{24}\right) - \left(1\frac{4}{15} - \frac{1}{4}\right)$

Hints

- Reduce any fraction that is not in simplest form. - When a minus sign is in front of parentheses, track the sign of every term inside. - Look for an equivalent grouping that creates whole numbers.

Solution

1. Simplify $\frac{18}{24}=\frac{3}{4}$ and distribute the subtraction: $6\frac{4}{15}+\frac{3}{4}-1\frac{4}{15}+\frac{1}{4}$. 2. Regroup: $\left(6\frac{4}{15}-1\frac{4}{15}\right)+\left(\frac{3}{4}+\frac{1}{4}\right)$. 3. The groups equal $5$ and $1$, so the value is $6$.

Answer

After simplifying and removing parentheses: $6\frac{4}{15}+\frac{3}{4}-1\frac{4}{15}+\frac{1}{4}$. One efficient regrouping is $\left(6\frac{4}{15}-1\frac{4}{15}\right)+\left(\frac{3}{4}+\frac{1}{4}\right)=6$.

Add and subtract fractions and decimals 74

Rewrite the subtraction as addition of opposites. Then show an equivalent rearranged and regrouped sum that makes the calculation efficient, evaluate it, and name the properties that justify changing the order and grouping. $\left(15\frac{5}{6} + 3\frac{1}{4}\right) - \left(2\frac{5}{6} + 1\frac{1}{2} + 1\frac{3}{4}\right)$

Hints

- Rewrite each subtraction as addition of the opposite before changing the order. - Keep each sign attached to its term when rearranging. - Look for groups whose fractional parts cancel or combine to whole numbers.

Solution

1. Rewrite subtraction as addition of opposites: $15\frac{5}{6}+3\frac{1}{4}+\left(-2\frac{5}{6}\right)+\left(-1\frac{1}{2}\right)+\left(-1\frac{3}{4}\right)$. 2. Use the commutative and associative properties of addition to regroup: $\left(15\frac{5}{6}-2\frac{5}{6}\right)+\left(3\frac{1}{4}-1\frac{1}{2}-1\frac{3}{4}\right)$. 3. The first group is $13$, and the second group is $0$. 4. Therefore, the value is $13$.

Answer

One valid regrouping is $\left(15\frac{5}{6}-2\frac{5}{6}\right)+\left(3\frac{1}{4}-1\frac{1}{2}-1\frac{3}{4}\right)=13+0=13$. The commutative and associative properties of addition justify the rearrangement and regrouping after subtraction is rewritten as addition of opposites.

Add and subtract fractions and decimals 75

Evaluate each expression. Before calculating, state which representation you will use for each nonmatching pair and show the first rewritten expression. a) $\left(\frac{1}{4} + 0.5\right) - 1$ b) $2.75 - \left(1\frac{1}{2} + 0.25\right)$ c) $\frac{1}{3} + 0.2 + \frac{1}{6}$

Hints

- Inspect whether a fraction terminates before deciding to convert it. - A useful representation should reduce, not increase, the amount of computation. - Your answer must show the rewritten expression that reflects your choice.

Solution

1. a) Use decimals: $(0.25+0.5)-1=-0.25$. 2. b) Use decimals: $2.75-(1.5+0.25)=1$. 3. c) Keep the thirds and sixths as fractions first: $\left(\frac{1}{3}+\frac{1}{6}\right)+0.2=0.5+0.2=0.7$.

Answer

a) Decimals: $(0.25+0.5)-1=-0.25$ b) Decimals: $2.75-(1.5+0.25)=1$ c) Fractions first: $\left(\frac{1}{3}+\frac{1}{6}\right)+0.2=0.7$

Add and subtract fractions and decimals 76

Lucas and Mia evaluate $0.125 + \frac{3}{4} + 0.25 + \frac{1}{8}$ in different ways. Lucas converts the fractions to decimals. Mia converts the decimals to fractions and rearranges the addends. Carry out both methods. Which method seems more efficient here? Explain.

Hints

- Convert only values whose equivalents you know exactly. - Look for pairs that combine to a whole number. - Rearranging addends does not change their sum.

Solution

1. Lucas’s method: $\frac{3}{4} = 0.75$ and $\frac{1}{8} = 0.125$. Then $0.125 + 0.75 + 0.25 + 0.125 = 1.25$. 2. Mia’s method: $0.125 = \frac{1}{8}$ and $0.25 = \frac{1}{4}$. Then $\frac{1}{8} + \frac{3}{4} + \frac{1}{4} + \frac{1}{8}$. 3. Regroup: $\left(\frac{1}{8} + \frac{1}{8}\right) + \left(\frac{3}{4} + \frac{1}{4}\right) = \frac{1}{4} + 1 = 1.25$. 4. Both methods are efficient. Mia’s grouping immediately creates a whole number from the fourths.

Answer

Both methods give $1.25$. Mia’s method is especially efficient because $\frac{3}{4} + \frac{1}{4} = 1$.

Add and subtract fractions and decimals 77

Evaluate the expression. Briefly explain why fractions or decimals are the better representation. $1\frac{1}{3} + 0.25 - \frac{7}{12}$

Hints

- Consider the decimal representation of $\frac{1}{3}$. - Convert the terminating decimal to a fraction. - Rewrite the fractions with a common denominator.

Solution

1. Fractions are more efficient because $\frac{1}{3}$ has a repeating decimal representation. 2. Convert $0.25 = \frac{1}{4}$ and $1\frac{1}{3} = \frac{4}{3}$. 3. Use denominator $12$: $\frac{16}{12} + \frac{3}{12} - \frac{7}{12} = \frac{12}{12} = 1$.

Answer

$1$. Fractions are preferable because $\frac{1}{3}$ is a repeating decimal.

Add and subtract fractions and decimals 78

Evaluate the expression. Follow the grouping symbols and choose an efficient representation for the numbers. $12.8 - \left(3\frac{1}{8} + 4.025\right) + 2\frac{1}{5}$

Hints

- Evaluate the grouped sum first. - The fractions $\frac{1}{8}$ and $\frac{1}{5}$ have terminating decimal forms. - Align decimal places carefully when adding and subtracting.

Solution

1. Convert the mixed numbers to decimals: $3\frac{1}{8} = 3.125$ and $2\frac{1}{5} = 2.2$. 2. Evaluate inside the parentheses: $3.125 + 4.025 = 7.15$. 3. Substitute and evaluate: $12.8 - 7.15 + 2.2 = 5.65 + 2.2 = 7.85$.

Answer

$7.85$

Add and subtract fractions and decimals 79

Evaluate the expression efficiently. Use an estimate to check the result. $\left(12\frac{3}{8} + 4.55\right) - \left(2.375 - 1\frac{9}{20}\right)$

Hints

- Convert the fractional parts to terminating decimals. - A subtraction sign before parentheses changes both signs inside. - Look for decimal pairs that combine to whole numbers.

Solution

1. Estimate: $(12 + 5) - (2 - 1) = 16$. 2. Convert the mixed numbers: $12\frac{3}{8} = 12.375$ and $1\frac{9}{20} = 1.45$. 3. Remove the parentheses and regroup: $12.375 + 4.55 - 2.375 + 1.45$. 4. Evaluate: $(12.375 - 2.375) + (4.55 + 1.45) = 10 + 6 = 16$. 5. The exact result agrees with the estimate.

Answer

Estimate: $16$ Exact value: $16$

Add and subtract fractions and decimals 80

Estimate first, then calculate the exact value. Decide whether fractions or decimals are more useful for the exact calculation. $4\frac{2}{3} - \left(1.5 + \frac{1}{6}\right) + 0.2$

Hints

- Use nearby decimals for a quick estimate. - For an exact result, avoid replacing repeating decimals with rounded values. - Convert the terminating decimals to fractions and use common denominators.

Solution

1. Estimate using nearby convenient values: $4.7 - 1.7 + 0.2 \approx 3.2$. 2. Fractions are useful because $\frac{2}{3}$ and $\frac{1}{6}$ have repeating decimal representations. 3. Convert the decimals: $1.5 = \frac{3}{2}$ and $0.2 = \frac{1}{5}$. 4. Evaluate the parentheses: $\frac{3}{2} + \frac{1}{6} = \frac{10}{6} = \frac{5}{3}$. 5. Calculate: $\frac{14}{3} - \frac{5}{3} + \frac{1}{5} = 3 + \frac{1}{5} = 3.2$.

Answer

Estimate: about $3.2$ Exact value: $3\frac{1}{5}$, or $3.2$

Add and subtract fractions and decimals 81

Consider $\frac{3}{4}+\frac{2}{3}-\frac{1}{2}$. a) Give a benchmark estimate that decides whether the result is greater than or less than $1$. Show the benchmark values you used. b) Calculate the exact value as a fraction. c) State whether the exact result is consistent with your estimate.

Hints

- Use simple benchmark decimal values rather than doing the exact fraction computation first. - For the exact calculation, choose a common denominator for $2$, $3$, and $4$. - Compare the exact result with the claim made by your estimate.

Solution

1. A benchmark estimate is $0.75+0.67-0.5\approx0.92$, so the result should be less than $1$. 2. With denominator $12$, the exact value is $\frac{9}{12}+\frac{8}{12}-\frac{6}{12}=\frac{11}{12}$. 3. Since $\frac{11}{12}<1$, the estimate is consistent.

Answer

a) For example, $0.75+0.67-0.5\approx0.92<1$. b) $\frac{11}{12}$ c) Yes. The exact value is less than $1$, as predicted.

Add and subtract fractions and decimals 82

Check the following work for an error: $\frac{4}{5}-\left(\frac{1}{2}-\frac{1}{10}\right)=\frac{4}{5}-\frac{1}{2}-\frac{1}{10}=\frac{8}{10}-\frac{5}{10}-\frac{1}{10}=\frac{1}{5}$. Explain whether the work is correct. If it is not, identify the error and calculate the correct result.

Hints

- Try evaluating the expression inside the parentheses before doing anything else. - What quantity is being subtracted from $\frac{4}{5}$? - Check each equality in the student's chain.

Solution

1. The work is incorrect. Subtracting the entire quantity $\frac{1}{2}-\frac{1}{10}$ is not the same as subtracting both terms separately. 2. Evaluate the parentheses first: $\frac{1}{2}-\frac{1}{10}=\frac{5}{10}-\frac{1}{10}=\frac{4}{10}$. 3. Then $\frac{4}{5}-\frac{4}{10}=\frac{8}{10}-\frac{4}{10}=\frac{4}{10}=\frac{2}{5}$.

Answer

The work is incorrect because the subtraction of the quantity in parentheses was handled incorrectly. The correct result is $\frac{2}{5}$.

Add and subtract fractions and decimals 83

Evaluate the expression: $\left(\frac{2}{3}-1.5\right)+\left(-\frac{5}{6}+0.25\right)$

Hints

- Decide whether fractions or decimals will be easier to use consistently. - Fractions need common denominators before addition or subtraction. - Evaluate the expressions inside parentheses first.

Solution

1. First parentheses: $\frac{2}{3}-1.5=\frac{2}{3}-\frac{3}{2}=\frac{4}{6}-\frac{9}{6}=-\frac{5}{6}$. 2. Second parentheses: $-\frac{5}{6}+0.25=-\frac{5}{6}+\frac{1}{4}=-\frac{10}{12}+\frac{3}{12}=-\frac{7}{12}$. 3. Add: $-\frac{5}{6}-\frac{7}{12}=-\frac{10}{12}-\frac{7}{12}=-\frac{17}{12}=-1 \frac{5}{12}$.

Answer

$-\frac{17}{12}$, or $-1 \frac{5}{12}$

Add and subtract fractions and decimals 84

Two students evaluate $4.8+2\frac{1}{4}+1.2+\frac{3}{4}$. - Jordan converts the fractions to decimals and adds from left to right. - Maya reorders and groups the addends as $(4.8+1.2)+(2\frac{1}{4}+\frac{3}{4})$. a) Evaluate the expression both ways. b) Explain which method is easier to do mentally.

Hints

- Complete both methods before comparing them. - Look for addends that combine to whole numbers. - Consider how reordering and regrouping affect the value of a sum.

Solution

1. Jordan’s method: $2\frac{1}{4}=2.25$ and $\frac{3}{4}=0.75$. Then $4.8+2.25=7.05$, $7.05+1.2=8.25$, and $8.25+0.75=9$. 2. Maya’s method: $4.8+1.2=6$ and $2\frac{1}{4}+\frac{3}{4}=3$. Then $6+3=9$. 3. Maya’s method is easier mentally because the commutative and associative properties create whole-number sums.

Answer

a) Both methods give $9$. b) Maya’s method is generally easier mentally because the grouped addends make $6$ and $3$.

Add and subtract fractions and decimals 85

Decide whether each statement is true or false. Justify your decision by evaluating both sides of the equation. a) $-12.5+7.5=-(12.5-7.5)$ b) $\frac{1}{2}-\left(\frac{1}{4}+\frac{1}{8}\right)=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}$

Hints

- Evaluate each side separately. - Apply parentheses first. - What happens to every term inside parentheses when a negative sign is distributed?

Solution

1. For a), the left side is $-12.5+7.5=-5$. The right side is $-(12.5-7.5)=-5$. The statement is true. 2. For b), the left side is $\frac{1}{2}-\left(\frac{1}{4}+\frac{1}{8}\right)=\frac{4}{8}-\frac{3}{8}=\frac{1}{8}$. 3. The right side is $\frac{1}{2}-\frac{1}{4}+\frac{1}{8}=\frac{4}{8}-\frac{2}{8}+\frac{1}{8}=\frac{3}{8}$. Since the two sides differ, the statement is false.

Answer

a) True; both sides equal $-5$. b) False; the left side is $\frac{1}{8}$ and the right side is $\frac{3}{8}$.

Add and subtract fractions and decimals 86

Evaluate the expression. Write the result as a fraction in simplest form or as a mixed number: $10-\left(3 \frac{1}{4}+2 \frac{5}{6}\right)-1 \frac{1}{12}$

Hints

- Evaluate the parentheses first. - When subtracting a mixed number from a whole number, you may need to regroup one whole. - Simplify the fractional part of the final answer. - After the parentheses are resolved, keep the remaining subtractions in their original order.

Solution

1. Evaluate the parentheses using denominator $12$: $3 \frac{1}{4}+2 \frac{5}{6}=3 \frac{3}{12}+2 \frac{10}{12}=5 \frac{13}{12}=6 \frac{1}{12}$. 2. Subtract from $10$: $10-6 \frac{1}{12}=3 \frac{11}{12}$. 3. Subtract the last mixed number: $3 \frac{11}{12}-1 \frac{1}{12}=2 \frac{10}{12}=2 \frac{5}{6}$.

Answer

$2 \frac{5}{6}$

Add and subtract fractions and decimals 87

Evaluate the expression. Write the answer as a fraction in simplest form or as a mixed number: $2 \frac{1}{4}-\frac{5}{6}-\frac{3}{8}$

Hints

- Rewrite the mixed number as an improper fraction. - Find one common denominator for all three fractions. - Perform the subtractions from left to right, then convert the result to a mixed number if needed. - Estimate whether the final value should be greater than $1$ as a reasonableness check.

Solution

1. Rewrite $2 \frac{1}{4}$ as $\frac{9}{4}$. 2. Use denominator $24$: $\frac{9}{4}=\frac{54}{24}$, $\frac{5}{6}=\frac{20}{24}$, and $\frac{3}{8}=\frac{9}{24}$. 3. Subtract from left to right: $\frac{54}{24}-\frac{20}{24}-\frac{9}{24}=\frac{25}{24}=1 \frac{1}{24}$.

Answer

$1 \frac{1}{24}$

Add and subtract fractions and decimals 88

Find the missing value of $a$, $b$, or $a+b$ in each part. a) $a=-4.5$, $b=2.1$, $a+b=\square$ b) $a=\square$, $b=-7.8$, $a+b=-3.2$ c) $a=\frac{3}{4}$, $b=\square$, $a+b=-\frac{1}{8}$

Hints

- When the sum and one addend are known, subtract the known addend to find the other one. - Subtraction is the inverse of addition. - For the fractions, first write them with a common denominator.

Solution

1. In a), add the two values: $-4.5+2.1=-2.4$. 2. In b), subtract the known addend from the sum: $a=-3.2-(-7.8)=-3.2+7.8=4.6$. 3. In c), subtract the known addend from the sum: $b=-\frac{1}{8}-\frac{3}{4}=-\frac{1}{8}-\frac{6}{8}=-\frac{7}{8}$.

Answer

a) $a+b=-2.4$ b) $a=4.6$ c) $b=-\frac{7}{8}$

Add and subtract fractions and decimals 89

Complete the subtraction table and find $x$ and $y$. Each entry is the value at the left of its row minus the value at the top of its column. <table> <tr><td>$-$</td><td>$1.2$</td><td>$y$</td><td>$-3.5$</td></tr> <tr><td>$4.5$</td><td>$3.3$</td><td>$7.2$</td><td></td></tr> <tr><td>$-2.8$</td><td></td><td>$-0.1$</td><td>$0.7$</td></tr> <tr><td>$x$</td><td>$-5.5$</td><td></td><td>$-0.8$</td></tr> </table>

Hints

- First determine how each table entry is calculated. - Use a simple equation to find a missing row or column heading. - Pay close attention when subtracting a negative number.

Solution

1. From the row headed by $4.5$, $4.5-y=7.2$. Therefore, $y=4.5-7.2=-2.7$. 2. From the row headed by $x$, $x-1.2=-5.5$. Therefore, $x=-5.5+1.2=-4.3$. 3. Fill the remaining cells: $4.5-(-3.5)=8.0$, $-2.8-1.2=-4.0$, and $-4.3-(-2.7)=-1.6$.

Answer

$x=-4.3$; $y=-2.7$ Completed table: <table> <tr><td>$-$</td><td>$1.2$</td><td>$-2.7$</td><td>$-3.5$</td></tr> <tr><td>$4.5$</td><td>$3.3$</td><td>$7.2$</td><td>$8.0$</td></tr> <tr><td>$-2.8$</td><td>$-4.0$</td><td>$-0.1$</td><td>$0.7$</td></tr> <tr><td>$-4.3$</td><td>$-5.5$</td><td>$-1.6$</td><td>$-0.8$</td></tr> </table>

Add and subtract fractions and decimals 90

For each pair, decide which expression has the greater value. Justify the choice using signs, direction on a number line, or relative magnitude; do not give the exact final values. a) $(-15)+(-20)$ or $(-15)-(-20)$ b) $0.4-0.6$ or $-0.4+0.6$ c) $-\frac{1}{8}-\frac{1}{4}$ or $-\frac{1}{8}+\frac{1}{4}$

Hints

- Look for a sign comparison that settles the question before exact calculation. - Rewrite subtraction of a negative as addition. - For the fraction pair, think about movement from the same starting point.

Solution

1. a) The first expression is a sum of negatives. The second rewrites as $-15+20$, which is positive, so the second is greater. 2. b) Subtracting a larger positive from a smaller gives a negative result, while adding $0.6$ to $-0.4$ gives a positive result, so the second is greater. 3. c) From $-\frac{1}{8}$, subtracting $\frac{1}{4}$ moves left while adding it moves right, so the second is greater.

Answer

a) $(-15)-(-20)$ — it is positive while the other expression is negative b) $-0.4+0.6$ — it is positive while $0.4-0.6$ is negative c) $-\frac{1}{8}+\frac{1}{4}$ — adding the positive fraction moves right while subtracting it moves left

Add and subtract fractions and decimals 91

Choose exactly three different numbers from the list whose sum is $0$. Find two different solutions. $0.7,\ -0.3,\ \frac{1}{5},\ -0.4,\ 0.1,\ -0.5$

Hints

- Convert the fraction to a decimal. - For each positive number, look for two numbers whose sum is its opposite. - Use each selected number only once in a solution.

Solution

1. Write $\frac{1}{5}=0.2$. 2. One solution is $0.7+(-0.3)+(-0.4)=0$. 3. A second solution is $0.2+0.1+(-0.3)=0$.

Answer

For example: $0.7, -0.3, -0.4$ $\frac{1}{5}, 0.1, -0.3$

Add and subtract fractions and decimals 92

Choose exactly four numbers from the list. Place a plus or minus sign between consecutive chosen numbers so that the result is $0.5$. $1.2,\ -0.5,\ \frac{1}{4},\ -0.8,\ 0.1,\ -0.2$

Hints

- Convert $\frac{1}{4}$ to a decimal if you use it. - Estimate combinations before calculating exactly. - Remember that subtracting a negative number adds its opposite.

Solution

1. One valid choice is $1.2, -0.8, 0.1, -0.2$. 2. Insert signs: $1.2+(-0.8)-0.1-(-0.2)$. 3. Evaluate: $1.2-0.8-0.1+0.2=0.5$.

Answer

One possible solution is $1.2+(-0.8)-0.1-(-0.2)=0.5$.

Add and subtract fractions and decimals 93

Two students evaluate $-0.5 + \frac{3}{4} - 0.25 + \frac{1}{2}$ in different ways. Mia works from left to right. Lucas groups values that combine easily. a) Evaluate the expression using Mia’s method. b) Show an efficient grouping Lucas could use. c) Compare the original expression with $-0.5 + \left(\frac{3}{4} - 0.25\right) + \frac{1}{2}$. Do the parentheses change the value? Explain.

Hints

- Convert a fraction to a decimal only when it helps with a particular step. - Look for additive inverses or pairs that make a simple decimal. - Parentheses preceded by a plus sign can be removed without changing the signs inside.

Solution

1. Mia’s method: $-0.5 + 0.75 = 0.25$, then $0.25 - 0.25 = 0$, and finally $0 + 0.5 = 0.5$. 2. Lucas can group $-0.5$ with $\frac{1}{2}$, and $\frac{3}{4}$ with $-0.25$: $\left(-0.5 + \frac{1}{2}\right) + \left(\frac{3}{4} - 0.25\right)$. 3. The groups equal $0$ and $0.5$, so the result is $0.5$. 4. In c), the parentheses are preceded by addition. Removing them does not change any signs, so the value remains $0.5$.

Answer

a) $0.5$ b) $\left(-0.5 + \frac{1}{2}\right) + \left(\frac{3}{4} - 0.25\right) = 0 + 0.5 = 0.5$ c) No. The parentheses are preceded by addition, so removing them does not change the value.

Add and subtract fractions and decimals 94

For each expression, write an equivalent grouping that combines compatible rational numbers before evaluating. a) $-\frac{3}{7} + \frac{5}{8} - \frac{4}{7} + \frac{3}{8}$ b) $1\frac{2}{5} - 3\frac{1}{4} + \frac{3}{5} + \frac{1}{4}$ c) $\frac{7}{10} - 0.45 + \frac{3}{10} - 0.55$

Hints

- Identify which terms can combine without introducing new denominators or representations. - Treat subtraction as addition of a signed term before regrouping. - Show the grouping that makes each calculation short.

Solution

1. a) $\left(-\frac{3}{7}-\frac{4}{7}\right)+\left(\frac{5}{8}+\frac{3}{8}\right)=-1+1=0$. 2. b) $\left(1\frac{2}{5}+\frac{3}{5}\right)+\left(-3\frac{1}{4}+\frac{1}{4}\right)=2-3=-1$. 3. c) $\left(\frac{7}{10}+\frac{3}{10}\right)+(-0.45-0.55)=1-1=0$.

Answer

a) $\left(-\frac{3}{7}-\frac{4}{7}\right)+\left(\frac{5}{8}+\frac{3}{8}\right)=0$ b) $\left(1\frac{2}{5}+\frac{3}{5}\right)+\left(-3\frac{1}{4}+\frac{1}{4}\right)=-1$ c) $\left(\frac{7}{10}+\frac{3}{10}\right)+(-0.45-0.55)=0$

Add and subtract fractions and decimals 95

Lina and Tim evaluate $12.5 - 8.3 + 7.5 - 1.7$ in different ways. Lina groups the positive terms and the amounts being subtracted: $(12.5 + 7.5) - (8.3 + 1.7)$. Tim groups consecutive pairs: $(12.5 - 8.3) + (7.5 - 1.7)$. a) Evaluate the expression using both methods. b) Explain why Lina’s method is especially efficient here.

Hints

- Carry out each proposed grouping separately. - Compare the intermediate values produced by the two methods. - An efficient mental method often creates whole numbers.

Solution

1. Lina’s method: $12.5 + 7.5 = 20$ and $8.3 + 1.7 = 10$, so $20 - 10 = 10$. 2. Tim’s method: $12.5 - 8.3 = 4.2$ and $7.5 - 1.7 = 5.8$, so $4.2 + 5.8 = 10$. 3. Lina’s grouping creates whole-number intermediate results, making the mental calculation simpler.

Answer

a) Both methods give $10$. b) Lina’s method creates the whole-number intermediate results $20$ and $10$.

Add and subtract fractions and decimals 96

Evaluate each expression using properties of rational-number addition. For each part, show the reordered or regrouped expression that makes the calculation efficient. a) $\frac{5}{9} - \frac{1}{3} + \frac{4}{9}$ b) $\frac{3}{4} - 2.5 - \frac{7}{4} + 1.5$ c) $-\frac{2}{7} + \frac{5}{6} + \frac{2}{7} - \frac{1}{6}$

Hints

- Search for equal denominators, additive inverses, or decimal pairs that make whole numbers. - Keep signs attached to terms while changing order. - The answer must show how the properties were used.

Solution

1. a) $\left(\frac{5}{9}+\frac{4}{9}\right)-\frac{1}{3}=1-\frac{1}{3}=\frac{2}{3}$. 2. b) $\left(\frac{3}{4}-\frac{7}{4}\right)+(-2.5+1.5)=-1-1=-2$. 3. c) $\left(-\frac{2}{7}+\frac{2}{7}\right)+\left(\frac{5}{6}-\frac{1}{6}\right)=\frac{2}{3}$.

Answer

a) $\left(\frac{5}{9}+\frac{4}{9}\right)-\frac{1}{3}=\frac{2}{3}$ b) $\left(\frac{3}{4}-\frac{7}{4}\right)+(-2.5+1.5)=-2$ c) $\left(-\frac{2}{7}+\frac{2}{7}\right)+\left(\frac{5}{6}-\frac{1}{6}\right)=\frac{2}{3}$

Add and subtract fractions and decimals 97

Rewrite each expression by removing parentheses or regrouping strategically, then evaluate. a) $-3.25-(1.75-8)$ b) $2\frac{1}{5}-\left(4.8+\frac{1}{5}\right)+0.8$ c) $\left(\frac{5}{8}+1.4\right)-\left(\frac{1}{8}-0.6\right)$

Hints

- A subtraction sign before parentheses changes the sign of every term inside. - Convert a fraction to a decimal when that creates easy pairs. - Regroup terms to form whole numbers.

Solution

1. For a), remove the parentheses: $-3.25-1.75+8=-5+8=3$. 2. For b), remove the parentheses and regroup: $2\frac{1}{5}-4.8-\frac{1}{5}+0.8=(2.2-0.2)+(-4.8+0.8)=2-4=-2$. 3. For c), remove the parentheses: $\frac{5}{8}+1.4-\frac{1}{8}+0.6$. Regroup: $\left(\frac{5}{8}-\frac{1}{8}\right)+(1.4+0.6)=0.5+2=2.5$.

Answer

a) $3$ b) $-2$ c) $2.5$

Add and subtract fractions and decimals 98

Evaluate each expression using properties of operations. For each part, show the cancellation or regrouping that makes the calculation short. a) $17.4 - (5.9 + 17.4) + 0.9$ b) $-\frac{2}{5} + (3.7 + 0.4) - 2.7$

Hints

- First rewrite any subtraction of a grouped sum carefully. - Look for additive inverses or pairs differing by a whole number. - Show the cancellation or regrouping, not only the result.

Solution

1. a) Remove the parentheses: $17.4-5.9-17.4+0.9$. Regroup as $(17.4-17.4)+(-5.9+0.9)=-5$. 2. b) Convert $-\frac{2}{5}=-0.4$. Then regroup as $(-0.4+0.4)+(3.7-2.7)=1$.

Answer

a) $(17.4-17.4)+(-5.9+0.9)=-5$ b) $(-0.4+0.4)+(3.7-2.7)=1$

Add and subtract fractions and decimals 99

Rewrite and evaluate each expression. Pay attention to the mixture of fractions and decimals. a) $\frac{1}{8}-0.25+0.875-1.75$ b) $\left(\frac{5}{6}+1.2\right)-\left(\frac{1}{6}-0.8\right)$

Hints

- Choose fraction or decimal forms that make useful pairs. - Combine fractions with the same denominator directly. - Simplify the final fraction.

Solution

1. For a), write $\frac{1}{8}=0.125$. Regroup: $(0.125+0.875)-(0.25+1.75)=1-2=-1$. 2. For b), remove the parentheses: $\frac{5}{6}+1.2-\frac{1}{6}+0.8$. 3. Regroup: $\left(\frac{5}{6}-\frac{1}{6}\right)+(1.2+0.8)=\frac{4}{6}+2=\frac{2}{3}+2=2\frac{2}{3}$.

Answer

a) $-1$ b) $2\frac{2}{3}$, or $\frac{8}{3}$

Add and subtract fractions and decimals 100

A weather station starts the day at $45.5\,{}^\circ\text{F}$. The temperature then changes by $-3.4\,{}^\circ\text{F}$, $+8.8\,{}^\circ\text{F}$, $-2.2\,{}^\circ\text{F}$, $-5.7\,{}^\circ\text{F}$, and $+6.1\,{}^\circ\text{F}$, in that order. a) What is the final temperature? b) What is the highest temperature reached during the day? c) What is the lowest temperature reached during the day?

Hints

- Make a running list of the temperature after each change. - Include the starting temperature when finding the highest and lowest values. - Pay attention to the signs of the changes.

Solution

1. Track the temperatures in order: $45.5$, $42.1$, $50.9$, $48.7$, $43.0$, and $49.1$, all in degrees Fahrenheit. 2. For a), the final temperature is $49.1\,{}^\circ\text{F}$. 3. For b), the highest value is $50.9\,{}^\circ\text{F}$. 4. For c), the lowest value is $42.1\,{}^\circ\text{F}$.

Answer

a) $49.1\,{}^\circ\text{F}$ b) $50.9\,{}^\circ\text{F}$ c) $42.1\,{}^\circ\text{F}$

Add and subtract fractions and decimals 101

A rain barrel holds $50$ gallons. It starts the week with $30.75$ gallons of water. During the week, the amount changes in this order: - Rain: $+8.50$ gallons - Watering: $-13.20$ gallons - Rain: $+3.85$ gallons - Watering: $-22.40$ gallons a) How much water is in the barrel at the end of the week? b) When the barrel reached its highest amount during the week, how many gallons of empty capacity remained?

Hints

- Update the amount after each addition or removal. - For part b), identify the greatest amount reached during the week. - Subtract that maximum amount from the barrel's total capacity.

Solution

1. Track the amounts: $30.75$, $39.25$, $26.05$, $29.90$, and $7.50$ gallons. 2. For a), the final amount is $7.50$ gallons. 3. The highest amount is $39.25$ gallons. 4. For b), the empty capacity at that point is $50.00-39.25=10.75$ gallons.

Answer

a) $7.50$ gallons b) $10.75$ gallons of empty capacity

Add and subtract fractions and decimals 102

A checking account starts with a balance of $\$250.00$. These transactions occur in order: - Withdrawal: $\$315.50$ - Deposit: $\$120.00$ - Withdrawal: $\$85.25$ a) What is the lowest balance reached during the sequence? b) What is the balance after the final transaction? c) How much must be deposited after the final transaction to bring the balance to exactly $\$100.00$?

Hints

- Update the account balance after each transaction in order. - A withdrawal decreases the balance, and the balance can become negative. - To move from a negative final balance to a positive target, find the difference between the two signed values.

Solution

1. Track the balances: $\$250.00$, $-\$65.50$, $\$54.50$, and $-\$30.75$. 2. For a), the lowest balance is $-\$65.50$. 3. For b), the final balance is $-\$30.75$. 4. For c), the required deposit is $100.00-(-30.75)=130.75$, so $\$130.75$ must be deposited.

Answer

a) $-\$65.50$ b) $-\$30.75$ c) $\$130.75$

Add and subtract fractions and decimals 103

Find the rational number that belongs in each blank. a) $\frac{2}{3}+\square=-\frac{1}{6}$ b) $\square-\frac{3}{5}=-1$ c) $-\frac{1}{2}-\square=\frac{1}{8}$

Hints

- Treat each blank as an unknown value and use the inverse operation. - Use the same inverse-operation reasoning you would use with positive numbers. - Rewrite rational numbers with common denominators before adding or subtracting. - In part c), pay close attention to the subtraction sign before the blank.

Solution

1. For a), the missing value is $-\frac{1}{6}-\frac{2}{3}=-\frac{1}{6}-\frac{4}{6}=-\frac{5}{6}$. 2. For b), the missing value is $-1+\frac{3}{5}=-\frac{5}{5}+\frac{3}{5}=-\frac{2}{5}$. 3. For c), the missing value is $-\frac{1}{2}-\frac{1}{8}=-\frac{4}{8}-\frac{1}{8}=-\frac{5}{8}$.

Answer

a) $-\frac{5}{6}$ b) $-\frac{2}{5}$ c) $-\frac{5}{8}$

Add and subtract fractions and decimals 104

Write each expression and find its value. a) Decrease $-18.5$ by the difference of $12.4$ and $15.9$. b) Add the opposite of $7.2$ to the sum of $-14.6$ and $21.3$. c) Subtract three times $4.5$ from the difference of $-10$ and $-25$.

Hints

- In a), decide which subtraction must stay inside parentheses. - In b), what is the signed number opposite to the one given? - In c), find the difference first, then subtract the stated product.

Solution

1. For a), $-18.5-(12.4-15.9)=-18.5-(-3.5)=-15$. 2. For b), $(-14.6+21.3)+(-7.2)=6.7-7.2=-0.5$. 3. For c), $(-10-(-25))-3\cdot4.5=15-13.5=1.5$.

Answer

a) $-18.5-(12.4-15.9)=-15$ b) $(-14.6+21.3)+(-7.2)=-0.5$ c) $(-10-(-25))-3\cdot4.5=1.5$

Add and subtract fractions and decimals 105

A weather station in Colorado records these temperature changes from one reading to the next. At midnight, the temperature is $27.5\,{}^\circ\text{F}$. <table> <tr><td>6:00 a.m.</td><td>$+7.2\,{}^\circ\text{F}$</td></tr> <tr><td>9:00 a.m.</td><td>$+6.8\,{}^\circ\text{F}$</td></tr> <tr><td>12:00 p.m.</td><td>$+4.4\,{}^\circ\text{F}$</td></tr> <tr><td>3:00 p.m.</td><td>$-2.5\,{}^\circ\text{F}$</td></tr> <tr><td>6:00 p.m.</td><td>$-8.6\,{}^\circ\text{F}$</td></tr> <tr><td>9:00 p.m.</td><td>$-4.3\,{}^\circ\text{F}$</td></tr> </table> a) Write one expression for the temperature at 9:00 p.m. and calculate it. b) Find the highest and lowest temperatures at the listed reading times, including midnight.

Hints

- Start with the midnight temperature and apply each signed change in order. - Keep a list of the temperature after every change. - Include the starting value when finding the maximum and minimum. - A positive change raises the temperature; a negative change lowers it.

Solution

1. For a), use $27.5+7.2+6.8+4.4-2.5-8.6-4.3=30.5$. The temperature at 9:00 p.m. is $30.5\,{}^\circ\text{F}$. 2. The successive temperatures are $27.5$, $34.7$, $41.5$, $45.9$, $43.4$, $34.8$, and $30.5$ degrees Fahrenheit. 3. For b), the highest temperature is $45.9\,{}^\circ\text{F}$, and the lowest is $27.5\,{}^\circ\text{F}$.

Answer

a) $27.5+7.2+6.8+4.4-2.5-8.6-4.3=30.5$, so the 9:00 p.m. temperature is $30.5\,{}^\circ\text{F}$. b) Highest: $45.9\,{}^\circ\text{F}$; lowest: $27.5\,{}^\circ\text{F}$

Add and subtract fractions and decimals 106

Lucas keeps track of money he has and money he owes. A negative balance means he owes money. His starting balance in April is $\$28.50$. He records these changes: <table> <tr><td>Apr. 4</td><td>Movie and popcorn</td><td>$-\$14.20$</td></tr> <tr><td>Apr. 10</td><td>Gift from Grandma</td><td>$+\$20.00$</td></tr> <tr><td>Apr. 15</td><td>New video game</td><td>$-\$35.00$</td></tr> <tr><td>Apr. 22</td><td>Mowing a neighbor's lawn</td><td>$+\$12.50$</td></tr> <tr><td>Apr. 28</td><td>Pizza with friends</td><td>$-\$6.80$</td></tr> </table> a) Write an expression and find the account balance at the end of April. b) Determine whether the account balance was ever negative during April. If so, give the lowest balance.

Hints

- Treat income as positive and spending as negative. - Apply the transactions in chronological order. - Keep each intermediate balance to check whether the account ever goes below $0$.

Solution

1. For a), use $28.50-14.20+20.00-35.00+12.50-6.80$. 2. The balances in order are $\$28.50$, $\$14.30$, $\$34.30$, $-\$0.70$, $\$11.80$, and $\$5.00$. 3. The end-of-month balance is $\$5.00$. 4. For b), the balance was negative after the Apr. 15 purchase, and the lowest balance was $-\$0.70$.

Answer

a) $28.50-14.20+20.00-35.00+12.50-6.80=5.00$, so the balance is $\$5.00$. b) Yes. The lowest balance was $-\$0.70$.

Add and subtract fractions and decimals 107

A bulk-food store has $34.0$ pounds of oatmeal on Monday morning. The table shows how many pounds it sells (negative) or receives (positive) each day. <table> <tr><td>Monday</td><td>$-7.0\,\text{lb}$</td></tr> <tr><td>Tuesday</td><td>$-10.0\,\text{lb}$</td></tr> <tr><td>Wednesday</td><td>$+44.0\,\text{lb}$ (delivery)</td></tr> <tr><td>Thursday</td><td>$-16.5\,\text{lb}$</td></tr> <tr><td>Friday</td><td>$-20.5\,\text{lb}$</td></tr> </table> a) How many pounds of oatmeal remain on Friday evening? Write an expression. b) What was the greatest amount of oatmeal in the store at one time during the week?

Hints

- Start with the Monday morning inventory and apply each signed change. - Keep a running list of the inventory after each day. - Compare all the inventory amounts to find the maximum.

Solution

1. For a), $34.0-7.0-10.0+44.0-16.5-20.5=24.0$. So $24.0$ pounds remain Friday evening. 2. The successive inventory amounts are $34.0$, $27.0$, $17.0$, $61.0$, $44.5$, and $24.0$ pounds. 3. For b), the greatest inventory amount was $61.0$ pounds, after Wednesday's delivery.

Answer

a) $34.0-7.0-10.0+44.0-16.5-20.5=24.0\,\text{lb}$ b) $61.0\,\text{lb}$

Add and subtract fractions and decimals 108

For each movement on a number line, find the missing value and write the corresponding addition equation. a) Start at $-5$ and move $8$ units right. b) Start at $3.5$ and move $6$ units left. c) You end at $-2$ after moving $4$ units left from an unknown starting point. Find the starting point.

Hints

- Moving right corresponds to adding a positive number. - Moving left corresponds to adding a negative number. - To find an unknown starting point, undo the movement.

Solution

1. For a), moving right means adding $8$: $-5+8=3$. 2. For b), moving left means adding $-6$: $3.5+(-6)=-2.5$. 3. For c), let the starting point be $s$. Then $s+(-4)=-2$, so $s=2$. The complete equation is $2+(-4)=-2$.

Answer

a) $3$; $-5+8=3$ b) $-2.5$; $3.5+(-6)=-2.5$ c) $2$; $2+(-4)=-2$

Add and subtract fractions and decimals 109

Analyze sums on a number line. a) Which sum lies farther left: $(-12)+7$ or $(-3)+(-3)$? Calculate and compare. b) A point is at $-4.8$. What number $x$ must be added to move it to $0$? c) A point at $-2.4$ is moved to its opposite by adding a number $y$. Find $y$ and write the complete equation.

Hints

- The value farther left on a number line is smaller. - A number plus its opposite equals $0$. - Find the change needed to move from $-2.4$ to $2.4$.

Solution

1. For a), $(-12)+7=-5$ and $(-3)+(-3)=-6$. Since $-6<-5$, $(-3)+(-3)$ lies farther left. 2. For b), add the opposite of $-4.8$: $-4.8+4.8=0$. Therefore, $x=4.8$. 3. For c), the opposite of $-2.4$ is $2.4$. The required change is $2.4-(-2.4)=4.8$, so $y=4.8$. The equation is $-2.4+4.8=2.4$.

Answer

a) $(-3)+(-3)$, because $-6<-5$ b) $x=4.8$ c) $y=4.8$; $-2.4+4.8=2.4$

Add and subtract fractions and decimals 110

Complete the table by finding each missing value of $x$, $y$, or $x+y$. <table> <thead> <tr> <th>Item</th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>5</th> </tr> </thead> <tbody> <tr> <td>$x$</td> <td>$-7.2$</td> <td>$3 \frac{1}{4}$</td> <td>$-0.8$</td> <td>$-\frac{5}{6}$</td> <td>$-4.15$</td> </tr> <tr> <td>$y$</td> <td>$4.5$</td> <td>$-5 \frac{1}{2}$</td> <td>$\square$</td> <td>$\frac{1}{3}$</td> <td>$4.15$</td> </tr> <tr> <td>$x+y$</td> <td>$\square$</td> <td>$\square$</td> <td>$-2$</td> <td>$\square$</td> <td>$\square$</td> </tr> </tbody> </table>

Hints

- For each column, decide whether you need to add two rational numbers or find a missing addend. - Fractions need a common denominator before they can be added. - Converting between fractions and decimals can make some calculations easier. - When a sum and one addend are known, subtract the known addend to find the other one.

Solution

1. For item 1, $-7.2+4.5=-2.7$. 2. For item 2, $3 \frac{1}{4}+\left(-5 \frac{1}{2}\right)=3 \frac{1}{4}-5 \frac{2}{4}=-2 \frac{1}{4}$. 3. For item 3, $-0.8+y=-2$, so $y=-2-(-0.8)=-1.2$. 4. For item 4, $-\frac{5}{6}+\frac{1}{3}=-\frac{5}{6}+\frac{2}{6}=-\frac{1}{2}$. 5. For item 5, $-4.15+4.15=0$.

Answer

1) $-2.7$ 2) $-2 \frac{1}{4}$ 3) $y=-1.2$ 4) $-\frac{1}{2}$ 5) $0$

Add and subtract fractions and decimals 111

Evaluate each expression, then order the results from least to greatest. a) $-12.5+7.8$ b) $-\frac{3}{5}-\frac{1}{2}$ c) $4 \frac{1}{3}+\left(-2 \frac{5}{6}\right)$ d) $-0.75-(-1.2)$

Hints

- Evaluate each expression before comparing the results. - Be careful when subtracting a negative number. - Converting all results to the same form can make them easier to compare.

Solution

1. For a), $-12.5+7.8=-4.7$. 2. For b), $-\frac{3}{5}-\frac{1}{2}=-\frac{6}{10}-\frac{5}{10}=-\frac{11}{10}=-1.1$. 3. For c), $4 \frac{1}{3}+\left(-2 \frac{5}{6}\right)=\frac{13}{3}-\frac{17}{6}=\frac{26}{6}-\frac{17}{6}=\frac{9}{6}=1.5$. 4. For d), $-0.75-(-1.2)=-0.75+1.2=0.45$. 5. Compare the results: $-4.7<-1.1<0.45<1.5$. Therefore, the order is a), b), d), c).

Answer

a) $-4.7$ b) $-1.1$ c) $1.5$ d) $0.45$ Order: $-4.7<-1.1<0.45<1.5$ (parts a), b), d), c)).

Add and subtract fractions and decimals 112

Evaluate each sum using the commutative and associative properties. For every part, write the regrouped expression you use before giving the result. 1) $(-24) + 15 + (-16) + 35$ 2) $6.7 + (-3.2) + 1.3 + (-1.8)$ 3) $-\frac{3}{4} + 2\frac{1}{5} - 0.25 + 0.8$

Hints

- Keep each signed term intact while changing order. - Search for pairings that produce whole numbers. - Convert a rational number only when the conversion creates a useful pair.

Solution

1. $(-24-16)+(15+35)=-40+50=10$. 2. $(6.7+1.3)+(-3.2-1.8)=8-5=3$. 3. Convert $-\frac{3}{4}=-0.75$, then $(-0.75-0.25)+(2.2+0.8)=-1+3=2$.

Answer

1) $(-24-16)+(15+35)=10$ 2) $(6.7+1.3)+(-3.2-1.8)=3$ 3) $(-0.75-0.25)+(2.2+0.8)=2$

Add and subtract fractions and decimals 113

Evaluate each sum by choosing and showing pairings that avoid converting every term to one representation. 1) $0.375 - 5\frac{1}{2} + \frac{5}{8} + 4.5$ 2) $-8.1 + 2\frac{3}{7} + 1.1 + 5\frac{4}{7}$ 3) $11\frac{1}{9} - 4.9 - 2\frac{1}{9} - 5.1$

Hints

- Look for compatible values already in the same representation. - Convert only the term needed to form a particularly simple pair. - Show the pairings in your answer.

Solution

1. Use $0.375=\frac{3}{8}$: $\left(\frac{3}{8}+\frac{5}{8}\right)+(-5.5+4.5)=1-1=0$. 2. $(-8.1+1.1)+\left(2\frac{3}{7}+5\frac{4}{7}\right)=-7+8=1$. 3. $\left(11\frac{1}{9}-2\frac{1}{9}\right)+(-4.9-5.1)=9-10=-1$.

Answer

1) $\left(\frac{3}{8}+\frac{5}{8}\right)+(-5.5+4.5)=0$ 2) $(-8.1+1.1)+\left(2\frac{3}{7}+5\frac{4}{7}\right)=1$ 3) $\left(11\frac{1}{9}-2\frac{1}{9}\right)+(-4.9-5.1)=-1$

Add and subtract fractions and decimals 114

Determine which value lies farther left on the number line. Value A: the sum of the opposite of $2\frac{3}{5}$ and the reciprocal of $0.5$ Value B: the result of subtracting the sum of $0.6$ and $-3\frac{1}{2}$ from $-1.4$

Hints

- A smaller number lies farther left on the number line. - Find the opposite and reciprocal in Value A. - Evaluate the grouped sum in Value B before subtracting it.

Solution

1. For Value A, the opposite of $2\frac{3}{5}=2.6$ is $-2.6$, and the reciprocal of $0.5$ is $2$. Thus A is $-2.6+2=-0.6$. 2. For Value B, first find the sum: $0.6+(-3.5)=-2.9$. Then $-1.4-(-2.9)=1.5$. 3. Since $-0.6<1.5$, Value A lies farther left.

Answer

Value A lies farther left because $-0.6<1.5$.

Add and subtract fractions and decimals 115

Find each requested value. a) Evaluate $(+7)-(-15)$. b) What number must be subtracted from $-3.8$ to get $-10$? c) Subtract $+1 \frac{3}{4}$ from $-2 \frac{1}{2}$.

Hints

- Subtracting a negative number is the same as adding its opposite. - For b), represent the unknown number with a variable or a blank. - For mixed numbers, a common denominator can make the subtraction easier.

Solution

1. For a), subtracting a negative is addition: $7-(-15)=7+15=22$. 2. For b), let the missing number be $x$. Then $-3.8-x=-10$, so $x=-3.8-(-10)=6.2$. 3. For c), use a common denominator: $-2 \frac{1}{2}-1 \frac{3}{4}=-\frac{10}{4}-\frac{7}{4}=-\frac{17}{4}=-4 \frac{1}{4}$.

Answer

a) $22$ b) $6.2$ c) $-4 \frac{1}{4}$, or $-4.25$

Add and subtract fractions and decimals 116

Explore how changing the order affects subtraction. Let $x=-6.2$ and $y=-3.4$. a) Find $x-y$. b) Find $y-x$. c) Compare your answers from a) and b). What do you notice? d) Kertu claims, “When you subtract one negative number from another negative number, the result is always positive.” Give a counterexample to show that the claim is false.

Hints

- Substitute the negative values carefully and pay attention to both operation signs and number signs. - Compare the absolute values and signs of the results in a) and b). - One counterexample is enough to disprove a statement that claims something is always true.

Solution

1. For a), $x-y=-6.2-(-3.4)=-6.2+3.4=-2.8$. 2. For b), $y-x=-3.4-(-6.2)=-3.4+6.2=2.8$. 3. The two results are opposites: they have the same absolute value and opposite signs. 4. One counterexample is $(-10)-(-2)=-10+2=-8$. Since the result is negative, the claim is false.

Answer

a) $-2.8$ b) $2.8$ c) The results are opposites. d) Example: $(-10)-(-2)=-8$. The result is negative, so the claim is false.

Add and subtract fractions and decimals 117

Evaluate each expression. Write each result as a simplified fraction or mixed number. 1) $\left(-\frac{7}{8}\right)-\left(+\frac{1}{8}\right)-(-3)$ 2) $\left(+\frac{2}{5}\right)-\left(-\frac{1}{2}\right)-\left(+\frac{9}{10}\right)$ 3) $\left(-2 \frac{1}{4}\right)-\left(-\frac{3}{4}\right)-(+1)$

Hints

- Subtracting a number is the same as adding its opposite. - Check whether the fractions already have a common denominator before finding a new one. - You can write whole numbers as fractions when that helps combine terms. - Converting mixed numbers to improper fractions can make some calculations easier.

Solution

1. For 1), rewrite subtraction of a negative as addition: $-\frac{7}{8}-\frac{1}{8}+3=-1+3=2$. 2. For 2), use denominator $10$: $\frac{4}{10}+\frac{5}{10}-\frac{9}{10}=0$. 3. For 3), $-2 \frac{1}{4}+\frac{3}{4}-1=-1 \frac{1}{2}-1=-2 \frac{1}{2}$.

Answer

1) $2$ 2) $0$ 3) $-2 \frac{1}{2}$

Add and subtract fractions and decimals 118

a) Write three different subtractions using positive fractions or decimals: one with a positive answer, one with zero, and one with a negative answer. You may choose new numbers each time. Show the calculations. b) Let $a$ and $b$ be positive rational numbers. When is $a-b$ positive, zero, or negative? Compare $a$ and $b$ using $<$, $=$, or $>$. c) If you add $0.5$ to both $a$ and $b$, will the difference change? Explain.

Hints

- Compare the two numbers in each example, not just the sign of the answer. - What changes when the first number is larger, equal to, or smaller than the second? - For c), try changing both numbers in one of your examples before explaining the general case.

Solution

1. For example, $5.8-2.3=3.5$ is positive, $1.5-1.5=0$, and $0.2-0.7=-0.5$ is negative. 2. The difference is positive when $a>b$, zero when $a=b$, and negative when $a<b$. 3. The difference does not change: $(a+0.5)-(b+0.5)=a-b$. Adding the same amount to both numbers leaves their difference unchanged.

Answer

a) Examples: positive $5.8-2.3=3.5$; zero $1.5-1.5=0$; negative $0.2-0.7=-0.5$. b) Positive: $a>b$; zero: $a=b$; negative: $a<b$. c) No. $(a+0.5)-(b+0.5)=a-b$; adding the same amount to both numbers does not change the difference.

Add and subtract fractions and decimals 119

The expressions are $A=a-b+c$ and $B=a-(b+c)$. Evaluate and compare them for each assignment. a) $a=10$, $b=-4$, $c=2$ b) $a=-3.5$, $b=1.5$, $c=-5$

Hints

- Use parentheses when substituting negative values. - Evaluate the parentheses in $B$ before subtracting. - Compare the final values on a number line.

Solution

1. For a), $A=10-(-4)+2=16$, while $B=10-((-4)+2)=12$. Thus, $A>B$. 2. For b), $A=-3.5-1.5+(-5)=-10$, while $B=-3.5-(1.5+(-5))=0$. Thus, $A<B$.

Answer

a) $A=16$, $B=12$, so $A>B$. b) $A=-10$, $B=0$, so $A<B$.

Add and subtract fractions and decimals 120

Evaluate $T=-x-(y-z)$ for each set of rational numbers. a) $x=\frac{2}{3}$, $y=-\frac{1}{6}$, $z=\frac{1}{2}$ b) $x=-2\frac{1}{4}$, $y=1\frac{3}{8}$, $z=-0.5$

Hints

- Convert mixed numbers and decimals to fractions. - Use common denominators for addition and subtraction. - Pay close attention to the negative sign in front of $x$. - Work inside the parentheses first.

Solution

1. For part a), $T=-\frac{2}{3}-\left(-\frac{1}{6}-\frac{1}{2}\right)=-\frac{2}{3}-\left(-\frac{2}{3}\right)=0$. 2. For part b), convert the values to fractions: $x=-\frac{9}{4}$, $y=\frac{11}{8}$, and $z=-\frac{1}{2}$. 3. Then $T=-\left(-\frac{9}{4}\right)-\left(\frac{11}{8}-\left(-\frac{1}{2}\right)\right)=\frac{18}{8}-\frac{15}{8}=\frac{3}{8}$.

Answer

a) $0$ b) $\frac{3}{8}$

Add and subtract fractions and decimals 121

Let $a$ and $b$ be rational numbers. a) Under what condition is $a - b > a$? Explain why and give a numerical example. b) Under what condition does $a - b = a + b$? Justify your answer algebraically.

Hints

- Subtract the common term $a$ from both sides. - Subtracting a negative number is equivalent to adding a positive number. - In part b), collect all terms containing $b$ on one side.

Solution

1. For a), subtract $a$ from both sides of $a - b > a$: $-b > 0$. This is equivalent to $b < 0$. 2. Subtracting a negative number increases the original value. For example, $7 - (-3) = 10$, and $10 > 7$. 3. For b), subtract $a$ from both sides: $-b = b$. 4. Add $b$: $0 = 2b$. Divide by $2$: $b = 0$. The value of $a$ can be any rational number.

Answer

a) $b<0$. Subtracting a negative number adds a positive amount. For example, $7-(-3)=10>7$. b) $b=0$, because $a-b=a+b$ implies $-b=b$, then $0=2b$, so $b=0$.

Add and subtract fractions and decimals 122

Use the number line. a) Write the value of each point $A$, $B$, and $C$ as both a decimal and a fraction in simplest form or a mixed number. b) Find the distance from $A$ to $B$ and from $B$ to $C$.

Hints

- Determine the value of one small interval. - Use the point’s position relative to $0$ to determine its sign. - Distance is the absolute value of the difference between two coordinates. - Convert quarters to decimals or decimals to fractions as needed.

Solution

1. There are $4$ equal intervals between consecutive integers, so each interval represents $0.25=\frac{1}{4}$. 2. Point $A$ is at $-1.75=-\frac{7}{4}=-1\frac{3}{4}$. 3. Point $B$ is at $-0.5=-\frac{1}{2}$. 4. Point $C$ is at $0.25=\frac{1}{4}$. 5. The distance from $A$ to $B$ is $\left|-0.5-(-1.75)\right|=1.25=\frac{5}{4}$. 6. The distance from $B$ to $C$ is $|0.25-(-0.5)|=0.75=\frac{3}{4}$.

Answer

a) $A=-1.75=-\frac{7}{4}$; $B=-0.5=-\frac{1}{2}$; $C=0.25=\frac{1}{4}$ b) $AB=1.25=\frac{5}{4}$; $BC=0.75=\frac{3}{4}$

Add and subtract fractions and decimals 123

Each number outside the triangle is the sum of the two corner numbers next to it. Find the two missing corner numbers and the missing side number. Show a subtraction for each missing corner.

Hints

- Each outside number joins two neighboring corner values. - Use a known sum and one corner to work backward to the other corner. - Once both bottom corners are known, calculate the bottom side.

Solution

1. The top corner is $-1.2$. The bottom-right corner is $-2.7-(-1.2)=-1.5$. 2. The bottom-left corner is $1.9-(-1.2)=3.1$. 3. The bottom side is the sum of the two bottom corners: $-1.5+3.1=1.6$.

Answer

Bottom-right corner: $-2.7-(-1.2)=-1.5$ Bottom-left corner: $1.9-(-1.2)=3.1$ Bottom side: $-1.5+3.1=1.6$

Add and subtract fractions and decimals 124

Divide the six number cards into two groups of three so that the sums of the groups are equal. $1.2,\ \frac{3}{10},\ -1.5,\ 0.8,\ -0.2,\ -\frac{3}{5}$

Hints

- Find the sum of all six numbers first. - Convert the fractions to decimals. - Look for three numbers that sum to half the total.

Solution

1. Write $\frac{3}{10}=0.3$ and $-\frac{3}{5}=-0.6$. 2. The sum of all six numbers is $0$, so each group must have sum $0$. 3. One group is $1.2, 0.3, -1.5$, whose sum is $0$. 4. The remaining group is $0.8, -0.2, -0.6$, whose sum is also $0$.

Answer

Group 1: $1.2, \frac{3}{10}, -1.5$ Group 2: $0.8, -0.2, -\frac{3}{5}$

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Add and subtract fractions and decimals

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