Representing parts of a whole

Using Models to Represent Fractions

In the following exercises, name the fraction of each figure that is shaded.

Exercise 1

Exercise 2

Using Models to Represent Fractions

In the following exercises, shade parts of circles or squares to model the following fractions.

1. $\Large\frac{1}{2}$
   Show Solution

   
2. $\Large\frac{1}{3}$
3. $\Large\frac{3}{4}$
   Show Solution

   
4. $\Large\frac{2}{5}$
5. $\Large\frac{5}{6}$
   Show Solution

   
6. $\Large\frac{7}{8}$
7. $\Large\frac{5}{8}$
   Show Solution

   
8. $\Large\frac{7}{10}$

Using Models to Represent Mixed Numbers

In the following exercises, use fraction circles to make wholes, if possible, with the following pieces.

1. $3$ thirds
   Show Solution

   
2. $8$ eighths
3. $7$ sixths
   Show Solution

   
4. $4$ thirds
5. $7$ fifths
   Show Solution

   
6. $7$ fourths

Using Models to Represent Mixed Numbers

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

Exercise 1

Exercise 2

Exercise 3

Using Models to Represent Mixed Numbers

In the following exercises, draw fraction circles to model the given fraction.

1. $\Large\frac{3}{3}$
2. $\Large\frac{4}{4}$
   Show Solution

   
3. $\Large\frac{7}{4}$
4. $\Large\frac{5}{3}$
   Show Solution

   
5. $\Large\frac{11}{6}$
6. $\Large\frac{13}{8}$
   Show Solution

   
7. $\Large\frac{10}{3}$
8. $\Large\frac{9}{4}$
   Show Solution

   

Converting Between Improper Fractions and Mixed Numbers

Write an Improper Fraction as a Mixed Number

In the following exercises, rewrite the improper fraction as a mixed number.

1. $\Large\frac{3}{2}$
2. $\Large\frac{5}{3}$
   Show Solution

   $1\Large\frac{2}{3}$
3. $\Large\frac{11}{4}$
4. $\Large\frac{13}{5}$
   Show Solution

   $2\Large\frac{3}{5}$
5. $\Large\frac{25}{6}$
6. $\Large\frac{28}{9}$
   Show Solution

   $3\Large\frac{1}{9}$
7. $\Large\frac{42}{13}$
8. $\Large\frac{47}{15}$
   Show Solution

   $3\Large\frac{2}{15}$

Write a Mixed Number as an Improper Fraction

In the following exercises, rewrite the mixed number as an improper fraction.

1. $1\Large\frac{2}{3}$
2. $1\Large\frac{2}{5}$
   Show Solution

   $\Large\frac{7}{5}$
3. $2\Large\frac{1}{4}$
4. $2\Large\frac{5}{6}$
   Show Solution

   $\Large\frac{17}{6}$
5. $2\Large\frac{7}{9}$
6. $2\Large\frac{5}{7}$
   Show Solution

   $\Large\frac{19}{7}$
7. $3\Large\frac{4}{7}$
8. $3\Large\frac{5}{9}$
   Show Solution

   $\Large\frac{32}{9}$

Modeling and Finding Equivalent Fractions

Modeling Equivalent Fractions

In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.

1. How many sixths equal one-third?
2. How many twelfths equal one-third?
   Show Solution

   4
3. How many eighths equal three-fourths?
4. How many twelfths equal three-fourths?
   Show Solution

   9
5. How many fourths equal three-halves?
6. How many sixths equal three-halves?
   Show Solution

   9

Finding Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

1. $\Large\frac{1}{4}$
2. $\Large\frac{1}{3}$
   Show Solution

   Answers may vary. Correct answers include $\Large\frac{2}{6},\Large\frac{3}{9},\Large\frac{4}{12}$.
3. $\Large\frac{3}{8}$
4. $\Large\frac{5}{6}$
   Show Solution

   Answers may vary. Correct answers include $\Large\frac{10}{12},\Large\frac{15}{18},\Large\frac{20}{24}$.
5. $\Large\frac{2}{7}$
6. $\Large\frac{5}{9}$
   Show Solution

   Answers may vary. Correct answers include $\Large\frac{10}{18},\Large\frac{15}{27},\Large\frac{20}{36}$.

Locating and Ordering Fractions and Mixed Numbers on the Number Line

Locating Fractions on the Number Line

In the following exercises, plot the numbers on a number line.

1. $\Large\frac{2}{3},\Large\frac{5}{4},\Large\frac{12}{5}$
2. $\Large\frac{1}{3},\Large\frac{7}{4},\Large\frac{13}{5}$
   Show Solution

   
3. $\Large\frac{1}{4},\Large\frac{9}{5},\Large\frac{11}{3}$
4. $\Large\frac{7}{10},\Large\frac{5}{2},\Large\frac{13}{8},\normalsize 3$
   Show Solution

   
5. $2\Large\frac{1}{3}\normalsize ,-2\Large\frac{1}{3}$
6. $1\Large\frac{3}{4}\normalsize ,-1\Large\frac{3}{5}$
   Show Solution

   
7. $\Large\frac{3}{4},-\Large\frac{3}{4}\normalsize ,1\Large\frac{2}{3}\normalsize ,-1\Large\frac{2}{3},\Large\frac{5}{2},-\Large\frac{5}{2}$
8. $\Large\frac{2}{5},-\Large\frac{2}{5}\normalsize ,1\Large\frac{3}{4}\normalsize ,-1\Large\frac{3}{4},\Large\frac{8}{3},-\Large\frac{8}{3}$
   Show Solution

   

Ordering Fractions on the Number Line

In the following exercises, order each of the following pairs of numbers, using $<$ or $>$.

1. $-1\text{ __}-\Large\frac{1}{4}$
2. $-1\text{ __}-\Large\frac{1}{3}$
   Show Solution

   <
3. $-2\Large\frac{1}{2}\normalsize\text{ __}-3$
4. $-1\Large\frac{3}{4}\normalsize\text{ __}-2$
   Show Solution

   >
5. $-\Large\frac{5}{12}\text{ __}-\Large\frac{7}{12}$
6. $-\Large\frac{9}{10}\text{ __}-\Large\frac{3}{10}$
   Show Solution

   <
7. $-3\text{ __}-\Large\frac{13}{5}$
8. $-4\text{ __}-\Large\frac{23}{6}$
   Show Solution

   <

Multiplying and Dividing Fractions

Simplify Fractions

In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.

1. $\Large\frac{7}{21}$
   Show Solution

   $\Large\frac{1}{3}$
2. $\Large\frac{8}{24}$
3. $\Large\frac{15}{20}$
   Show Solution

   $\Large\frac{3}{4}$
4. $\Large\frac{12}{18}$
5. $-\Large\frac{40}{88}$
   Show Solution

   $-\Large\frac{5}{11}$
6. $-\Large\frac{63}{99}$
7. $-\Large\frac{108}{63}$
   Show Solution

   $-\Large\frac{12}{7}$
8. $-\Large\frac{104}{48}$
9. $\Large\frac{120}{252}$
   Show Solution

   $\Large\frac{10}{21}$
10. $\Large\frac{182}{294}$
11. $-\Large\frac{168}{192}$
    Show Solution

    $-\Large\frac{7}{8}$
12. $-\Large\frac{140}{224}$
13. $\Large\frac{11x}{11y}$
    Show Solution

    $\Large\frac{x}{y}$
14. $\Large\frac{15a}{15b}$
15. $-\Large\frac{3x}{12y}$
    Show Solution

    $-\Large\frac{x}{4y}$
16. $-\Large\frac{4x}{32y}$
17. $\Large\frac{14{x}^{2}}{21y}$
    Show Solution

    $\Large\frac{2{x}^{2}}{3y}$
18. $\Large\frac{24a}{32{b}^{2}}$

Multiply Fractions

In the following exercises, use a diagram to model.

1. $\Large\frac{1}{2}\cdot\Large\frac{2}{3}$
2. $\Large\frac{1}{3}$
3. $\Large\frac{1}{2}\cdot\Large\frac{5}{8}$
4. $\Large\frac{1}{3}\cdot\Large\frac{5}{6}$
5. $\Large\frac{5}{18}$
6. $\Large\frac{1}{3}\cdot\Large\frac{2}{5}$

Multiply and Simplify Fractions

In the following exercises, multiply, and write the answer in simplified form.

1. $\Large\frac{2}{5}\cdot\Large\frac{1}{3}$
   Show Solution

   $\Large\frac{2}{15}$
2. $\Large\frac{1}{2}\cdot\Large\frac{3}{8}$
3. $\Large\frac{3}{4}\cdot\Large\frac{9}{10}$
   Show Solution

   $\Large\frac{27}{40}$
4. $\Large\frac{4}{5}\cdot\Large\frac{2}{7}$
5. $-\Large\frac{2}{3}\left(-\Large\frac{3}{8}\right)$
   Show Solution

   $\Large\frac{1}{4}$
6. $-\Large\frac{3}{4}\left(-\Large\frac{4}{9}\right)$
7. $-\Large\frac{5}{9}\cdot\Large\frac{3}{10}$
   Show Solution

   $-\Large\frac{1}{6}$
8. $-\Large\frac{3}{8}\cdot\Large\frac{4}{15}$
9. $\Large\frac{7}{12}\left(-\Large\frac{8}{21}\right)$
   Show Solution

   $-\Large\frac{2}{9}$
10. $\Large\frac{5}{12}\left(-\Large\frac{8}{15}\right)$$\left(-\Large\frac{14}{15}\right)\left(\Large\frac{9}{20}\right)$
    Show Solution

    $-\Large\frac{21}{50}$
11. $\left(-\Large\frac{9}{10}\right)\left(\Large\frac{25}{33}\right)$
12. $\left(-\Large\frac{63}{84}\right)\left(-\Large\frac{44}{90}\right)$
    Show Solution

    $\Large\frac{11}{30}$
13. $\left(-\Large\frac{33}{60}\right)\left(-\Large\frac{40}{88}\right)$
14. $4\cdot\Large\frac{5}{11}$
15. $\Large\frac{20}{11}$
16. $5\cdot\Large\frac{8}{3}$
17. $\Large\frac{3}{7}\normalsize\cdot 21n$
    Show Solution

    9n
18. $\Large\frac{5}{6}\normalsize\cdot 30m$
19. $-28p\left(-\Large\frac{1}{4}\right)$
    Show Solution

    7p
20. $-51q\left(-\Large\frac{1}{3}\right)$
21. $-8\Large\left(\frac{17}{4}\right)$
    Show Solution

    −34
22. $\Large\frac{14}{5}\normalsize\left(-15\right)$
23. $-1\Large\left(-\frac{3}{8}\right)$
24. $\Large\frac{3}{8}$
25. $\left(-1\right)\Large\left(-\frac{6}{7}\right)$
26. ${\Large\left(\frac{2}{3}\right)}^{3}$
27. $\Large\frac{8}{27}$
28. ${\Large\left(\frac{4}{5}\right)}^{2}$
29. ${\Large\left(\frac{6}{5}\right)}^{4}$
30. $\Large\frac{1296}{625}$
31. ${\Large\left(\frac{4}{7}\right)}^{4}$

Find Reciprocals

In the following exercises, find the reciprocal.

1. $\Large\frac{3}{4}$
   Show Solution

   $\Large\frac{4}{3}$
2. $\Large\frac{2}{3}$
3. $-\Large\frac{5}{17}$
   Show Solution

   $-\Large\frac{17}{5}$
4. $-\Large\frac{6}{19}$
5. $\Large\frac{11}{8}$
   Show Solution

   $\Large\frac{8}{11}$
6. $-13$
7. $-19$
   Show Solution

   $-\Large\frac{1}{19}$
8. $-1$
   Show Solution

   $1$

Find Reciprocals

Exercise 1

Fill in the chart.

Opposite	Absolute Value	Reciprocal
$-\Large\frac{7}{11}$
$\Large\frac{4}{5}$
$\Large\frac{10}{7}$
$-\large 8$

Exercise 2

Fill in the chart.

Opposite	Absolute Value	Reciprocal
$-\Large\frac{3}{13}$
$\Large\frac{9}{14}$
$\Large\frac{15}{7}$
$-\large 9$

Divide Fractions

In the following exercises, model each fraction division.

1. $\Large\frac{1}{2}\normalsize\div\Large\frac{1}{4}$
2. $\Large\frac{1}{2}\normalsize\div\Large\frac{1}{8}$
   Show Solution

   4
3. $2\div\Large\frac{1}{5}$
4. $3\div\Large\frac{1}{4}$
   Show Solution

   12

Divide and Simplify Fractions

In the following exercises, divide, and write the answer in simplified form.

1. $\Large\frac{1}{2}\normalsize\div\Large\frac{1}{4}$
2. $\Large\frac{1}{2}\normalsize\div\Large\frac{1}{8}$
   Show Solution

   4
3. $\Large\frac{3}{4}\normalsize\div\Large\frac{2}{3}$
4. $\Large\frac{4}{5}\normalsize\div\Large\frac{3}{4}$
5. $\Large\frac{16}{15}$
6. $-\Large\frac{4}{5}\normalsize\div\Large\frac{4}{7}$
7. $-\Large\frac{3}{4}\normalsize\div\Large\frac{3}{5}$
8. $-\Large\frac{5}{4}$
9. $-\Large\frac{7}{9}\normalsize\div\Large\left(-\frac{7}{9}\right)$
10. $-\Large\frac{5}{6}\normalsize\div\Large\left(-\frac{5}{6}\right)$
    Show Solution

    1
11. $\Large\frac{3}{4}\normalsize\div\Large\frac{x}{11}$
12. $\Large\frac{2}{5}\normalsize\div\Large\frac{y}{9}$
13. $\Large\frac{18}{5y}$
14. $\Large\frac{5}{8}\normalsize\div\Large\frac{a}{10}$
15. $\Large\frac{5}{6}\normalsize\div\Large\frac{c}{15}$
16. $\Large\frac{25}{2c}$
17. $\Large\frac{5}{18}\normalsize\div\Large\left(-\frac{15}{24}\right)$
18. $\Large\frac{7}{18}\normalsize\div\Large\left(-\frac{14}{27}\right)$
19. $-\Large\frac{3}{4}$
20. $\Large\frac{7p}{12}\normalsize\div\Large\frac{21p}{8}$
21. $\Large\frac{5q}{12}\normalsize\div\Large\frac{15q}{8}$
22. $\Large\frac{2}{9}$
23. $\Large\frac{8u}{15}\normalsize\div\Large\frac{12v}{25}$
24. $\Large\frac{12r}{25}\normalsize\div\Large\frac{18s}{35}$
25. $\Large\frac{14r}{15s}$
26. $-5\normalsize\div\Large\frac{1}{2}$
27. $-3\normalsize\div\Large\frac{1}{4}$
    Show Solution

    −12
28. $\Large\frac{3}{4}\normalsize\div\left(-12\right)$
29. $\Large\frac{2}{5}\normalsize\div\left(-10\right)$
30. $-\Large\frac{1}{25}$
31. $-18\normalsize\div\Large\left(-\frac{9}{2}\right)$
32. $-15\normalsize\div\Large\left(-\frac{5}{3}\right)$
    Show Solution

    9
33. $\Large\frac{1}{2}\normalsize\div\Large\left(-\frac{3}{4}\right)\normalsize\div\Large\frac{7}{8}$
34. $\Large\frac{11}{2}\normalsize\div\Large\frac{7}{8}\normalsize\cdot\Large\frac{2}{11}$
35. $\Large\frac{8}{7}$

Multiplying and Dividing Mixed Numbers and Complex Fractions

Multiply Mixed Numbers

In the following exercises, multiply and write the answer in simplified form.

1. $4\Large\frac{3}{8}\normalsize\cdot\Large\frac{7}{10}$
2. $2\Large\frac{4}{9}\normalsize\cdot\Large\frac{6}{7}$
   Show Solution

   $\Large\frac{44}{21}$
3. $\Large\frac{15}{22}\normalsize\cdot 3\Large\frac{3}{5}$
4. $\Large\frac{25}{36}\normalsize\cdot 6\Large\frac{3}{10}$
   Show Solution

   $\Large\frac{35}{8}$
5. $4\Large\frac{2}{3}\left(\normalsize -1\Large\frac{1}{8}\right)$
6. $2\Large\frac{2}{5}\left(\normalsize -2\Large\frac{2}{9}\right)$
   Show Solution

   $-\Large\frac{16}{3}$
7. $-4\Large\frac{4}{9}\normalsize\cdot 5\Large\frac{13}{16}$
8. $-1\Large\frac{7}{20}\normalsize\cdot 2\Large\frac{11}{12}$
   Show Solution

   $-\Large\frac{63}{16}$

Divide Mixed Numbers

In the following exercises, divide, and write your answer in simplified form.

1. $5\Large\frac{1}{3}\normalsize\div 4$
2. $13\Large\frac{1}{2}\normalsize\div 9$
   Show Solution

   $\Large\frac{3}{2}$
3. $-12\Large\normalsize\div 3\Large\frac{3}{11}$
4. $-7\div 5\Large\frac{1}{4}$
   Show Solution

   $-\Large\frac{4}{3}$
5. $6\Large\frac{3}{8}\normalsize\div 2\Large\frac{1}{8}$
6. $2\Large\frac{1}{5}\normalsize\div 1\Large\frac{1}{10}$
   Show Solution

   2
7. $-9\Large\frac{3}{5}\normalsize\div\Large\left(\normalsize -1\Large\frac{3}{5}\right)$
8. $-18\Large\frac{3}{4}\normalsize\div\Large\left(\normalsize -3\Large\frac{3}{4}\right)$
   Show Solution

   5

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

1. the quotient of $5u$ and $11$
2. the quotient of $7v$ and $13$
   Show Solution

   $\Large\frac{7v}{13}$
3. the quotient of $p$ and $q$
4. the quotient of $a$ and $b$
   Show Solution

   $\Large\frac{a}{b}$
5. the quotient of $r$ and the sum of $s$ and $10$
6. the quotient of $A$ and the difference of $3$ and $B$
   Show Solution

   $\Large\frac{A}{3-B}$

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

1. $\displaystyle\Large\frac{\Large\frac{2}{3}}{\Large\frac{8}{9}}$
2. $\displaystyle\Large\frac{\Large\frac{4}{5}}{\Large\frac{8}{15}}$
   Show Solution

   $\displaystyle\frac{3}{2}$
3. $\displaystyle\Large\frac{-\Large\frac{8}{21}}{\Large\frac{12}{35}}$
4. $\displaystyle\Large\frac{-\Large\frac{9}{16}}{\Large\frac{33}{40}}$
   Show Solution

   $-\Large\frac{15}{22}$
5. $\displaystyle\frac{-\Large\frac{4}{5}}{2}$
6. $\displaystyle\frac{-\Large\frac{9}{10}}{3}$
   Show Solution

   $-\Large\frac{3}{10}$
7. $\displaystyle\frac{\Large\frac{2}{5}}{8}$
8. $\displaystyle\frac{\Large\frac{5}{3}}{10}$
   Show Solution

   $\displaystyle\frac{1}{6}$
9. $\displaystyle\Large\frac{\Large\frac{m}{3}}{\Large\frac{n}{2}}$
10. $\displaystyle\Large\frac{\Large\frac{r}{5}}{\Large\frac{s}{3}}$
    Show Solution

    $\displaystyle\frac{3r}{5s}$
11. $\displaystyle\Large\frac{-\Large\frac{x}{6}}{-\Large\frac{8}{9}}$
12. $\displaystyle\Large\frac{-\Large\frac{3}{8}}{-\Large\frac{y}{12}}$
    Show Solution

    $\displaystyle\frac{9}{2y}$
13. $\displaystyle\frac{2\Large\frac{4}{5}}{\Large\frac{1}{10}}$
14. $\displaystyle\frac{4\Large\frac{2}{3}}{\Large\frac{1}{6}}$
    Show Solution

    28
15. $\displaystyle\Large\frac{\Large\frac{7}{9}}{\normalsize -2\Large\frac{4}{5}}$
16. $\displaystyle\Large\frac{\Large\frac{3}{8}}{\normalsize -6\Large\frac{3}{4}}$
    Show Solution

    $-\Large\frac{1}{18}$

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

1. Which of the following fractions are equivalent to $\Large\frac{5}{-11}?$
   $\Large\frac{-5}{-11},\Large\frac{-5}{11},\Large\frac{5}{11},-\Large\frac{5}{11}$
2. Which of the following fractions are equivalent to $\Large\frac{-4}{9}?$
   $\Large\frac{-4}{-9},\Large\frac{-4}{9},\Large\frac{4}{9},-\Large\frac{4}{9}$
   Show Solution

   $\Large\frac{-4}{9},\text{}-\Large\frac{4}{9}$
3. Which of the following fractions are equivalent to $-\Large\frac{11}{3}?$
   $\Large\frac{-11}{3},\Large\frac{11}{3},\Large\frac{-11}{-3},\text{}\Large\frac{11}{-3}$
4. Which of the following fractions are equivalent to $-\Large\frac{13}{6}?$
   $\Large\frac{13}{6},\Large\frac{13}{-6},\Large\frac{-13}{-6},\Large\frac{-13}{6}$
   Show Solution

   $\Large\frac{13}{-6},\Large\frac{-13}{6}$

Simplify Fractions

In the following exercises, simplify.

1. $\Large\frac{4+11}{8}$
2. $\Large\frac{9+3}{7}$
   Show Solution

   $\Large\frac{12}{7}$
3. $\Large\frac{22+3}{10}$
4. $\Large\frac{19 - 4}{6}$
   Show Solution

   $\Large\frac{5}{2}$
5. $\Large\frac{48}{24 - 15}$
6. $\Large\frac{46}{4+4}$
   Show Solution

   $\Large\frac{23}{4}$
7. $\Large\frac{-6+6}{8+4}$
8. $\Large\frac{-6+3}{17 - 8}$
   Show Solution

   $-\Large\frac{1}{3}$
9. $\Large\frac{22 - 14}{19 - 13}$
10. $\Large\frac{15+9}{18+12}$
    Show Solution

    $\Large\frac{4}{5}$
11. $\Large\frac{5\cdot 8}{-10}$
12. $\Large\frac{3\cdot 4}{-24}$
    Show Solution

    $-\Large\frac{1}{2}$
13. $\Large\frac{4\cdot 3}{6\cdot 6}$
14. $\Large\frac{6\cdot 6}{9\cdot 2}$
    Show Solution

    2
15. $\Large\frac{{4}^{2}-1}{25}$
16. $\Large\frac{{7}^{2}+1}{60}$
    Show Solution

    $\Large\frac{5}{6}$
17. $\Large\frac{8\cdot 3+2\cdot 9}{14+3}$
18. $\Large\frac{9\cdot 6 - 4\cdot 7}{22+3}$
    Show Solution

    $\Large\frac{26}{25}$
19. $\Large\frac{15\cdot 5-{5}^{2}}{2\cdot 10}$
20. $\Large\frac{12\cdot 9-{3}^{2}}{3\cdot 18}$
    Show Solution

    $\Large\frac{11}{6}$
21. $\Large\frac{5\cdot 6 - 3\cdot 4}{4\cdot 5 - 2\cdot 3}$
22. $\Large\frac{8\cdot 9 - 7\cdot 6}{5\cdot 6 - 9\cdot 2}$
    Show Solution

    $\Large\frac{5}{2}$
23. $\Large\frac{{5}^{2}-{3}^{2}}{3 - 5}$
24. $\Large\frac{{6}^{2}-{4}^{2}}{4 - 6}$
    Show Solution

    −10
25. $\Large\frac{2+4\left(3\right)}{-3-{2}^{2}}$
26. $\Large\frac{7+3\left(5\right)}{-2-{3}^{2}}$
    Show Solution

    −2
27. $\Large\frac{7\cdot 4 - 2\left(8 - 5\right)}{9.3 - 3.5}$
28. $\Large\frac{9\cdot 7 - 3\left(12 - 8\right)}{8.7 - 6.6}$
    Show Solution

    $\Large\frac{51}{20}$
29. $\Large\frac{9\left(8 - 2\right)-3\left(15 - 7\right)}{6\left(7 - 1\right)-3\left(17 - 9\right)}$
30. $\Large\frac{8\left(9 - 2\right)-4\left(14 - 9\right)}{7\left(8 - 3\right)-3\left(16 - 9\right)}$
    Show Solution

    $\Large\frac{18}{7}$

Adding and Subtracting Fractions With Common Denominators

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

1. $\Large\frac{2}{5}+\Large\frac{1}{5}$
2. $\Large\frac{3}{10}+\Large\frac{4}{10}$
   Show Solution

   
3. $\Large\frac{7}{10}$
4. $\Large\frac{1}{6}+\Large\frac{3}{6}$
5. $\Large\frac{3}{8}+\Large\frac{3}{8}$
   Show Solution

   
6. $\Large\frac{3}{4}$

Add Fractions with a Common Denominator

In the following exercises, find each sum.

1. $\Large\frac{4}{9}+\Large\frac{1}{9}$
2. $\Large\frac{2}{9}+\Large\frac{5}{9}$
   Show Solution

   $\Large\frac{7}{9}$
3. $\Large\frac{6}{13}+\Large\frac{7}{13}$
4. $\Large\frac{9}{15}+\Large\frac{7}{15}$
   Show Solution

   $\Large\frac{16}{15}$
5. $\Large\frac{x}{4}+\Large\frac{3}{4}$
6. $\Large\frac{y}{3}+\Large\frac{2}{3}$
   Show Solution

   $\Large\frac{y+2}{3}$
7. $\Large\frac{7}{p}+\Large\frac{9}{p}$
8. $\Large\frac{8}{q}+\Large\frac{6}{q}$
   Show Solution

   $\Large\frac{14}{q}$
9. $\Large\frac{8b}{9}+\Large\frac{3b}{9}$
10. $\Large\frac{5a}{7}+\Large\frac{4a}{7}$
    Show Solution

    $\Large\frac{9a}{7}$
11. $\Large\frac{-12y}{8}+\Large\frac{3y}{8}$
12. $\Large\frac{-11x}{5}+\Large\frac{7x}{5}$
    Show Solution

    $\Large\frac{-4x}{5}$
13. $-\Large\frac{1}{8}+\Large\left(-\frac{3}{8}\right)$
14. $-\Large\frac{1}{8}+\Large\left(-\frac{5}{8}\right)$
    Show Solution

    $-\Large\frac{3}{4}$
15. $-\Large\frac{3}{16}+\Large\left(-\frac{7}{16}\right)$
16. $-\Large\frac{5}{16}+\Large\left(-\frac{9}{16}\right)$
    Show Solution

    $-\Large\frac{7}{8}$
17. $-\Large\frac{8}{17}+\Large\frac{15}{17}$
18. $-\Large\frac{9}{19}+\Large\frac{17}{19}$
    Show Solution

    $\Large\frac{8}{19}$
19. $\Large\frac{6}{13}+\Large\left(-\frac{10}{13}\right)+\Large\left(-\frac{12}{13}\right)$
20. $\Large\frac{5}{12}+\Large\left(-\frac{7}{12}\right)+\Large\left(-\frac{11}{12}\right)$
    Show Solution

    $-\Large\frac{13}{12}$

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

1. $\Large\frac{5}{8}-\Large\frac{2}{8}$
2. $\Large\frac{5}{6}-\Large\frac{2}{6}$
   Show Solution

   
3. $\Large\frac{1}{2}$

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

1. $\Large\frac{4}{5}-\Large\frac{1}{5}$
2. $\Large\frac{4}{5}-\Large\frac{3}{5}$
   Show Solution

   $\Large\frac{1}{5}$
3. $\Large\frac{11}{15}-\Large\frac{7}{15}$
4. $\Large\frac{9}{13}-\Large\frac{4}{13}$
   Show Solution

   $\Large\frac{5}{13}$
5. $\Large\frac{11}{12}-\Large\frac{5}{12}$
6. $\Large\frac{7}{12}-\Large\frac{5}{12}$
   Show Solution

   $\Large\frac{1}{6}$
7. $\Large\frac{4}{21}-\Large\frac{19}{21}$
8. $-\Large\frac{8}{9}-\Large\frac{16}{9}$
   Show Solution

   $-\Large\frac{8}{3}$
9. $\Large\frac{y}{17}-\Large\frac{9}{17}$
10. $\Large\frac{x}{19}-\Large\frac{8}{19}$
    Show Solution

    $\Large\frac{x - 8}{19}$
11. $\Large\frac{5y}{8}-\Large\frac{7}{8}$
12. $\Large\frac{11z}{13}-\Large\frac{8}{13}$
    Show Solution

    $\Large\frac{11z - 8}{13}$
13. $-\Large\frac{8}{d}-\Large\frac{3}{d}$
14. $-\Large\frac{7}{c}-\Large\frac{7}{c}$
    Show Solution

    $-\Large\frac{14}{c}$
15. $-\Large\frac{23}{u}-\Large\frac{15}{u}$
16. $-\Large\frac{29}{v}-\Large\frac{26}{v}$
    Show Solution

    $-\Large\frac{55}{v}$
17. $\Large\frac{6c}{7}-\Large\frac{5c}{7}$
18. $\Large\frac{12d}{11}-\Large\frac{9d}{11}$
    Show Solution

    $\Large\frac{3d}{11}$
19. $\Large\frac{-4r}{13}-\Large\frac{5r}{13}$
20. $\Large\frac{-7s}{3}-\Large\frac{7s}{3}$
    Show Solution

    $-\Large\frac{14s}{3}$
21. $-\Large\frac{3}{5}-\Large\left(-\frac{4}{5}\right)$
22. $-\Large\frac{3}{7}-\Large\left(-\frac{5}{7}\right)$
    Show Solution

    $\Large\frac{2}{7}$
23. $-\Large\frac{7}{9}-\Large\left(-\frac{5}{9}\right)$
24. $-\Large\frac{8}{11}-\Large\left(-\frac{5}{11}\right)$
    Show Solution

    $-\Large\frac{3}{11}$

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

1. $-\Large\frac{5}{18}\cdot\Large\frac{9}{10}$
2. $-\Large\frac{3}{14}\cdot\Large\frac{7}{12}$
   Show Solution

   $-\Large\frac{1}{8}$
3. $\Large\frac{n}{5}-\Large\frac{4}{5}$
4. $\Large\frac{6}{11}-\Large\frac{s}{11}$
   Show Solution

   $\Large\frac{6-s}{11}$
5. $-\Large\frac{7}{24}+\Large\frac{2}{24}$
6. $-\Large\frac{5}{18}+\Large\frac{1}{18}$
   Show Solution

   $-\Large\frac{2}{9}$
7. $\Large\frac{8}{15}\div\Large\frac{12}{5}$
8. $\Large\frac{7}{12}\div\Large\frac{9}{28}$
   Show Solution

   $\Large\frac{49}{27}$

Adding and Subtracting Fractions with Different Denominators

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

1. $\Large\frac{2}{3}\normalsize\text{ and }\Large\frac{3}{4}$
2. $\Large\frac{3}{4}\normalsize\text{ and }\Large\frac{2}{5}$
   Show Solution

   20
3. $\Large\frac{7}{12}\normalsize\text{ and }\Large\frac{5}{8}$
4. $\Large\frac{9}{16}\normalsize\text{ and }\Large\frac{7}{12}$
   Show Solution

   48
5. $\Large\frac{13}{30}\normalsize\text{ and }\Large\frac{25}{42}$
6. $\Large\frac{23}{30}\normalsize\text{ and }\Large\frac{5}{48}$
   Show Solution

   240
7. $\Large\frac{21}{35}\normalsize\text{ and }\Large\frac{39}{56}$
8. $\Large\frac{18}{35}\normalsize\text{ and }\Large\frac{33}{49}$
   Show Solution

   245
9. $\Large\frac{2}{3}\normalsize\text{,}\Large\frac{1}{6}\normalsize,\text{ and }\Large\frac{3}{4}$
10. $\Large\frac{2}{3}\normalsize\text{,}\Large\frac{1}{4}\normalsize,\text{ and }\Large\frac{3}{5}$
    Show Solution

    60

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.

1. $\Large\frac{1}{3}\normalsize\text{ and }\Large\frac{1}{4}\normalsize,\text{LCD}=12$
2. $\Large\frac{1}{4}\normalsize\text{ and }\Large\frac{1}{5}\normalsize,\text{LCD}=20$
   Show Solution

   $\Large\frac{5}{20},\Large\frac{4}{20}$
3. $\Large\frac{5}{12}\normalsize\text{ and }\Large\frac{7}{8}\normalsize,\text{LCD}=24$
4. $\Large\frac{7}{12}\normalsize\text{ and }\Large\frac{5}{8}\normalsize,\text{LCD}=24$
   Show Solution

   $\Large\frac{14}{24},\Large\frac{15}{24}$
5. $\Large\frac{13}{16}\normalsize\text{ and }\text{-}\Large\frac{11}{12}\normalsize,\text{LCD}=48$
6. $\Large\frac{11}{16}\normalsize\text{ and }\text{-}\Large\frac{5}{12}\normalsize,\text{LCD}=48$
   Show Solution

   $\Large\frac{33}{48},-\Large\frac{20}{48}$
7. $\Large\frac{1}{3},\Large\frac{5}{6}\normalsize,\text{ and }\Large\frac{3}{4}\normalsize,\text{LCD}=12$
8. $\Large\frac{1}{3},\Large\frac{3}{4}\normalsize,\text{ and }\Large\frac{3}{5}\normalsize,\text{LCD}=60$
   Show Solution

   $\Large\frac{20}{60},\Large\frac{45}{60},\Large\frac{36}{60}$

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

1. $\Large\frac{1}{3}+\Large\frac{1}{5}$
2. $\Large\frac{1}{4}+\Large\frac{1}{5}$
   Show Solution

   $\Large\frac{9}{20}$
3. $\Large\frac{1}{2}+\Large\frac{1}{7}$
4. $\Large\frac{1}{3}+\Large\frac{1}{8}$
   Show Solution

   $\Large\frac{11}{24}$
5. $\Large\frac{1}{3}-\Large\left(-\frac{1}{9}\right)$
6. $\Large\frac{1}{4}-\Large\left(-\frac{1}{8}\right)$
   Show Solution

   $\Large\frac{3}{8}$
7. $\Large\frac{1}{5}-\Large\left(-\frac{1}{10}\right)$
8. $\Large\frac{1}{2}-\Large\left(-\frac{1}{6}\right)$
   Show Solution

   $\Large\frac{2}{3}$
9. $\Large\frac{2}{3}+\Large\frac{3}{4}$
10. $\Large\frac{3}{4}+\Large\frac{2}{5}$
    Show Solution

    $\Large\frac{23}{20}$
11. $\Large\frac{7}{12}+\Large\frac{5}{8}$
12. $\Large\frac{5}{12}+\Large\frac{3}{8}$
    Show Solution

    $\Large\frac{19}{24}$
13. $\Large\frac{7}{12}-\Large\frac{9}{16}$
14. $\Large\frac{7}{16}-\Large\frac{5}{12}$
    Show Solution

    $\Large\frac{1}{48}$
15. $\Large\frac{11}{12}-\Large\frac{3}{8}$
16. $\Large\frac{5}{8}-\Large\frac{7}{12}$
    Show Solution

    $\Large\frac{1}{24}$
17. $\Large\frac{2}{3}-\Large\frac{3}{8}$
18. $\Large\frac{5}{6}-\Large\frac{3}{4}$
    Show Solution

    $\Large\frac{1}{12}$
19. $-\Large\frac{11}{30}+\Large\frac{27}{40}$
20. $-\Large\frac{9}{20}+\Large\frac{17}{30}$
    Show Solution

    $\Large\frac{7}{60}$
21. $-\Large\frac{13}{30}+\Large\frac{25}{42}$
22. $-\Large\frac{23}{30}+\Large\frac{5}{48}$
    Show Solution

    $-\Large\frac{53}{80}$
23. $-\Large\frac{39}{56}-\Large\frac{22}{35}$
24. $-\Large\frac{33}{49}-\Large\frac{18}{35}$
    Show Solution

    $-\Large\frac{291}{245}$
25. $-\Large\frac{2}{3}-\Large\left(-\frac{3}{4}\right)$
26. $-\Large\frac{3}{4}-\Large\left(-\frac{4}{5}\right)$
    Show Solution

    $\Large\frac{1}{20}$
27. $-\Large\frac{9}{16}-\Large\left(-\frac{4}{5}\right)$
28. $-\Large\frac{7}{20}-\Large\left(-\frac{5}{8}\right)$
    Show Solution

    $\Large\frac{11}{40}$
29. $1+\Large\frac{7}{8}$
30. $1+\Large\frac{5}{6}$
    Show Solution

    $\Large\frac{11}{6}$
31. $1-\Large\frac{5}{9}$
32. $1-\Large\frac{3}{10}$
    Show Solution

    $\Large\frac{7}{10}$
33. $\Large\frac{x}{3}+\Large\frac{1}{4}$
34. $\Large\frac{y}{2}+\Large\frac{2}{3}$
    Show Solution

    $\Large\frac{3y+4}{6}$
35. $\Large\frac{y}{4}-\Large\frac{3}{5}$
36. $\Large\frac{x}{5}-\Large\frac{1}{4}$
    Show Solution

    $\Large\frac{4x - 5}{20}$

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

1. $\Large\frac{3}{4}+\Large\frac{1}{6}$
2. $\Large\frac{3}{4}\div\Large\frac{1}{6}$
3. $\Large\frac{2}{3}+\Large\frac{1}{6}$
   Show Solution

   $\Large\frac{5}{6}$
4. $\Large\frac{2}{3}\div\Large\frac{1}{6}$
   Show Solution

   $4$
5. $\text{-}\Large\frac{2}{5}-\Large\frac{1}{8}$
6. $\text{-}\Large\frac{2}{5}\cdot\Large\frac{1}{8}$
7. $\text{-}\Large\frac{4}{5}-\Large\frac{1}{8}$
   Show Solution

   $-\Large\frac{37}{40}$
8. $\text{-}\Large\frac{4}{5}\cdot\Large\frac{1}{8}$
   Show Solution

   $-\Large\frac{1}{10}$
9. $\Large\frac{5}{n}\div\Large\frac{8}{15}$
10. $\Large\frac{5}{n}-\Large\frac{8}{15}$
11. $\Large\frac{3}{a}\div\Large\frac{7}{12}$
    Show Solution

    $\Large\frac{9a}{14}$
12. $\Large\frac{3}{a}-\Large\frac{7}{12}$
    Show Solution

    $\Large\frac{9a - 14}{24}$
13. $\Large\frac{9}{10}\cdot\Large\left(-\frac{11}{d}\right)$
14. $\Large\frac{9}{10}+\Large\left(-\frac{11}{d}\right)$
15. $\Large\frac{4}{15}\cdot\Large\left(-\frac{5}{q}\right)$
    Show Solution

    $-\Large\frac{4}{3q}$
16. $\Large\frac{4}{15}+\Large\left(-\frac{5}{q}\right)$
    Show Solution

    $\Large\frac{12 - 25q}{45}$
17. $-\Large\frac{3}{8}\div\Large\left(-\frac{3}{10}\right)$
18. $-\Large\frac{5}{12}\div\Large\left(-\frac{5}{9}\right)$
19. $-\Large\frac{3}{8}+\Large\frac{5}{12}$
20. $-\Large\frac{1}{8}+\Large\frac{7}{12}$
    Show Solution

    $\Large\frac{11}{24}$
21. $\Large\frac{5}{6}-\Large\frac{1}{9}$
22. $\Large\frac{5}{9}-\Large\frac{1}{6}$
    Show Solution

    $\Large\frac{7}{18}$
23. $\Large\frac{3}{8}\cdot\Large\left(-\frac{10}{21}\right)$
24. $\Large\frac{7}{12}\cdot\Large\left(-\frac{8}{35}\right)$
    Show Solution

    $-\Large\frac{2}{15}$
25. $-\Large\frac{7}{15}-\Large\frac{y}{4}$
26. $-\Large\frac{3}{8}-\Large\frac{x}{11}$
    Show Solution

    $\Large\frac{-33 - 8x}{88}$
27. $\Large\frac{11}{12a}\cdot\Large\frac{9a}{16}$
28. $\Large\frac{10y}{13}\cdot\Large\frac{8}{15y}$
    Show Solution

    $\Large\frac{16}{39}$

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

1. $\displaystyle\frac{{\Large\left(\frac{1}{5}\right)}^{2}}{2+{3}^{2}}$
2. $\displaystyle\frac{{\Large\left(\frac{1}{3}\right)}^{2}}{5+{2}^{2}}$
   Show Solution

   $\displaystyle\frac{1}{81}$
3. $\displaystyle\frac{{2}^{3}+{4}^{2}}{{\Large\left(\frac{2}{3}\right)}^{2}}$
4. $\displaystyle\frac{{3}^{3}-{3}^{2}}{{\Large\left(\frac{3}{4}\right)}^{2}}$
   Show Solution

   32
5. $\displaystyle\frac{{\Large\left(\frac{3}{5}\right)}^{2}}{{\Large\left(\frac{3}{7}\right)}^{2}}$
6. $\displaystyle\frac{{\Large\left(\frac{3}{4}\right)}^{2}}{{\Large\left(\frac{5}{8}\right)}^{2}}$
   Show Solution

   $\displaystyle\frac{36}{25}$
7. $\displaystyle\frac{2}{\Large\frac{1}{3}+\frac{1}{5}}$
8. $\displaystyle\frac{5}{\Large\frac{1}{4}+\frac{1}{3}}$
   Show Solution

   $\displaystyle\frac{60}{7}$
9. $\displaystyle\frac{\Large\frac{2}{3}+\frac{1}{2}}{\Large\frac{3}{4}-\frac{2}{3}}$
10. $\displaystyle\frac{\Large\frac{3}{4}+\frac{1}{2}}{\Large\frac{5}{6}-\frac{2}{3}}$
    Show Solution

    $\displaystyle\frac{15}{2}$
11. $\displaystyle\frac{\Large\frac{7}{8}-\frac{2}{3}}{\Large\frac{1}{2}+\frac{3}{8}}$
12. $\displaystyle\frac{\Large\frac{3}{4}-\frac{3}{5}}{\Large\frac{1}{4}+\frac{2}{5}}$
    Show Solution

    $\displaystyle\frac{3}{13}$

Mixed Practice

In the following exercises, simplify.

1. $\Large\frac{1}{2}+\frac{2}{3}\cdot \frac{5}{12}$
2. $\Large\frac{1}{3}+\frac{2}{5}\cdot \frac{3}{4}$
   Show Solution

   $\Large\frac{19}{30}$
3. $1-\Large\frac{3}{5}\div \frac{1}{10}$
4. $1-\Large\frac{5}{6}\div \frac{1}{12}$
   Show Solution

   −9
5. $\Large\frac{2}{3}+\frac{1}{6}+\frac{3}{4}$
6. $\Large\frac{2}{3}+\frac{1}{4}+\frac{3}{5}$
   Show Solution

   $\Large\frac{91}{60}$
7. $\Large\frac{3}{8}-\frac{1}{6}+\frac{3}{4}$
8. $\Large\frac{2}{5}+\frac{5}{8}-\frac{3}{4}$
   Show Solution

   $\Large\frac{11}{40}$
9. $12\Large\left(\frac{9}{20}-\frac{4}{15}\right)$
10. $8\Large\left(\frac{15}{16}-\frac{5}{6}\right)$
    Show Solution

    $\Large\frac{5}{6}$
11. $\Large\frac{\LARGE\frac{5}{8}+\LARGE\frac{1}{6}}{\LARGE\frac{19}{24}}$
12. $\LARGE\frac{\LARGE\frac{1}{6}+\LARGE\frac{3}{10}}{\LARGE\frac{14}{30}}$
    Show Solution

    1
13. $\Large\left(\frac{5}{9}+\frac{1}{6}\right)\div \left(\frac{2}{3}-\frac{1}{2}\right)$
14. $\Large\left(\frac{3}{4}+\frac{1}{6}\right)\div \left(\frac{5}{8}-\frac{1}{3}\right)$
    Show Solution

    $\Large\frac{22}{7}$

Mixed Practice

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

Exercise 1

$x+\Large\frac{1}{2}$ when

1. $x=-\Large\frac{1}{8}$
2. $x=-\Large\frac{1}{2}$

Exercise 2

$x+\Large\frac{2}{3}$ when

1. $x=-\Large\frac{1}{6}$
2. $x=-\Large\frac{5}{3}$

Exercise 3

$x+\Large\left(-\frac{5}{6}\right)$ when

1. $x=\Large\frac{1}{3}$
2. $x=-\Large\frac{1}{6}$

Exercise 4

$x+\Large\left(-\frac{11}{12}\right)$ when

1. $x=\Large\frac{11}{12}$
2. $x=\Large\frac{3}{4}$

Exercise 5

$x-\Large\frac{2}{5}$ when

1. $x=\Large\frac{3}{5}$
2. $x=-\Large\frac{3}{5}$

Exercise 6

$x-\Large\frac{1}{3}$ when

1. $x=\Large\frac{2}{3}$
2. $x=-\Large\frac{2}{3}$

Exercise 7

$\Large\frac{7}{10}\normalsize-w$ when

1. $w=\Large\frac{1}{2}$
2. $w=-\Large\frac{1}{2}$

Exercise 8

$\Large\frac{5}{12}\normalsize-w$ when

1. $w=\Large\frac{1}{4}$
2. $w=-\Large\frac{1}{4}$

Mixed Practice

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

1. $4{p}^{2}q$ when $p=-\Large\frac{1}{2}\normalsize\text{ and }q=\Large\frac{5}{9}$
2. $5{m}^{2}n\text{ when }m=-\Large\frac{2}{5}\normalsize\text{ and }n=\Large\frac{1}{3}$
   Show Solution

   $\Large\frac{4}{15}$
3. $2{x}^{2}{y}^{3}\text{ when }x=-\Large\frac{2}{3}\normalsize\text{ and }y=-\Large\frac{1}{2}$
4. $8{u}^{2}{v}^{3}\text{ when }u=-\Large\frac{3}{4}\normalsize\text{ and }v=-\Large\frac{1}{2}$
   Show Solution

   $-\Large\frac{9}{16}$
5. $\Large\frac{u+v}{w}\normalsize\text{ when }u=-4,v=-8,w=2$
6. $\Large\frac{m+n}{p}\normalsize\text{ when }m=-6,n=-2,p=4$
   Show Solution

   −2
7. $\Large\frac{a+b}{a-b}\normalsize\text{ when }a=-3,b=8$
8. $\Large\frac{r-s}{r+s}\normalsize\text{ when }r=10,s=-5$
   Show Solution

   3

Adding and Subtracting Mixed Numbers

Model Addition of Mixed Numbers

In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.

1. $1\Large\frac{1}{5}\normalsize+3\Large\frac{1}{5}$
2. $2\Large\frac{1}{3}\normalsize+1\Large\frac{1}{3}$
   Show Solution

   
3. $3\Large\frac{2}{3}$
4. $1\Large\frac{3}{8}\normalsize+1\Large\frac{7}{8}$
5. $1\Large\frac{5}{6}\normalsize+1\Large\frac{5}{6}$
   Show Solution

   
6. $3\Large\frac{2}{3}$

Add Mixed Numbers with a Common Denominator

In the following exercises, add.

1. $5\Large\frac{1}{3}\normalsize+6\Large\frac{1}{3}$
2. $2\Large\frac{4}{9}\normalsize+5\Large\frac{1}{9}$
   Show Solution

   $7\Large\frac{5}{9}$
3. $4\Large\frac{5}{8}\normalsize+9\Large\frac{3}{8}$
4. $7\Large\frac{9}{10}\normalsize+3\Large\frac{1}{10}$
   Show Solution

   11
5. $3\Large\frac{4}{5}\normalsize+6\Large\frac{4}{5}$
6. $9\Large\frac{2}{3}\normalsize+1\Large\frac{2}{3}$
   Show Solution

   $11\Large\frac{1}{3}$
7. $6\Large\frac{9}{10}\normalsize+8\Large\frac{3}{10}$
8. $8\Large\frac{4}{9}\normalsize+2\Large\frac{8}{9}$
   Show Solution

   $11\Large\frac{1}{3}$

Model Subtraction of Mixed Numbers

In the following exercises, use a model to find the difference. Draw a picture to illustrate your model.

1. $1\Large\frac{1}{6}-\Large\frac{5}{6}$
2. $1\Large\frac{1}{8}-\Large\frac{5}{8}$
   Show Solution

   
3. $\Large\frac{1}{2}$

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.

1. $2\Large\frac{7}{8}\normalsize-1\Large\frac{3}{8}$
2. $2\Large\frac{7}{12}\normalsize-1\Large\frac{5}{12}$
   Show Solution

   $1\Large\frac{1}{6}$
3. $8\Large\frac{17}{20}\normalsize-4\Large\frac{9}{20}$
4. $19\Large\frac{13}{15}\normalsize-13\Large\frac{7}{15}$
   Show Solution

   $6\Large\frac{2}{5}$
5. $8\Large\frac{3}{7}\normalsize-4\Large\frac{4}{7}$
6. $5\Large\frac{2}{9}\normalsize-3\Large\frac{4}{9}$
   Show Solution

   $1\Large\frac{7}{9}$
7. $2\Large\frac{5}{8}\normalsize-1\Large\frac{7}{8}$
8. $2\Large\frac{5}{12}\normalsize-1\Large\frac{7}{12}$
   Show Solution

   $\Large\frac{5}{6}$

Add and Subtract Mixed Numbers with Different Denominators

In the following exercises, write the sum or difference as a mixed number in simplified form.

1. $3\Large\frac{1}{4}\normalsize+6\Large\frac{1}{3}$
2. $2\Large\frac{1}{6}\normalsize+5\Large\frac{3}{4}$
   Show Solution

   $7\Large\frac{11}{12}$
3. $1\Large\frac{5}{8}\normalsize+4\Large\frac{1}{2}$
4. $7\Large\frac{2}{3}\normalsize+8\Large\frac{1}{2}$
   Show Solution

   $16\Large\frac{1}{6}$
5. $9\Large\frac{7}{10}\normalsize-2\Large\frac{1}{3}$
6. $6\Large\frac{4}{5}\normalsize-1\Large\frac{1}{4}$
   Show Solution

   $5\Large\frac{11}{20}$
7. $2\Large\frac{2}{3}\normalsize-3\Large\frac{1}{2}$
8. $2\Large\frac{7}{8}\normalsize-4\Large\frac{1}{3}$
   Show Solution

   $-1\Large\frac{11}{24}$

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

1. $2\Large\frac{5}{8}\normalsize\cdot 1\Large\frac{3}{4}$
2. $1\Large\frac{2}{3}\normalsize\cdot 4\Large\frac{1}{6}$
   Show Solution

   $6\Large\frac{17}{18}$
3. $\Large\frac{2}{7}+\Large\frac{4}{7}$
4. $\Large\frac{2}{9}+\Large\frac{5}{9}$
   Show Solution

   $\Large\frac{7}{9}$
5. $1\Large\frac{5}{12}\div\Large\frac{1}{12}$
6. $2\Large\frac{3}{10}\div\Large\frac{1}{10}$
   Show Solution

   23
7. $13\Large\frac{5}{12}\normalsize-9\Large\frac{7}{12}$
8. $15\Large\frac{5}{8}\normalsize-6\Large\frac{7}{8}$
   Show Solution

   $8\Large\frac{3}{4}$
9. $\Large\frac{5}{9}-\Large\frac{4}{9}$
10. $\Large\frac{11}{15}-\Large\frac{7}{15}$
    Show Solution

    $\Large\frac{4}{15}$
11. $4-\Large\frac{3}{4}$
12. $6-\Large\frac{2}{5}$
    Show Solution

    $5\Large\frac{3}{5}$
13. $\Large\frac{9}{20}\div\Large\frac{3}{4}$
14. $\Large\frac{7}{24}\div\Large\frac{14}{3}$
    Show Solution

    $\Large\frac{1}{16}$
15. $9\Large\frac{6}{11}\normalsize+7\Large\frac{10}{11}$
16. $8\Large\frac{5}{13}\normalsize+4\Large\frac{9}{13}$
    Show Solution

    $13\Large\frac{1}{13}$
17. $3\Large\frac{2}{5}\normalsize+5\Large\frac{3}{4}$
18. $2\Large\frac{5}{6}\normalsize+4\Large\frac{1}{5}$
    Show Solution

    $7\Large\frac{1}{30}$
19. $\Large\frac{8}{15}\cdot\Large\frac{10}{19}$
20. $\Large\frac{5}{12}\cdot\Large\frac{8}{9}$
    Show Solution

    $\Large\frac{10}{27}$
21. $6\Large\frac{7}{8}\normalsize-2\Large\frac{1}{3}$
22. $6\Large\frac{5}{9}\normalsize-4\Large\frac{2}{5}$
    Show Solution

    $2\Large\frac{7}{45}$
23. $5\Large\frac{2}{9}\normalsize-4\Large\frac{4}{5}$
24. $4\Large\frac{3}{8}\normalsize-3\Large\frac{2}{3}$
    Show Solution

    $\Large\frac{17}{24}$

Solving Equations That Contain Fractions

Determine Whether a Fraction is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

Exercise 1

$x-\Large\frac{2}{5}=\Large\frac{1}{10}\normalsize\text{:}$

1. $x=1$
2. $x=\Large\frac{1}{2}$
3. $x=-\Large\frac{1}{2}$

Exercise 2

$y-\Large\frac{1}{3}=\Large\frac{5}{12}\normalsize\text{:}$

1. $y=1$
2. $y=\Large\frac{3}{4}$
3. $y=-\Large\frac{3}{4}$

Exercise 3

$h+\Large\frac{3}{4}=\Large\frac{2}{5}\normalsize\text{:}$

1. $h=1$
2. $h=\Large\frac{7}{20}$
3. $h=-\Large\frac{7}{20}$

Exercise 4

$k+\Large\frac{2}{5}=\Large\frac{5}{6}\normalsize\text{:}$

1. $k=1$
2. $k=\Large\frac{13}{30}$
3. $k=-\Large\frac{13}{30}$

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

1. $y+\Large\frac{1}{3}=\Large\frac{4}{3}$
2. $m+\Large\frac{3}{8}=\Large\frac{7}{8}$
   Show Solution

   $m=\Large\frac{1}{2}$
3. $f+\Large\frac{9}{10}=\Large\frac{2}{5}$
4. $h+\Large\frac{5}{6}=\Large\frac{1}{6}$
   Show Solution

   $h=-\Large\frac{2}{3}$
5. $a-\Large\frac{5}{8}=-\Large\frac{7}{8}$
6. $c-\Large\frac{1}{4}=-\Large\frac{5}{4}$
   Show Solution

   c = −1
7. $x-\Large\left(-\frac{3}{20}\right)=-\Large\frac{11}{20}$
8. $z-\Large\left(-\frac{5}{12}\right)=-\Large\frac{7}{12}$
   Show Solution

   z = −1
9. $n-\Large\frac{1}{6}=\Large\frac{3}{4}$
10. $p-\Large\frac{3}{10}=\Large\frac{5}{8}$
    Show Solution

    $p=\Large\frac{37}{40}$
11. $s+\Large\left(-\frac{1}{2}\right)=-\Large\frac{8}{9}$
12. $k+\Large\left(-\frac{1}{3}\right)=-\Large\frac{4}{5}$
    Show Solution

    $k=-\Large\frac{7}{15}$
13. $5j=17$
14. $7k=18$
    Show Solution

    $k=\Large\frac{18}{7}$
15. $-4w=26$
16. $-9v=33$
    Show Solution

    $v=-\Large\frac{11}{3}$

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

1. $\Large\frac{f}{4}\normalsize=-20$
2. $\Large\frac{b}{3}\normalsize=-9$
   Show Solution

   b = −27
3. $\Large\frac{y}{7}\normalsize=-21$
4. $\Large\frac{x}{8}\normalsize=-32$
   Show Solution

   x = −256
5. $\Large\frac{p}{-5}\normalsize=-40$
6. $\Large\frac{q}{-4}\normalsize=-40$
   Show Solution

   q = 160
7. $\Large\frac{r}{-12}\normalsize=-6$
8. $\Large\frac{s}{-15}\normalsize=-3$
   Show Solution

   s = 45
9. $-x=23$
10. $-y=42$
    Show Solution

    y = −42
11. $-h=-\Large\frac{5}{12}$
12. $-k=-\Large\frac{17}{20}$
    Show Solution

    $k=\Large\frac{17}{20}$
13. $\Large\frac{4}{5}\normalsize n=20$
14. $\Large\frac{3}{10}\normalsize p=30$
    Show Solution

    p = 100
15. $\Large\frac{3}{8}\normalsize q=-48$
16. $\Large\frac{5}{2}\normalsize m=-40$
    Show Solution

    m = −16
17. $-\Large\frac{2}{9}\normalsize a=16$
18. $-\Large\frac{3}{7}\normalsize b=9$
    Show Solution

    b = −21
19. $-\Large\frac{6}{11}\normalsize u=-24$
20. $-\Large\frac{5}{12}\normalsize v=-15$
    Show Solution

    v = 36

Mixed Practice

In the following exercises, solve.

1. $3x=0$
2. $8y=0$
   Show Solution

   y = 0
3. $4f=\Large\frac{4}{5}$
4. $7g=\Large\frac{7}{9}$
   Show Solution

   $g=\Large\frac{1}{9}$
5. $p+\Large\frac{2}{3}=\Large\frac{1}{12}$
6. $q+\Large\frac{5}{6}=\Large\frac{1}{12}$
   Show Solution

   $q=-\Large\frac{3}{4}$
7. $\Large\frac{7}{8}\normalsize m=\Large\frac{1}{10}$
8. $\Large\frac{1}{4}\normalsize n=\Large\frac{7}{10}$
   Show Solution

   $n=\Large\frac{14}{5}$
9. $-\Large\frac{2}{5}\normalsize=x+\Large\frac{3}{4}$
10. $-\Large\frac{2}{3}\normalsize=y+\Large\frac{3}{8}$
    Show Solution

    $y=-\Large\frac{25}{24}$
11. $\Large\frac{11}{20}\normalsize=\text{-}\mathit{\text{f}}$
12. $\Large\frac{8}{15}\normalsize=\text{-}\mathit{\text{d}}$
    Show Solution

    $d=-\Large\frac{8}{15}$

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

1. $n$ divided by eight is $-16$.
2. $n$ divided by six is $-24$.
   Show Solution

   $\Large\frac{n}{6}\normalsize=-24;n=-144$
3. $m$ divided by $-9$ is $-7$.
4. $m$ divided by $-7$ is $-8$.
   Show Solution

   $\Large\frac{m}{-7}\normalsize=-8;m=56$
5. The quotient of $f$ and $-3$ is $-18$.
6. The quotient of $f$ and $-4$ is $-20$.
   Show Solution

   $\Large\frac{f}{-4}\normalsize=-20;f=80$
7. The quotient of $g$ and twelve is $8$.
8. The quotient of $g$ and nine is $14$.
   Show Solution

   $\Large\frac{g}{9}\normalsize=14;g=126$
9. Three-fourths of $q$ is $12$.
10. Two-fifths of $q$ is $20$.
    Show Solution

    $\Large\frac{2}{5}\normalsize q=20;q=50$
11. Seven-tenths of $p$ is $-63$.
12. Four-ninths of $p$ is $-28$.
    Show Solution

    $\Large\frac{4}{9}\normalsize p=-28;p=-63$
13. $m$ divided by $4$ equals negative $6$.
14. The quotient of $h$ and $2$ is $43$.
    Show Solution

    $\Large\frac{h}{2}\normalsize=43$
15. Three-fourths of $z$ is the same as $15$.
16. The quotient of $a$ and $\Large\frac{2}{3}$ is $\Large\frac{3}{4}$.
    Show Solution

    $\Large\frac{a}{\LARGE\frac{2}{3}}=\Large\frac{3}{4}$
17. The sum of five-sixths and $x$ is $\Large\frac{1}{2}$.
18. The sum of three-fourths and $x$ is $\Large\frac{1}{8}$.
    Show Solution

    $\Large\frac{3}{4}\normalsize+x=\Large\frac{1}{8}\normalsize ;x=-\Large\frac{5}{8}$
19. The difference of $y$ and one-fourth is $-\Large\frac{1}{8}$.
20. The difference of $y$ and one-third is $-\Large\frac{1}{6}$.
    Show Solution

    $y-\Large\frac{1}{3}=-\Large\frac{1}{6}\normalsize ;y=\Large\frac{1}{6}$

Chapter Review Exercises