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# 4.E: Fractions (Exercises)

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### 4.1 - Visualize Fractions

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

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

In the following exercises, convert the improper fraction to a mixed number.

1. $$\frac{58}{15}$$
2. $$\frac{63}{11}$$

In the following exercises, convert the mixed number to an improper fraction.

1. $$12 \frac{1}{4}$$
2. $$9 \frac{4}{5}$$
3. Find three fractions equivalent to $$\frac{2}{5}$$. Show your work, using figures or algebra.
4. Find three fractions equivalent to $$− \frac{4}{3}$$. Show your work, using figures or algebra.

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

1. $$\frac{5}{8}, \frac{4}{3}, 3 \frac{3}{4}$$, 4
2. $$\frac{1}{4}, − \frac{1}{4}, 1 \frac{1}{3}, −1 \frac{1}{3}, \frac{7}{2}, − \frac{7}{2}$$

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

1. −1___$$− \frac{2}{5}$$
2. $$−2 \frac{1}{2}$$___−3

### 4.2 - Multiply and Divide Fractions

In the following exercises, simplify.

1. $$− \frac{63}{84}$$
2. $$− \frac{90}{120}$$
3. $$− \frac{14a}{14b}$$
4. $$− \frac{8x}{8y}$$

In the following exercises, multiply.

1. $$\frac{2}{5} \cdot \frac{8}{13}$$
2. $$− \frac{1}{3} \cdot \frac{12}{7}$$
3. $$\frac{2}{9} \cdot \left(− \dfrac{45}{32}\right)$$
4. 6m $$\cdot \frac{4}{11}$$
5. $$− \frac{1}{4}$$ (−32)
6. $$3 \frac{1}{5} \cdot 1 \frac{7}{8}$$

In the following exercises, find the reciprocal.

1. $$\frac{2}{9}$$
2. $$\frac{15}{4}$$
3. 3
4. $$− \frac{1}{4}$$
5. Fill in the chart.
Opposite Absolute Value Reciprocal
$$- \frac{5}{13}$$
$$\frac{3}{10}$$
$$\frac{9}{4}$$
-12

In the following exercises, divide.

1. $$\frac{2}{3} \div \frac{1}{6}$$
2. $$\left(− \dfrac{3x}{5}\right) \div \left(− \dfrac{2y}{3}\right)$$
3. $$\frac{4}{5} \div$$ 3
4. 8 $$\div 2 \frac{2}{3}$$
5. $$8 \frac{2}{3} \div 1 \frac{1}{12}$$

### 4.3 - Multiply and Divide Mixed Numbers and Complex Fractions

In the following exercises, perform the indicated operation.

1. $$3 \frac{1}{5} \cdot 1 \frac{7}{8}$$
2. $$−5 \frac{7}{12} \cdot 4 \frac{4}{11}$$
3. 8 $$\div 2 \frac{2}{3}$$
4. $$8 \frac{2}{3} \div 1 \frac{1}{12}$$

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

1. the quotient of 8 and y
2. the quotient of V and the difference of h and 6

In the following exercises, simplify the complex fraction.

1. $$\frac{\frac{5}{8}}{\frac{4}{5}}$$
2. $$\frac{\frac{8}{9}}{−4}$$
3. $$\frac{\frac{n}{4}}{\frac{3}{8}}$$
4. $$\frac{−1 \frac{5}{6}}{− \frac{1}{12}}$$

In the following exercises, simplify.

1. $$\frac{5 + 16}{5}$$
2. $$\frac{8 \cdot 4 − 5^{2}}{3 \cdot 12}$$
3. $$\frac{8 \cdot 7 + 5(8 − 10)}{9 \cdot 3 − 6 \cdot 4}$$

### 4.4 - Add and Subtract Fractions with Common Denominators

1. $$\frac{3}{8} + \frac{2}{8}$$
2. $$\frac{4}{5} + \frac{1}{5}$$
3. $$\frac{2}{5} + \frac{1}{5}$$
4. $$\frac{15}{32} + \frac{9}{32}$$
5. $$\frac{x}{10} + \frac{7}{10}$$

In the following exercises, subtract.

1. $$\frac{8}{11} − \frac{6}{11}$$
2. $$\frac{11}{12} − \frac{5}{12}$$
3. $$\frac{4}{5} − \frac{y}{5}$$
4. $$− \frac{31}{30} − \frac{7}{30}$$
5. $$\frac{3}{2} − \left(\dfrac{3}{2}\right)$$
6. $$\frac{11}{15} − \frac{5}{15} − \left(− \dfrac{2}{15}\right)$$

### 4.5 - Add and Subtract Fractions with Different Denominators

In the following exercises, find the least common denominator.

1. $$\frac{1}{3}$$ and $$\frac{1}{12}$$
2. $$\frac{1}{3}$$ and $$\frac{4}{5}$$
3. $$\frac{8}{15}$$ and $$\frac{11}{20}$$
4. $$\frac{3}{4}, \frac{1}{6}$$, and $$\frac{5}{10}$$

In the following exercises, change to equivalent fractions using the given LCD.

1. $$\frac{1}{3}$$ and $$\frac{1}{5}$$, LCD = 15
2. $$\frac{3}{8}$$ and $$\frac{5}{6}$$, LCD = 24
3. $$− \frac{9}{16}$$ and $$\frac{5}{12}$$, LCD = 48
4. $$\frac{1}{3}, \frac{3}{4}$$ and $$\frac{4}{5}$$, LCD = 60

In the following exercises, perform the indicated operations and simplify.

1. $$\frac{1}{5} + \frac{2}{3}$$
2. $$\frac{11}{12} − \frac{2}{3}$$
3. $$− \frac{9}{10} − \frac{3}{4}$$
4. $$− \frac{11}{36} − \frac{11}{20}$$
5. $$− \frac{22}{25} + \frac{9}{40}$$
6. $$\frac{y}{10} − \frac{1}{3}$$
7. $$\frac{2}{5} + \left(− \dfrac{5}{9}\right)$$
8. $$\frac{4}{11} \div \frac{2}{7d}$$
9. $$\frac{2}{5} + \left(− \dfrac{3n}{8}\right) \left(− \dfrac{2}{9n}\right)$$
10. $$\frac{\left(\dfrac{2}{3}\right)^{2}}{\left(\dfrac{5}{8}\right)^{2}}$$
11. $$\left(\dfrac{11}{12} + \dfrac{3}{8}\right) \div \left(\dfrac{5}{6} − \dfrac{1}{10}\right)$$

In the following exercises, evaluate.

1. y − $$\frac{4}{5}$$ when (a) y = $$− \frac{4}{5}$$ (b) y = $$\frac{1}{4}$$
2. 6mn2 when m = $$\frac{3}{4}$$ and n = $$− \frac{1}{3}$$

### 4.6 - Add and Subtract Mixed Numbers

In the following exercises, perform the indicated operation.

1. $$4 \frac{1}{3} + 9 \frac{1}{3}$$
2. $$6 \frac{2}{5} + 7 \frac{3}{5}$$
3. $$5 \frac{8}{11} + 2 \frac{4}{11}$$
4. $$3 \frac{5}{8} + 3 \frac{7}{8}$$
5. $$9 \frac{13}{20} − 4 \frac{11}{20}$$
6. $$2 \frac{3}{10} − 1 \frac{9}{10}$$
7. $$2 \frac{11}{12} − 1 \frac{7}{12}$$
8. $$8 \frac{6}{11} − 2 \frac{9}{11}$$

### 4.7 - Solve Equations with Fractions

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

1. x − $$\frac{1}{2}$$ = $$\frac{1}{6}$$:
1. x = 1
2. x = $$\frac{2}{3}$$
3. x = $$− \frac{1}{3}$$
2. y + $$\frac{3}{5}$$ = $$\frac{5}{9}$$:
1. y = $$\frac{1}{2}$$
2.  y = $$\frac{52}{45}$$
3. y = $$− \frac{2}{45}$$

In the following exercises, solve the equation.

1. n + $$\frac{9}{11}$$ = $$\frac{4}{11}$$
2. x − $$\frac{1}{6}$$ = $$\frac{7}{6}$$
3. h − $$\left(- \dfrac{7}{8}\right)$$ = $$− \frac{2}{5}$$
4. $$\frac{x}{5}$$ = −10
5. −z = 23

In the following exercises, translate and solve.

1. The sum of two-thirds and n is $$− \frac{3}{5}$$.
2. The difference of q and one-tenth is $$\frac{1}{2}$$.
3. The quotient of p and −4 is −8.
4. Three-eighths of y is 24.

### PRACTICE TEST

Convert the improper fraction to a mixed number.

1. $$\frac{19}{5}$$

Convert the mixed number to an improper fraction.

1. $$3 \frac{2}{7}$$

Locate the numbers on a number line.

1. $$\frac{1}{2}, 1 \frac{2}{3}, −2 \frac{3}{4}$$, and $$\frac{9}{4}$$

In the following exercises, simplify.

1. $$\frac{5}{20}$$
2. $$\frac{18r}{27s}$$
3. $$\frac{1}{3} \cdot \frac{3}{4}$$
4. $$\frac{3}{5} \cdot$$ 15
5. −36u$$\left(− \dfrac{4}{9}\right)$$
6. $$−5 \frac{7}{12} \cdot 4 \frac{4}{11}$$
7. $$− \frac{5}{6} \div \frac{5}{12}$$
8. $$\frac{7}{11} \div \left(− \dfrac{7}{11}\right)$$
9. $$\frac{9a}{10} \div \frac{15a}{8}$$
10. $$−6 \frac{2}{5} \div$$ 4
11. $$\left(−15 \dfrac{5}{6}\right) \div \left(−3 \dfrac{1}{6}\right)$$
12. $$\frac{−6}{\frac{6}{11}}$$
13. $$\frac{\frac{p}{2}}{\frac{q}{5}}$$
14. $$\frac{− \frac{4}{15}}{−2 \frac{2}{3}}$$
15. $$\frac{9^{2} − 4^{2}}{9 − 4}$$
16. $$\frac{2}{d} + \frac{9}{d}$$
17. $$− \frac{3}{13} + \left(− \dfrac{4}{13}\right)$$
18. $$− \frac{22}{25} + \frac{9}{40}$$
19. $$\frac{2}{5} + \left(− \dfrac{7}{5}\right)$$
20. $$− \frac{3}{10} + \left(- \dfrac{5}{8}\right)$$
21. $$− \frac{3}{4} \div \frac{x}{3}$$
22. $$\frac{2^{3} − 2^{2}}{\left(\dfrac{3}{4}\right)^{2}}$$
23. $$\frac{\frac{5}{14} + \frac{1}{8}}{\frac{9}{56}}$$

Evaluate.

1. x + $$\frac{1}{3}$$ when (a) x = $$\frac{2}{3}$$ (b) x = $$− \frac{5}{6}$$

In the following exercises, solve the equation.

1. y + $$\frac{3}{5}$$ = $$\frac{7}{5}$$
2. a − $$\frac{3}{10}$$ = $$− \frac{9}{10}$$
3. f + $$\left(− \dfrac{2}{3}\right)$$ = $$\frac{5}{12}$$
4. $$\frac{m}{−2}$$ = −16
5. $$− \frac{2}{3}$$c = 18
6. Translate and solve: The quotient of p and −4 is −8. Solve for p.