# 1.6E: Exercises


## Practice Makes Perfect

Find Equivalent Fractions

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

##### Exercise $$\PageIndex{1}$$

$$\dfrac{3}{8}$$

$$\dfrac{6}{16}$$, $$\dfrac{9}{24}$$, $$\dfrac{12}{32}$$, answers may vary

##### Exercise $$\PageIndex{2}$$

$$\dfrac{5}{8}$$

##### Exercise $$\PageIndex{3}$$

$$\dfrac{5}{9}$$

$$\dfrac{10}{18}$$, $$\dfrac{15}{27}$$, $$\dfrac{20}{36}$$, answers may vary

##### Exercise $$\PageIndex{4}$$

$$\dfrac{1}{8}$$

Simplify Fractions

In the following exercises, simplify.

##### Exercise $$\PageIndex{5}$$

$$-\dfrac{40}{88}$$

$$-\dfrac{5}{11}$$

##### Exercise $$\PageIndex{6}$$

$$-\dfrac{63}{99}$$

##### Exercise $$\PageIndex{7}$$

$$-\dfrac{108}{63}$$

$$-\dfrac{12}{7}$$

##### Exercise $$\PageIndex{8}$$

$$-\dfrac{104}{48}$$

##### Exercise $$\PageIndex{9}$$

$$\dfrac{120}{252}$$

$$\dfrac{10}{21}$$

##### Exercise $$\PageIndex{10}$$

$$\dfrac{182}{294}$$

##### Exercise $$\PageIndex{11}$$

$$-\dfrac{3x}{12y}$$

$$-\dfrac{x}{4y}$$

##### Exercise $$\PageIndex{12}$$

$$-\dfrac{4x}{32y}$$

##### Exercise $$\PageIndex{13}$$

$$\dfrac{14x^{2}}{21y}$$

$$\dfrac{2x^{2}}{3y}$$

##### Exercise $$\PageIndex{14}$$

$$\dfrac{24a}{32b^{2}}$$

Multiply Fractions

In the following exercises, multiply.

##### Exercise $$\PageIndex{15}$$

$$\dfrac{3}{4}\cdot \dfrac{9}{10}$$

$$\dfrac{27}{40}$$

##### Exercise $$\PageIndex{16}$$

$$\dfrac{4}{5}\cdot \dfrac{2}{7}$$

##### Exercise $$\PageIndex{17}$$

$$-\dfrac{2}{3}\cdot -\dfrac{3}{8}$$

$$\dfrac{1}{4}$$

##### Exercise $$\PageIndex{18}$$

$$-\dfrac{3}{4}\left(-\dfrac{4}{9}\right)$$

##### Exercise $$\PageIndex{19}$$

$$-\dfrac{5}{9}\cdot \dfrac{3}{10}$$

$$-\dfrac{1}{6}$$

##### Exercise $$\PageIndex{20}$$

$$-\dfrac{3}{8}\cdot \dfrac{4}{15}$$

##### Exercise $$\PageIndex{21}$$

$$\left(-\dfrac{14}{15}\right)\left(\dfrac{9}{20}\right)$$

$$-\dfrac{21}{50}$$

##### Exercise $$\PageIndex{22}$$

$$\left(-\dfrac{9}{10}\right)\left(\dfrac{25}{33}\right)$$

##### Exercise $$\PageIndex{23}$$

$$\left(-\dfrac{63}{84}\right)\left(-\dfrac{44}{90}\right)$$

$$\dfrac{11}{30}$$

##### Exercise $$\PageIndex{24}$$

$$\left(-\dfrac{63}{60}\right)\left(-\dfrac{40}{88}\right)$$

##### Exercise $$\PageIndex{25}$$

$$4\cdot \dfrac{5}{11}$$

$$\dfrac{20}{11}$$

##### Exercise $$\PageIndex{26}$$

$$5\cdot \dfrac{8}{3}$$

##### Exercise $$\PageIndex{27}$$

$$\dfrac{3}{7}\cdot 21n$$

9n

##### Exercise $$\PageIndex{28}$$

$$\dfrac{5}{6}\cdot 30m$$

##### Exercise $$\PageIndex{29}$$

$$-8\cdot\dfrac{17}{4}$$

−34

##### Exercise $$\PageIndex{30}$$

$$(-1)\left(-\dfrac{6}{7}\right)$$

Divide Fractions

In the following exercises, divide.

##### Exercise $$\PageIndex{31}$$

$$\dfrac{3}{4}\div \dfrac{2}{3}$$

$$\dfrac{9}{8}$$

##### Exercise $$\PageIndex{32}$$

$$\dfrac{4}{5}\div \dfrac{3}{4}$$

##### Exercise $$\PageIndex{33}$$

$$-\dfrac{7}{9}\div \left(-\dfrac{7}{4}\right)$$

1

##### Exercise $$\PageIndex{34}$$

$$-\dfrac{5}{6}\div \left(-\dfrac{5}{6}\right)$$

##### Exercise $$\PageIndex{35}$$

$$\dfrac{3}{4}\div \dfrac{x}{11}$$

$$\dfrac{33}{4x}$$

##### Exercise $$\PageIndex{36}$$

$$\dfrac{2}{5}\div \dfrac{y}{9}$$

##### Exercise $$\PageIndex{37}$$

$$\dfrac{5}{18}\div -\dfrac{15}{24}$$

$$-\dfrac{4}{9}$$

##### Exercise $$\PageIndex{38}$$

$$\dfrac{7}{18}\div \left(-\dfrac{14}{27}\right)$$

##### Exercise $$\PageIndex{39}$$

$$\dfrac{8u}{15} \div \dfrac{12v}{25}$$

$$\dfrac{10u}{9v}$$

##### Exercise $$\PageIndex{40}$$

$$\dfrac{12r}{25}\div \dfrac{18s}{35}$$

##### Exercise $$\PageIndex{41}$$

$$-5\div \dfrac{1}{2}$$

-10

##### Exercise $$\PageIndex{42}$$

$$-3\div \dfrac{1}{4}$$

##### Exercise $$\PageIndex{43}$$

$$\dfrac{3}{4}\div (-12)$$

$$\dfrac{1}{16}$$

##### Exercise $$\PageIndex{44}$$

$$-15\div -\dfrac{5}{3}$$

In the following exercises, simplify.

##### Exercise $$\PageIndex{45}$$

$$\dfrac{-\dfrac{8}{21}}{\dfrac{12}{35}}$$

$$-\dfrac{10}{9}$$

##### Exercise $$\PageIndex{46}$$

$$\dfrac{-\dfrac{9}{16}}{\dfrac{33}{40}}$$

##### Exercise $$\PageIndex{47}$$

$$\dfrac{-\dfrac{4}{5}}{2}$$

$$-\dfrac{2}{5}$$

##### Exercise $$\PageIndex{48}$$

$$\dfrac{5}{\dfrac{3}{10}}$$

##### Exercise $$\PageIndex{49}$$

$$\dfrac{\dfrac{m}{3}}{\dfrac{n}{2}}$$

$$\dfrac{2m}{3n}$$

##### Exercise $$\PageIndex{50}$$

$$\dfrac{-\dfrac{3}{8}}{-\dfrac{y}{12}}$$

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

##### Exercise $$\PageIndex{51}$$

$$\dfrac{22 + 3}{10}$$

$$\dfrac{5}{2}$$

##### Exercise $$\PageIndex{52}$$

$$\dfrac{19 - 4}{6}$$

##### Exercise $$\PageIndex{53}$$

$$\dfrac{48}{24 - 15}$$

$$\dfrac{16}{3}$$

##### Exercise $$\PageIndex{54}$$

$$\dfrac{46}{4 + 4}$$

##### Exercise $$\PageIndex{55}$$

$$\dfrac{-6 + 6}{8 + 4}$$

0

##### Exercise $$\PageIndex{56}$$

$$\dfrac{-6 + 3}{17 - 8}$$

##### Exercise $$\PageIndex{57}$$

$$\dfrac{4\cdot 3}{6\cdot 6}$$

$$\dfrac{1}{3}$$

##### Exercise $$\PageIndex{58}$$

$$\dfrac{6\cdot 6}{9\cdot 2}$$

##### Exercise $$\PageIndex{59}$$

$$\dfrac{4^{2} - 1}{25}$$

$$\dfrac{3}{5}$$

##### Exercise $$\PageIndex{60}$$

$$\dfrac{7^{2} + 1}{60}$$

##### Exercise $$\PageIndex{61}$$

$$\dfrac{8\cdot 3 + 2\cdot 9}{14 + 3}$$

$$2\dfrac{8}{17}$$

##### Exercise $$\PageIndex{62}$$

$$\dfrac{9\cdot 6 - 4\cdot 7}{22 + 3}$$

##### Exercise $$\PageIndex{63}$$

$$\dfrac{5\cdot 6 - 3\cdot 4}{4\cdot 5 -2\cdot 3}$$

$$\dfrac{9}{7}$$

##### Exercise $$\PageIndex{64}$$

$$\dfrac{8\cdot 9 - 7\cdot 6}{5\cdot 6 - 9\cdot 2}$$

##### Exercise $$\PageIndex{65}$$

$$\dfrac{5^{2} - 3^{2}}{3 - 5}$$

$$-8$$

##### Exercise $$\PageIndex{66}$$

$$\dfrac{6^{2} - 4^{2}}{4 - 6}$$

##### Exercise $$\PageIndex{67}$$

$$\dfrac{7\cdot 4 - 2(8 - 5)}{9\cdot 3 - 3\cdot 5}$$

$$\dfrac{11}{6}$$

##### Exercise $$\PageIndex{68}$$

$$\dfrac{9\cdot 7 - 3(12- 8)}{8\cdot 7- 6\cdot 6}$$

##### Exercise $$\PageIndex{69}$$

$$\dfrac{9(8-2)-3(15-7)}{6(7-1) - 3(17-9)}$$

$$\dfrac{5}{2}$$

##### Exercise $$\PageIndex{70}$$

$$\dfrac{8(9-2) - 4(14 - 9)}{7(8-3)-3(16 -9)}$$

Translate Phrases to Expressions with Fractions

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

##### Exercise $$\PageIndex{71}$$

the quotient of $$r$$ and the sum of $$s$$ and $$10$$

$$\dfrac{r}{s + 10}$$

##### Exercise $$\PageIndex{72}$$

the quotient of $$A$$ and the difference of $$3$$ and $$B$$

##### Exercise $$\PageIndex{73}$$

the quotient of the difference of $$x$$ and $$y$$, and $$−3$$

$$\dfrac{x - y}{-3}$$

##### Exercise $$\PageIndex{74}$$

the quotient of the sum of $$m$$ and $$n$$, and $$4q$$

## Everyday Math

##### Exercise $$\PageIndex{75}$$

Baking. A recipe for chocolate chip cookies calls for $$\frac{3}{4}$$ cup brown sugar. Imelda wants to double the recipe.

1. How much brown sugar will Imelda need? Show your calculation.
2. Measuring cups usually come in sets of $$\frac{1}{4}$$, $$\frac{1}{3}$$, $$\frac{1}{2}$$, and $$1$$ cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.
1. $$1\frac{1}{2}$$ cups
##### Exercise $$\PageIndex{76}$$

Baking. Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs $$\frac{2}{3}$$ cup of condensed milk.

1. How much condensed milk will Nina need? Show your calculation.
2. Measuring cups usually come in sets of $$\frac{1}{4}$$, $$\frac{1}{3}$$, $$\frac{1}{2}$$, and $$1$$ cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for $$4$$ pans of fudge.
##### Exercise $$\PageIndex{77}$$

Portions Don purchased a bulk package of candy that weighs $$5$$ pounds. He wants to sell the candy in little bags that hold $$\frac{1}{4}$$ pound. How many little bags of candy can he fill from the bulk package?

$$20$$ bags

##### Exercise $$\PageIndex{78}$$

Portions Kristen has $$\frac{3}{4}$$ yards of ribbon that she wants to cut into $$6$$ equal parts to make hair ribbons for her daughter’s $$6$$ dolls. How long will each doll’s hair ribbon be?

## Writing Exercises

##### Exercise $$\PageIndex{79}$$

Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into $$6$$ or $$8$$ slices. Would he prefer $$3$$ out of $$6$$ slices or $$4$$ out of $$8$$ slices? Rafael replied that since he wasn’t very hungry, he would prefer $$3$$ out of $$6$$ slices. Explain what is wrong with Rafael’s reasoning.

##### Exercise $$\PageIndex{80}$$

Give an example from everyday life that demonstrates how $$\dfrac{1}{2}\cdot \dfrac{2}{3}$$ is $$\dfrac{1}{3}$$.

##### Exercise $$\PageIndex{81}$$

Explain how you find the reciprocal of a fraction.

##### Exercise $$\PageIndex{82}$$

Explain how you find the reciprocal of a negative number.

## Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

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