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 $\PageIndex{1}$

$\dfrac{3}{8}$

Answer

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

Exercise $\PageIndex{2}$

$\dfrac{5}{8}$

Exercise $\PageIndex{3}$

$\dfrac{5}{9}$

Answer

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

Exercise $\PageIndex{4}$

$\dfrac{1}{8}$

Simplify Fractions

In the following exercises, simplify.

Exercise $\PageIndex{5}$

$-\dfrac{40}{88}$

Answer

$-\dfrac{5}{11}$

Exercise $\PageIndex{6}$

$-\dfrac{63}{99}$

Exercise $\PageIndex{7}$

$-\dfrac{108}{63}$

Answer

$-\dfrac{12}{7}$

Exercise $\PageIndex{8}$

$-\dfrac{104}{48}$

Exercise $\PageIndex{9}$

$\dfrac{120}{252}$

Answer

$\dfrac{10}{21}$

Exercise $\PageIndex{10}$

$\dfrac{182}{294}$

Exercise $\PageIndex{11}$

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

Answer

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

Exercise $\PageIndex{12}$

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

Exercise $\PageIndex{13}$

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

Answer

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

Exercise $\PageIndex{14}$

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

Multiply Fractions

In the following exercises, multiply.

Exercise $\PageIndex{15}$

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

Answer

$\dfrac{27}{40}$

Exercise $\PageIndex{16}$

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

Exercise $\PageIndex{17}$

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

Answer

$\dfrac{1}{4}$

Exercise $\PageIndex{18}$

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

Exercise $\PageIndex{19}$

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

Answer

$-\dfrac{1}{6}$

Exercise $\PageIndex{20}$

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

Exercise $\PageIndex{21}$

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

Answer

$-\dfrac{21}{50}$

Exercise $\PageIndex{22}$

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

Exercise $\PageIndex{23}$

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

Answer

$\dfrac{11}{30}$

Exercise $\PageIndex{24}$

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

Exercise $\PageIndex{25}$

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

Answer

$\dfrac{20}{11}$

Exercise $\PageIndex{26}$

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

Exercise $\PageIndex{27}$

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

Answer

9n

Exercise $\PageIndex{28}$

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

Exercise $\PageIndex{29}$

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

Answer

−34

Exercise $\PageIndex{30}$

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

Divide Fractions

In the following exercises, divide.

Exercise $\PageIndex{31}$

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

Answer

$\dfrac{9}{8}$

Exercise $\PageIndex{32}$

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

Exercise $\PageIndex{33}$

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

Answer

$\dfrac{4}{9}$

Exercise $\PageIndex{34}$

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

Exercise $\PageIndex{35}$

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

Answer

$\dfrac{33}{4x}$

Exercise $\PageIndex{36}$

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

Exercise $\PageIndex{37}$

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

Answer

$-\dfrac{4}{9}$

Exercise $\PageIndex{38}$

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

Exercise $\PageIndex{39}$

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

Answer

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

Exercise $\PageIndex{40}$

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

Exercise $\PageIndex{41}$

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

Answer

-10

Exercise $\PageIndex{42}$

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

Exercise $\PageIndex{43}$

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

Answer

$\dfrac{1}{16}$

Exercise $\PageIndex{44}$

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

In the following exercises, simplify.

Exercise $\PageIndex{45}$

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

Answer

$-\dfrac{10}{9}$

Exercise $\PageIndex{46}$

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

Exercise $\PageIndex{47}$

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

Answer

$-\dfrac{2}{5}$

Exercise $\PageIndex{48}$

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

Exercise $\PageIndex{49}$

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

Answer

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

Exercise $\PageIndex{50}$

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

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

Exercise $\PageIndex{51}$

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

Answer

$\dfrac{5}{2}$

Exercise $\PageIndex{52}$

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

Exercise $\PageIndex{53}$

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

Answer

$\dfrac{16}{3}$

Exercise $\PageIndex{54}$

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

Exercise $\PageIndex{55}$

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

Answer

0

Exercise $\PageIndex{56}$

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

Exercise $\PageIndex{57}$

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

Answer

$\dfrac{1}{3}$

Exercise $\PageIndex{58}$

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

Exercise $\PageIndex{59}$

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

Answer

$\dfrac{3}{5}$

Exercise $\PageIndex{60}$

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

Exercise $\PageIndex{61}$

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

Answer

$\dfrac{42}{17}$

Exercise $\PageIndex{62}$

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

Exercise $\PageIndex{63}$

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

Answer

$\dfrac{9}{7}$

Exercise $\PageIndex{64}$

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

Exercise $\PageIndex{65}$

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

Answer

$-8$

Exercise $\PageIndex{66}$

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

Exercise $\PageIndex{67}$

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

Answer

$\dfrac{11}{6}$

Exercise $\PageIndex{68}$

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

Exercise $\PageIndex{69}$

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

Answer

$\dfrac{5}{2}$

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 $10$

Answer

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

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 $−3$

Answer

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

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.

Answer

1. $1\frac{1}{2}$ cups
2. answers will vary

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?

Answer

$20$ 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.

Answer

Answers may vary

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.

Answer

Answers may vary

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?