4.7 Solve equations with fractions  (Page 5/5)

The quotient of $g$ and twelve is $8.$

The quotient of $g$ and nine is $14.$

Three-fourths of $q$ is $12.$

Two-fifths of $q$ is $20.$

Seven-tenths of $p$ is $\mathrm{-63}.$

Four-ninths of $p$ is $\mathrm{-28}.$

The quotient of $h$ and $2$ is $43.$

The quotient of $a$ and $\frac{2}{3}$ is $\frac{3}{4}.$

The sum of five-sixths and $x$ is $\frac{1}{2}.$

The sum of three-fourths and $x$ is $\frac{1}{8}.$

Shopping Teresa bought a pair of shoes on sale for $\text{\$48}.$ The sale price was $\frac{2}{3}$ of the regular price. Find the regular price of the shoes by solving the equation $\frac{2}{3}p=48$

Playhouse The table in a child’s playhouse is $\frac{3}{5}$ of an adult-size table. The playhouse table is $18$ inches high. Find the height of an adult-size table by solving the equation $\frac{3}{5}h=18.$

[link] describes three methods to solve the equation $-y=15.$ Which method do you prefer? Why?

Richard thinks the solution to the equation $\frac{3}{4}x=24$ is $16.$ Explain why Richard is wrong.

$\frac{58}{15}$

$\frac{63}{11}$

$12\frac{1}{4}$

$9\frac{4}{5}$

Find three fractions equivalent to $\frac{2}{5}.$ Show your work, using figures or algebra.

Find three fractions equivalent to $-\frac{4}{3}.$ Show your work, using figures or algebra.

$\frac{5}{8},\frac{4}{3},3\frac{3}{4},4$

$\frac{1}{4},-\frac{1}{4},1\frac{1}{3},\mathrm{-1}\frac{1}{3},\frac{7}{2},-\frac{7}{2}$

$\mathrm{-1}\_\_\_-\frac{2}{5}$

$\mathrm{-2}\frac{1}{2}\_\_\_\mathrm{-3}$

$-\frac{63}{84}$

$-\frac{90}{120}$

$-\frac{14a}{14b}$

$-\frac{8x}{8y}$

$\frac{2}{5}·\frac{8}{13}$

$-\frac{1}{3}·\frac{12}{7}$

$\frac{2}{9}·\left(-\frac{45}{32}\right)$

$6m·\frac{4}{11}$

$-\frac{1}{4}\left(\mathrm{-32}\right)$

$3\frac{1}{5}·1\frac{7}{8}$

$\frac{2}{9}$

$\frac{15}{4}$

$3$

$-\frac{1}{4}$

$\frac{2}{3}÷\frac{1}{6}$

$\left(-\frac{3x}{5}\right)÷\left(-\frac{2y}{3}\right)$

$\frac{4}{5}÷3$

$8÷2\frac{2}{3}$

$8\frac{2}{3}÷1\frac{1}{12}$

$3\frac{1}{5}·1\frac{7}{8}$

$\mathrm{-5}\frac{7}{12}·4\frac{4}{11}$

$8÷2\frac{2}{3}$

$8\frac{2}{3}÷1\frac{1}{12}$

the quotient of $8$ and $y$

the quotient of $V$ and the difference of $h$ and $6$

$\frac{\frac{5}{8}}{\frac{4}{5}}$

$\frac{\frac{8}{9}}{\mathrm{-4}}$

$\frac{\frac{n}{4}}{\frac{3}{8}}$

$\frac{\mathrm{-1}\frac{5}{6}}{-\frac{1}{12}}$

$\frac{5+16}{5}$

$\frac{8·4-5^2}{3·12}$

$\frac{8·7+5\left(8-10\right)}{9·3-6·4}$

$\frac{3}{8}+\frac{2}{8}$

$\frac{4}{5}+\frac{1}{5}$

$\frac{2}{5}+\frac{1}{5}$

$\frac{15}{32}+\frac{9}{32}$

$\frac{x}{10}+\frac{7}{10}$

$\frac{8}{11}-\frac{6}{11}$

$\frac{11}{12}-\frac{5}{12}$

$\frac{4}{5}-\frac{y}{5}$

$-\frac{31}{30}-\frac{7}{30}$

$\frac{3}{2}-\left(\frac{3}{2}\right)$

$\frac{11}{15}-\frac{5}{15}-(-\frac{2}{15})$

$\frac{1}{3}\text{and}\frac{1}{12}$

$\frac{1}{3}\text{and}\frac{4}{5}$

$\frac{8}{15}\text{and}\frac{11}{20}$

$\frac{3}{4},\frac{1}{6},\text{and}\frac{5}{10}$

$\frac{1}{3}\text{and}\frac{1}{5},\text{LCD}=15$

$\frac{3}{8}\text{and}\frac{5}{6},\text{LCD}=24$

$-\frac{9}{16}\text{and}\frac{5}{12},\text{LCD}=48$

$\frac{1}{3}\text{,}\frac{3}{4}\text{and}\frac{4}{5},\text{LCD}=60$

$\frac{1}{5}+\frac{2}{3}$

$\frac{11}{12}-\frac{2}{3}$

$-\frac{9}{10}-\frac{3}{4}$

$-\frac{11}{36}-\frac{11}{20}$

$-\frac{22}{25}+\frac{9}{40}$

$\frac{y}{10}-\frac{1}{3}$

$\frac{2}{5}+\left(-\frac{5}{9}\right)$

$\frac{4}{11}÷\frac{2}{7d}$

$\frac{2}{5}+\left(-\frac{3n}{8}\right)\left(-\frac{2}{9n}\right)$

$\frac{{\left(\frac{2}{3}\right)}^2}{{\left(\frac{5}{8}\right)}^2}$

$\left(\frac{11}{12}+\frac{3}{8}\right)÷\left(\frac{5}{6}-\frac{1}{10}\right)$

In the following exercises, evaluate.

$y-\frac{4}{5}$ when

1. ⓐ $y=-\frac{4}{5}$
2. ⓑ $y=\frac{1}{4}$

$6m{n}^2$ when $m=\frac{3}{4}\text{and}n=-\frac{1}{3}$

$4\frac{1}{3}+9\frac{1}{3}$

$6\frac{2}{5}+7\frac{3}{5}$

$5\frac{8}{11}+2\frac{4}{11}$

$3\frac{5}{8}+3\frac{7}{8}$

$9\frac{13}{20}-4\frac{11}{20}$

$2\frac{3}{10}-1\frac{9}{10}$

$2\frac{11}{12}-1\frac{7}{12}$

$8\frac{6}{11}-2\frac{9}{11}$

$x-\frac{1}{2}=\frac{1}{6}\text{:}$

• ⓐ $x=1$
• ⓑ $x=\frac{2}{3}$
• ⓒ $x=-\frac{1}{3}$

$y+\frac{3}{5}=\frac{5}{9}\text{:}$

1. ⓐ $y=\frac{1}{2}$
2. ⓑ $y=\frac{52}{45}$
3. ⓒ $y=-\frac{2}{45}$

In the following exercises, solve the equation.

$n+\frac{9}{11}=\frac{4}{11}$

$x-\frac{1}{6}=\frac{7}{6}$

$h-\left(-\frac{7}{8}\right)=-\frac{2}{5}$

$\frac{x}{5}=\mathrm{-10}$

$-z=23$

The sum of two-thirds and $n$ is $-\frac{3}{5}.$

The difference of $q$ and one-tenth is $\frac{1}{2}.$

The quotient of $p$ and $\mathrm{-4}$ is $\mathrm{-8}.$

Three-eighths of $y$ is $24.$

$\frac{19}{5}$

Convert the mixed number to an improper fraction.

$3\frac{2}{7}$

$\frac{1}{2},1\frac{2}{3},\mathrm{-2}\frac{3}{4},\text{and}\frac{9}{4}$

$\frac{5}{20}$

$\frac{18r}{27s}$

$\frac{1}{3}·\frac{3}{4}$

$\frac{3}{5}·15$

$\mathrm{-36}u\left(-\frac{4}{9}\right)$

$\mathrm{-5}\frac{7}{12}·4\frac{4}{11}$

$-\frac{5}{6}÷\frac{5}{12}$

$\frac{7}{11}÷\left(-\frac{7}{11}\right)$

$\frac{9a}{10}÷\frac{15a}{8}$

$\mathrm{-6}\frac{2}{5}÷4$

$\left(\mathrm{-15}\frac{5}{6}\right)÷\left(\mathrm{-3}\frac{1}{6}\right)$

$\frac{\mathrm{-6}}{\frac{6}{11}}$

$\frac{\frac{p}{2}}{\frac{q}{5}}$

$\frac{-\frac{4}{15}}{\mathrm{-2}\frac{2}{3}}$

$\frac{9^2-4^2}{9-4}$

$\frac{2}{d}+\frac{9}{d}$

$-\frac{3}{13}+\left(-\frac{4}{13}\right)$

$-\frac{22}{25}+\frac{9}{40}$

$\frac{2}{5}+\left(-\frac{7}{5}\right)$

$-\frac{3}{10}+\left(-\frac{5}{8}\right)$

$-\frac{3}{4}÷\frac{x}{3}$

$\frac{2^3-2^2}{{\left(\frac{3}{4}\right)}^2}$

$\frac{\frac{5}{14}+\frac{1}{8}}{\frac{9}{56}}$

$x+\frac{1}{3}$ when

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

$y+\frac{3}{5}=\frac{7}{5}$

$a-\frac{3}{10}=-\frac{9}{10}$

$f+\left(-\frac{2}{3}\right)=\frac{5}{12}$

$\frac{m}{\mathrm{-2}}=\mathrm{-16}$

$-\frac{2}{3}c=18$

Translate and solve: The quotient of $p$ and $\mathrm{-4}$ is $\mathrm{-8}.$ Solve for $p.$