# 2.5E: Exercises

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## Practice Makes Perfect

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

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

$$\frac{1}{4} x-\frac{1}{2}=-\frac{3}{4}$$

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

$$\frac{3}{4} x-\frac{1}{2}=\frac{1}{4}$$

x=1

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

$$\frac{5}{6} y-\frac{2}{3}=-\frac{3}{2}$$

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

$$\frac{5}{6} y-\frac{1}{3}=-\frac{7}{6}$$

$$y=-1$$

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

$$\frac{1}{2} a+\frac{3}{8}=\frac{3}{4}$$

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

$$\frac{5}{8} b+\frac{1}{2}=-\frac{3}{4}$$

$$b=-2$$

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

$$2=\frac{1}{3} x-\frac{1}{2} x+\frac{2}{3} x$$

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

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

$$x=3$$

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

$$\frac{1}{4} m-\frac{4}{5} m+\frac{1}{2} m=-1$$

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

$$\frac{5}{6} n-\frac{1}{4} n-\frac{1}{2} n=-2$$

$$n=-24$$

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

$$x+\frac{1}{2}=\frac{2}{3} x-\frac{1}{2}$$

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

$$x+\frac{3}{4}=\frac{1}{2} x-\frac{5}{4}$$

$$x=-4$$

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

$$\frac{1}{3} w+\frac{5}{4}=w-\frac{1}{4}$$

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

$$\frac{3}{2} z+\frac{1}{3}=z-\frac{2}{3}$$

$$z=-2$$

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

$$\frac{1}{2} x-\frac{1}{4}=\frac{1}{12} x+\frac{1}{6}$$

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

$$\frac{1}{2} a-\frac{1}{4}=\frac{1}{6} a+\frac{1}{12}$$

$$a=1$$

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

$$\frac{1}{3} b+\frac{1}{5}=\frac{2}{5} b-\frac{3}{5}$$

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

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

$$x=-6$$

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

$$1=\frac{1}{6}(12 x-6)$$

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

$$1=\frac{1}{5}(15 x-10)$$

$$x=1$$

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

$$\frac{1}{4}(p-7)=\frac{1}{3}(p+5)$$

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

$$\frac{1}{5}(q+3)=\frac{1}{2}(q-3)$$

$$q=7$$

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

$$\frac{1}{2}(x+4)=\frac{3}{4}$$

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

$$\frac{1}{3}(x+5)=\frac{5}{6}$$

$$x=-\frac{5}{2}$$

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

$$\frac{5 q-8}{5}=\frac{2 q}{10}$$

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

$$\frac{4 m+2}{6}=\frac{m}{3}$$

$$m=-1$$

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

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

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

$$\frac{3 p+6}{3}=\frac{p}{2}$$

$$p=-4$$

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

$$\frac{u}{3}-4=\frac{u}{2}-3$$

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

$$\frac{v}{10}+1=\frac{v}{4}-2$$

$$v=20$$

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

$$\frac{c}{15}+1=\frac{c}{10}-1$$

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

$$\frac{d}{6}+3=\frac{d}{8}+2$$

$$d=-24$$

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

$$\frac{3 x+4}{2}+1=\frac{5 x+10}{8}$$

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

$$\frac{10 y-2}{3}+3=\frac{10 y+1}{9}$$

$$y=-1$$

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

$$\frac{7 u-1}{4}-1=\frac{4 u+8}{5}$$

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

$$\frac{3 v-6}{2}+5=\frac{11 v-4}{5}$$

$$v=4$$

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

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

$$0.6 y+3=9$$

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

$$0.4 y-4=2$$

$$y=15$$

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

$$3.6 j-2=5.2$$

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

$$2.1 k+3=7.2$$

$$k=2$$

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

$$0.4 x+0.6=0.5 x-1.2$$

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

$$0.7 x+0.4=0.6 x+2.4$$

$$x=20$$

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

$$0.23 x+1.47=0.37 x-1.05$$

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

$$0.48 x+1.56=0.58 x-0.64$$

$$x=22$$

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

$$0.9 x-1.25=0.75 x+1.75$$

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

$$1.2 x-0.91=0.8 x+2.29$$

$$x=8$$

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

$$0.05 n+0.10(n+8)=2.15$$

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

$$0.05 n+0.10(n+7)=3.55$$

$$n=19$$

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

$$0.10 d+0.25(d+5)=4.05$$

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

$$0.10 d+0.25(d+7)=5.25$$

$$d=10$$

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

$$0.05(q-5)+0.25 q=3.05$$

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

$$0.05(q-8)+0.25 q=4.10$$

$$q=15$$

## Everyday Math

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

Coins Taylor has $$\ 200$$ in dimes and pennies. The number of pennies is 2 more than the number of dimes. Solve the equation $$0.10 d+0.01(d+2)=2$$ for $$d$$, the number of dimes.

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

Stamps Paula bought \$22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 8 less than the number of 49-cent stamps. Solve the equation $$0.49 s+0.21(s-8)=22.82$$ for s, to find the number of 49-cent stamps Paula bought.

$$s=35$$

## Writing Exercises

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

Explain how you find the least common denominator of $$\frac{3}{8}, \frac{1}{6},$$ and $$\frac{2}{3}$$

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

If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?

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

If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?

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

In the equation $$0.35 x+2.1=3.85$$ what is the LCD? How do you know?