2.5E: Exercises

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 $\PageIndex{2}$

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

Answer

x=1

Exercise $\PageIndex{3}$

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

Exercise $\PageIndex{4}$

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

Answer

$y=-1$

Exercise $\PageIndex{5}$

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

Exercise $\PageIndex{6}$

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

Answer

$b=-2$

Exercise $\PageIndex{7}$

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

Exercise $\PageIndex{8}$

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

Answer

$x=3$

Exercise $\PageIndex{9}$

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

Exercise $\PageIndex{10}$

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

Answer

$n=-24$

Exercise $\PageIndex{11}$

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

Exercise $\PageIndex{12}$

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

Answer

$x=-4$

Exercise $\PageIndex{13}$

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

Exercise $\PageIndex{14}$

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

Answer

$z=-2$

Exercise $\PageIndex{15}$

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

Exercise $\PageIndex{16}$

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

Answer

$a=1$

Exercise $\PageIndex{17}$

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

Exercise $\PageIndex{18}$

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

Answer

$x=-6$

Exercise $\PageIndex{19}$

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

Exercise $\PageIndex{20}$

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

Answer

$x=1$

Exercise $\PageIndex{21}$

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

Exercise $\PageIndex{22}$

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

Answer

$q=7$

Exercise $\PageIndex{23}$

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

Exercise $\PageIndex{24}$

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

Answer

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

Exercise $\PageIndex{25}$

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

Exercise $\PageIndex{26}$

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

Answer

$m=-1$

Exercise $\PageIndex{27}$

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

Exercise $\PageIndex{28}$

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

Answer

$p=-4$

Exercise $\PageIndex{29}$

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

Exercise $\PageIndex{30}$

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

Answer

$v=20$

Exercise $\PageIndex{31}$

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

Exercise $\PageIndex{32}$

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

Answer

$d=-24$

Exercise $\PageIndex{33}$

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

Exercise $\PageIndex{34}$

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

Answer

$y=-1$

Exercise $\PageIndex{35}$

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

Exercise $\PageIndex{36}$

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

Answer

$v=4$

Solve Equations with Decimal Coefficients

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

Exercise $\PageIndex{37}$

$0.6 y+3=9$

Exercise $\PageIndex{38}$

$0.4 y-4=2$

Answer

$y=15$

Exercise $\PageIndex{39}$

$3.6 j-2=5.2$

Exercise $\PageIndex{40}$

$2.1 k+3=7.2$

Answer

$k=2$

Exercise $\PageIndex{41}$

$0.4 x+0.6=0.5 x-1.2$

Exercise $\PageIndex{42}$

$0.7 x+0.4=0.6 x+2.4$

Answer

$x=20$

Exercise $\PageIndex{43}$

$0.23 x+1.47=0.37 x-1.05$

Exercise $\PageIndex{44}$

$0.48 x+1.56=0.58 x-0.64$

Answer

$x=22$

Exercise $\PageIndex{45}$

$0.9 x-1.25=0.75 x+1.75$

Exercise $\PageIndex{46}$

$1.2 x-0.91=0.8 x+2.29$

Answer

$x=8$

Exercise $\PageIndex{47}$

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

Exercise $\PageIndex{48}$

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

Answer

$n=19$

Exercise $\PageIndex{49}$

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

Exercise $\PageIndex{50}$

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

Answer

$d=10$

Exercise $\PageIndex{51}$

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

Exercise $\PageIndex{52}$

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

Answer

$q=15$

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.

Answer

$s=35$

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?

Answer

Answers will vary.

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?

Answer

100. Justifications will vary.

Self Check

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

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