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Mathematics LibreTexts

2.5E: Exercises

  • Page ID
    30369
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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 \(\PageIndex{2}\)

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

    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.

    This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads: “solve equations with fraction coefficients,” and “solve equations with decimal coefficients.” The rest of the cells are blank.

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