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Mathematics LibreTexts

2.2E: Exercises

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    30363
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    Practice Makes Perfect

    Solve Equations Using the Division and Multiplication Properties of Equality

    In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.

    Exercise \(\PageIndex{1}\)

    \(8x=56\)

    Answer

    \(x=7\)

    Exercise \(\PageIndex{2}\)

    \(7 p=63\)

    Exercise \(\PageIndex{3}\)

    \(-5 c=55\)

    Answer

    \(c=-11\)

    Exercise \(\PageIndex{4}\)

    \(-9 x=-27\)

    Exercise \(\PageIndex{5}\)

    \(-809=15 y\)

    Answer

    \(y = -\frac{809}{15}\)

    Exercise \(\PageIndex{6}\)

    \(-731=19 y\)

    Exercise \(\PageIndex{7}\)

    \(-37 p=-541\)

    Answer

    \(p=-\frac{541}{37}\)

    Exercise \(\PageIndex{8}\)

    \(-19 m=-586\)

    Exercise \(\PageIndex{9}\)

    \(0.25 z=3.25\)

    Answer

    z= 13

    Exercise \(\PageIndex{10}\)

    \(0.75 a=11.25\)

    Exercise \(\PageIndex{11}\)

    \(-13x=0\)

    Answer

    \(x=0\)

    Exercise \(\PageIndex{12}\)

    \(24x=0\)

    Exercise \(\PageIndex{13}\)

    \(\frac{x}{4} = 35\)

    Answer

    \(x=140\)

    Exercise \(\PageIndex{14}\)

    \(\frac{z}{2}=54\)

    Exercise \(\PageIndex{15}\)

    \(-20=\frac{q}{-5}\)

    Answer

    \(q=100\)

    Exercise \(\PageIndex{16}\)

    \(\frac{c}{-3}=-12\)

    Exercise \(\PageIndex{17}\)

    \(\frac{y}{9}=-16\)

    Answer

    \(y=-144\)

    Exercise \(\PageIndex{18}\)

    \(\frac{q}{6}=-38\)

    Exercise \(\PageIndex{19}\)

    \(\frac{m}{-12}=45\)

    Answer

    \(m=-540\)

    Exercise \(\PageIndex{20}\)

    \(-24=\frac{p}{-20}\)

    Exercise \(\PageIndex{21}\)

    \(-y=6\)

    Answer

    \(y=-6\)

    Exercise \(\PageIndex{22}\)

    \(-u=15\)

    Exercise \(\PageIndex{23}\)

    \(-v=-72\)

    Answer

    \(v=72\)

    Exercise \(\PageIndex{24}\)

    \(-x=-39\)

    Exercise \(\PageIndex{25}\)

    \(\frac{2}{3} y=48\)

    Answer

    \(y=72\)

    Exercise \(\PageIndex{26}\)

    \(\frac{3}{5} r=75\)

    Exercise \(\PageIndex{27}\)

    \(-\frac{5}{8} w=40\)

    Answer

    \(w=-64\)

    Exercise \(\PageIndex{28}\)

    \(24=-\frac{3}{4} x\)

    Exercise \(\PageIndex{29}\)

    \(-\frac{2}{5}=\frac{1}{10} a\)

    Answer

    \(a=-4\)

    Exercise \(\PageIndex{30}\)

    \(-\frac{1}{3} q=-\frac{5}{6}\)

    Exercise \(\PageIndex{31}\)

    \(-\frac{7}{10} x=-\frac{14}{3}\)

    Answer

    \(x=\frac{20}{3}\)

    Exercise \(\PageIndex{32}\)

    \(\frac{3}{8} y=-\frac{1}{4}\)

    Exercise \(\PageIndex{33}\)

    \(\frac{7}{12}=-\frac{3}{4} p\)

    Answer

    \(p=-\frac{7}{9}\)

    Exercise \(\PageIndex{34}\)

    \(\frac{11}{18}=-\frac{5}{6} q\)

    Exercise \(\PageIndex{35}\)

    \(-\frac{5}{18}=-\frac{10}{9} u\)

    Answer

    \(u=\frac{1}{4}\)

    Exercise \(\PageIndex{36}\)

    \(-\frac{7}{20}=-\frac{7}{4} v\)

    Solve Equations That Require Simplification

    In the following exercises, solve each equation requiring simplification.

    Exercise \(\PageIndex{37}\)

    \(100-16=4 p-10 p-p\)

    Answer

    \(p=-12\)

    Exercise \(\PageIndex{38}\)

    \(-18-7=5 t-9 t-6 t\)

    Exercise \(\PageIndex{39}\)

    \(\frac{7}{8} n-\frac{3}{4} n=9+2\)

    Answer

    \(n=88\)

    Exercise \(\PageIndex{40}\)

    \(\frac{5}{12} q+\frac{1}{2} q=25-3\)

    Exercise \(\PageIndex{41}\)

    \(0.25 d+0.10 d=6-0.75\)

    Answer

    d=15

    Exercise \(\PageIndex{42}\)

    \(0.05 p-0.01 p=2+0.24\)

    Exercise \(\PageIndex{43}\)

    \(-10(q-4)-57=93\)

    Answer

    \(q=-11\)

    Exercise \(\PageIndex{44}\)

    \(-12(d-5)-29=43\)

    Exercise \(\PageIndex{45}\)

    \(-10(x+4)-19=85\)

    Answer

    \(x=-\frac{72}{5}\)

    Exercise \(\PageIndex{46}\)

    \(-15(z+9)-11=75\)

    Mixed Practice

    In the following exercises, solve each equation.

    Exercise \(\PageIndex{47}\)

    \(\frac{9}{10} x=90\)

    Answer

    \(x=100\)

    Exercise \(\PageIndex{48}\)

    \(\frac{5}{12} y=60\)

    Exercise \(\PageIndex{49}\)

    \(y+46=55\)

    Answer

    \(y=9\)

    Exercise \(\PageIndex{50}\)

    \(x+33=41\)

    Exercise \(\PageIndex{51}\)

    \(\frac{w}{-2}=99\)

    Answer

    \(w=-198\)

    Exercise \(\PageIndex{52}\)

    \(\frac{s}{-3}=-60\)

    Exercise \(\PageIndex{53}\)

    \(27=6 a\)

    Answer

    \(a=\frac{9}{2}\)

    Exercise \(\PageIndex{54}\)

    \(-a=7\)

    Exercise \(\PageIndex{55}\)

    \(-x=2\)

    Answer

    \(x=-2\)

    Exercise \(\PageIndex{56}\)

    \(z-16=-59\)

    Exercise \(\PageIndex{57}\)

    \(m-41=-14\)

    Answer

    \(m=27\)

    Exercise \(\PageIndex{58}\)

    \(0.04 r=52.60\)

    Exercise \(\PageIndex{59}\)

    \(63.90=0.03 p\)

    Answer

    \(p=2130\)

    Exercise \(\PageIndex{60}\)

    \(-15 x=-120\)

    Exercise \(\PageIndex{61}\)

    \(84=-12 z\)

    Answer

    \(y=-7\)

    Exercise \(\PageIndex{62}\)

    \(19.36=x-0.2 x\)

    Exercise \(\PageIndex{63}\)

    \(c-0.3 c=35.70\)

    Answer

    \(c=51\)

    Exercise \(\PageIndex{64}\)

    \(-y=-9\)

    Exercise \(\PageIndex{65}\)

    \(-x=-8\)

    Answer

    \(x=8\)

    Translate to an Equation and Solve

    In the following exercises, translate to an equation and then solve.

    Exercise \(\PageIndex{66}\)

    187 is the product of \(-17\) and \(m\)

    Exercise \(\PageIndex{67}\)

    133 is the product of \(-19\) and \(n\)

    Answer

    \(133=-19 n ; n=-7\)

    Exercise \(\PageIndex{68}\)

    \(-184\) is the product of 23 and \(p\)

    Exercise \(\PageIndex{69}\)

    \(-152\) is the product of 8 and \(q\)

    Answer

    \(-152=8 q ; q=-19\)

    Exercise \(\PageIndex{70}\)

    \(u\) divided by 7 is equal to \(-49\)

    Exercise \(\PageIndex{71}\)

    \(r\) divided by 12 is equal to \(-48\)

    Answer

    \(\frac{r}{12}=-48 ; r=-576\)

    Exercise \(\PageIndex{72}\)

    \(h\) divided by \(-13\) is equal to \(-65\)

    Exercise \(\PageIndex{73}\)

    \(j\) divided by \(-20\) is equal to \(-80\)

    Answer

    \(\frac{j}{-20}=-80 ; j=1,600\)

    Exercise \(\PageIndex{74}\)

    The quotient \(c\) and \(-19\) is \(38 .\)

    Exercise \(\PageIndex{75}\)

    The quotient of \(b\) and \(-6\) is 18

    Answer

    \(\frac{b}{-6}=18 ; b=-108\)

    Exercise \(\PageIndex{76}\)

    The quotient of \(h\) and 26 is \(-52\)

    Exercise \(\PageIndex{77}\)

    The quotient \(k\) and 22 is \(-66\)

    Answer

    \(\frac{k}{22}=-66 ; k=-1,452\)

    Exercise \(\PageIndex{78}\)

    Five-sixths of \(y\) is 15

    Exercise \(\PageIndex{79}\)

    Three-tenths of \(x\) is 15

    Answer

    \(\frac{3}{10} x=15 ; x=50\)

    Exercise \(\PageIndex{80}\)

    Four-thirds of \(w\) is 36

    Exercise \(\PageIndex{81}\)

    Five-halves of \(v\) is 50

    Answer

    \(\frac{5}{2} v=50 ; v=20\)

    Exercise \(\PageIndex{82}\)

    The sum of nine-tenths and \(g\) is two-thirds.

    Exercise \(\PageIndex{83}\)

    The sum of two-fifths and \(f\) is one-half.

    Answer

    \(\frac{2}{5}+f=\frac{1}{2} ; f=\frac{1}{10}\)

    Exercise \(\PageIndex{84}\)

    The difference of \(p\) and one-sixth is two-thirds.

    Exercise \(\PageIndex{85}\)

    The difference of \(q\) and one-eighth is three-fourths.

    Answer

    \(q-\frac{1}{8}=\frac{3}{4} ; q=\frac{7}{8}\)

    Translate and Solve Applications

    In the following exercises, translate into an equation and solve.

    Exercise \(\PageIndex{86}\)

    Kindergarten Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many children will she put in each group?

    Exercise \(\PageIndex{87}\)

    Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. How many balloons did she use in each bunch?

    Answer

    6 balloons

    Exercise \(\PageIndex{88}\)

    Tickets Mollie paid $36.25 for 5 movie tickets. What was the price of each ticket?

    Exercise \(\PageIndex{89}\)

    Shopping Serena paid $12.96 for a pack of 12 pairs of sport socks. What was the price of pair of sport socks?

    Answer

    $1.08

    Exercise \(\PageIndex{90}\)

    Sewing Nancy used 14 yards of fabric to make flags for one-third of the drill team. How much fabric, would Nancy need to make flags for the whole team?

    Exercise \(\PageIndex{91}\)

    MPG John’s SUV gets 18 miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?

    Answer

    36 mpg

    Exercise \(\PageIndex{92}\)

    Height Aiden is 27 inches tall. He is \(\frac{3}{8}\) as tall as his father. How tall is his father?

    Exercise \(\PageIndex{92}\)

    Real estate Bea earned \(\$ 11,700\) commission for selling a house, calculated as \(\frac{6}{100}\) of the selling price. What was the selling
    price of the house?

    Answer

    $195,000

    Everyday Math

    Exercise \(\PageIndex{93}\)

    Commissions Every week Perry gets paid \(\$150\) plus 12% of his total sales amount. Solve the equation
    \(840=150+0.12(a-1250)\) for \(a\) to find the total amount Perry must sell in order to be paid \(\$ 840\) one week.

    Exercise \(\PageIndex{94}\)

    Stamps Travis bought $9.45 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 5 less than the number of 49-cent stamps. Solve the equation 0.49s+0.21(s−5)=9.45 for s, to find the number of 49-cent stamps Travis bought.

    Answer

    15 49-cent stamps

    Writing Exercises

    Exercise \(\PageIndex{95}\)

    Frida started to solve the equation −3x=36 by adding 3 to both sides. Explain why Frida’s method will not solve the equation.

    Exercise \(\PageIndex{96}\)

    Emiliano thinks \(x=40\) is the solution to the equation \(\frac{1}{2} x=80 .\) Explain why he is wrong.

    Answer

    Answers 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 five 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 “1) solve equations using the Division and Multiplication Properties of equality,” “2) solve equations that require simplification,” “3) translate to an equation and solve,” and “4) translate and solve applications.” The rest of the cells are blank.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?