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Andreas bought a used car for $12,000. Because the car was 4-years old, its price was $\frac{3}{4}$ of the original price, when the car was new. What was the original price of the car?
$\begin{array}{cccc}\text{What are you asked to find?}\hfill & & & \text{The original price of the car}\hfill \\ \\ \\ \text{Assign a variable.}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}p=\phantom{\rule{0.2em}{0ex}}\text{the original price}.\hfill \\ \\ \\ \begin{array}{c}\text{Write a sentence that gives the}\hfill \\ \text{information to find it.}\hfill \end{array}\hfill & & & \text{\$12,000 is}\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{of the original price.}\hfill \\ \\ \\ \begin{array}{c}\text{Translate into an equation.}\hfill \\ \\ \\ \\ \\ \\ \\ \text{Solve.}\hfill \end{array}\hfill & & & \begin{array}{ccc}\hfill \mathrm{12,000}& =\hfill & \frac{3}{4}p\hfill \\ \\ \\ \hfill \frac{4}{3}\left(\mathrm{12,000}\right)& =\hfill & \frac{4}{3}\xb7\frac{3}{4}p\hfill \\ \hfill \mathrm{16,000}& =\hfill & p\hfill \end{array}\hfill \\ & & & \text{The original cost of the car was \$16,000.}\hfill \\ \\ \\ \text{Check: Is}\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{of \$16,000 equal to \$12,000?}\hfill & & & \\ \\ \\ \phantom{\rule{1em}{0ex}}\begin{array}{ccc}\hfill \frac{3}{4}\xb7\mathrm{16,000}& \stackrel{?}{=}\hfill & \mathrm{12,000}\hfill \\ \hfill \mathrm{12,000}& =\hfill & \mathrm{12,000}\u2713\hfill \end{array}\hfill & & & \end{array}$
Translate and solve:
The annual property tax on the Mehta’s house is $1,800, calculated as $\frac{15}{\mathrm{1,000}}$ of the assessed value of the house. What is the assessed value of the Mehta’s house?
$120,000
Translate and solve:
Stella planted 14 flats of flowers in $\frac{2}{3}$ of her garden. How many flats of flowers would she need to fill the whole garden?
21 flats
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.
$\mathrm{-9}x=\mathrm{-27}$
$\mathrm{-731}=19y$
$\mathrm{-19}m=\mathrm{-586}$
$0.75a=11.25$
$\frac{z}{2}=54$
$\frac{c}{\mathrm{-3}}=\mathrm{-12}$
$\frac{q}{6}=\mathrm{-38}$
$\mathrm{-24}=\frac{p}{\mathrm{-20}}$
$\text{\u2212}u=15$
$\text{\u2212}x=\mathrm{-39}$
$\frac{3}{5}r=75$
$24=-\frac{3}{4}x$
$-\frac{1}{3}q=-\frac{5}{6}$
$\frac{3}{8}y=-\frac{1}{4}$
$\frac{11}{18}=-\frac{5}{6}q$
$-\frac{7}{20}=-\frac{7}{4}v$
Solve Equations That Require Simplification
In the following exercises, solve each equation requiring simplification.
$\mathrm{-18}-7=5t-9t-6t$
$\frac{5}{12}q+\frac{1}{2}q=25-3$
$0.05p-0.01p=2+0.24$
$\mathrm{-12}(d-5)-29=43$
$\mathrm{-15}(z+9)-11=75$
Mixed Practice
In the following exercises, solve each equation.
$\frac{5}{12}y=60$
$x+33=41$
$\frac{s}{\mathrm{-3}}=\mathrm{-60}$
$\text{\u2212}a=7$
$z-16=\mathrm{-59}$
$0.04r=52.60$
$\mathrm{-15}x=\mathrm{-120}$
$19.36=x-0.2x$
$\text{\u2212}y=\mathrm{-9}$
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
187 is the product of $\mathrm{-17}$ and m .
133 is the product of $\mathrm{-19}$ and n .
$133=\mathrm{-19}n;n=\mathrm{-7}$
$\mathrm{-184}$ is the product of 23 and p .
$\mathrm{-152}$ is the product of 8 and q .
$\mathrm{-152}=8q;q=\mathrm{-19}$
u divided by 7 is equal to $\mathrm{-49}$ .
r divided by 12 is equal to $\mathrm{-48}$ .
$\frac{r}{12}=\mathrm{-48};r=\mathrm{-576}$
h divided by $\mathrm{-13}$ is equal to $\mathrm{-65}$ .
j divided by $\mathrm{-20}$ is equal to $\mathrm{-80}$ .
$\frac{j}{\mathrm{-20}}=\mathrm{-80};j=\mathrm{1,600}$
The quotient $c$ and $\mathrm{-19}$ is 38.
The quotient of $b$ and $\mathrm{-6}$ is 18.
$\frac{b}{\mathrm{-6}}=18;b=\mathrm{-108}$
The quotient of $h$ and 26 is $\mathrm{-52}$ .
The quotient $k$ and 22 is $\mathrm{-66}$ .
$\frac{k}{22}=\mathrm{-66};k=\mathrm{-1,452}$
Five-sixths of y is 15.
Four-thirds of w is 36.
The sum of nine-tenths and g is two-thirds.
The sum of two-fifths and f is one-half.
$\frac{2}{5}+f=\frac{1}{2};f=\frac{1}{10}$
The difference of p and one-sixth is two-thirds.
The difference of q and one-eighth is three-fourths.
$q-\frac{1}{8}=\frac{3}{4};q=\frac{7}{8}$
Translate and Solve Applications
In the following exercises, translate into an equation and solve.
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?
Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. How many balloons did she use in each bunch?
6 balloons
Tickets Mollie paid $36.25 for 5 movie tickets. What was the price of each ticket?
Shopping Serena paid $12.96 for a pack of 12 pairs of sport socks. What was the price of pair of sport socks?
$1.08
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?
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?
36 mpg
Height Aiden is 27 inches tall. He is $\frac{3}{8}$ as tall as his father. How tall is his father?
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?
$195,000
Commission 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.
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 $0.49s+0.21\text{}(s-5)\text{}\text{}=9.45$ for s , to find the number of 49-cent stamps Travis bought.
15 49-cent stamps
Frida started to solve the equation $\mathrm{-3}x=36$ by adding 3 to both sides. Explain why Frida’s method will not solve the equation.
Emiliano thinks $x=40$ is the solution to the equation $\frac{1}{2}x=80$ . Explain why he is wrong.
Answers will vary.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
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