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Andreas bought a used car for $12,000. Because the car was 4-years old, its price was 3 4 of the original price, when the car was new. What was the original price of the car?

Solution

What are you asked to find? The original price of the car Assign a variable. Let p = the original price . Write a sentence that gives the information to find it. $12,000 is 3 4 of the original price. Translate into an equation. Solve. 12,000 = 3 4 p 4 3 ( 12,000 ) = 4 3 · 3 4 p 16,000 = p The original cost of the car was $16,000. Check: Is 3 4 of $16,000 equal to $12,000? 3 4 · 16,000 = ? 12,000 12,000 = 12,000

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Translate and solve:

The annual property tax on the Mehta’s house is $1,800, calculated as 15 1,000 of the assessed value of the house. What is the assessed value of the Mehta’s house?

$120,000

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Translate and solve:

Stella planted 14 flats of flowers in 2 3 of her garden. How many flats of flowers would she need to fill the whole garden?

21 flats

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Key concepts

  • The Division Property of Equality —For any numbers a , b , and c , and c 0 , if a = b , then a c = b c .
    When you divide both sides of an equation by any non-zero number, you still have equality.
  • The Multiplication Property of Equality —For any numbers a , b , and c , if a = b , then a c = b c .
    If you multiply both sides of an equation by the same number, you still have equality.

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.

−809 = 15 y

y = 809 15

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−37 p = −541

p = 541 37

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5 8 w = 40

w = −64

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2 5 = 1 10 a

a = −4

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7 10 x = 14 3

x = 20 3

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7 12 = 3 4 p

p = 7 9

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5 18 = 10 9 u

u = 1 4

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Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

100 16 = 4 p 10 p p

p = −12

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−18 7 = 5 t 9 t 6 t

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7 8 n 3 4 n = 9 + 2

n = 88

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5 12 q + 1 2 q = 25 3

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0.25 d + 0.10 d = 6 0.75

d = 15

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0.05 p 0.01 p = 2 + 0.24

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−10 ( q 4 ) 57 = 93

q = −11

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−12 ( d 5 ) 29 = 43

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−10 ( x + 4 ) 19 = 85

x = 72 5

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−15 ( z + 9 ) 11 = 75

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Mixed Practice

In the following exercises, solve each equation.

c 0.3 c = 35.70

c = 51

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Translate to an Equation and Solve

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

187 is the product of −17 and m .

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133 is the product of −19 and n .

133 = −19 n ; n = −7

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−184 is the product of 23 and p .

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−152 is the product of 8 and q .

−152 = 8 q ; q = −19

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u divided by 7 is equal to −49 .

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r divided by 12 is equal to −48 .

r 12 = −48 ; r = −576

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h divided by −13 is equal to −65 .

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j divided by −20 is equal to −80 .

j −20 = −80 ; j = 1,600

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The quotient c and −19 is 38.

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The quotient of b and −6 is 18.

b −6 = 18 ; b = −108

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The quotient of h and 26 is −52 .

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The quotient k and 22 is −66 .

k 22 = −66 ; k = −1,452

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Three-tenths of x is 15.

3 10 x = 15 ; x = 50

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Five-halves of v is 50.

5 2 v = 50 ; v = 20

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

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

2 5 + f = 1 2 ; f = 1 10

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

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

q 1 8 = 3 4 ; q = 7 8

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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?

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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

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

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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

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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?

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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

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Height Aiden is 27 inches tall. He is 3 8 as tall as his father. How tall is his father?

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

$195,000

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Everyday math

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.

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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 0.49 s + 0.21 ( s 5 ) = 9.45 for s , to find the number of 49-cent stamps Travis bought.

15 49-cent stamps

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Writing exercises

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

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Emiliano thinks x = 40 is the solution to the equation 1 2 x = 80 . Explain why he is wrong.

Answers will vary.

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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?

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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