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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?

Questions & Answers

integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
how
muqtaar
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
No; 65m/hr
albert
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
hi
albert
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce

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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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