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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

a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Ok cool answer peggie
Frank
thanks
Ramil
copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x
Sandra Reply
What is c+4=8
Penny Reply
2
Letha
4
Lolita
4
Rich
4
thinking
C+4=8 -4 -4 C =4
thinking
I need to study
Letha
4+4=8
William
During two years in college, a student earned $9,500. The second year, she earned $500 more than twice the amount she earned the first year.
Nicole Reply
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp
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.
Josh Reply
Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
Ida Reply
hey
Juan
Sup
patrick
The sum of two numbers is 155. The difference is 23. Find the numbers
Michelle Reply
The sum of two numbers is 155. Their difference is 23. Find the numbers
Michelle
The difference between 89 and 66 is 23
Ciid
Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?
Amber Reply
hello
bismark
I need a math tutor BAD
Stacie
Me too
Letha
me too
Xavier
ok
Bishal
teet
Bishal
Sue and Deb work together writing a book that takes them 90 days. If Sue worked alone, it would take her 120 days. How long would it take Deb to write the book alone?
Hailey Reply
Tell Deb to write the book alone and report back on how long it took her. Deb could get bored and stop, she could get sick next week and prolong the time, she could die next month. This question is impossible to answer.
vV
right*
tyler
is there a thumbs up mathematical symbol in algebra? 👍²= vV 90days x 120days =10800 120days - 90days = 30 10800days/30days= 360days? (ad) derall=1440 mins (90x ad )+(120 x ad )/(120 x ad ))-((90d x ad)= you go glen coco! I'll see my self out
Nick
Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
genevieve Reply
6.25
Ciid
6.25
Big
?
Twila
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Katherine Reply
complexed. could you please help me figure out?
Ciid
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
help
Twila
draw a diagram first
John
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
Marisol Reply
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid

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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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