1. Using Equations to Solve Business Problems
CHAPTER 5
Using Equations to Solve Business Problems
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2. PERFORMANCE OBJECTIVES
Section I Solving Basic Equations
5-1: Understanding the concept, terminology, and rules of equations
5-2: Solving equations for the unknown and proving the solution
5-3: Writing expressions and equations from written statements
Section II Using Equations to Solve Business-Related Word Problems
5-4: Setting up and solving business-related word problems by using equations
5-5: Understanding and solving ratio and proportion problems
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3. Understanding Equations
Formula
A mathematical representation of a fact, rule, principle, or other logical relation in which letters represent number quantities.
Equation
A mathematical statement expressing a relationship of equality; usually written as a series of symbols that are separated into left and right sides and joined by an equal sign. X + 7 = 10 is an equation.
Expression
A mathematical operation or a quantity stated in symbolic form, not containing an equal sign. X + 7 is an expression.
Variables (Unknowns)
The part of an equation that is not given. In equations, the unknowns are variables (letters of the alphabet), which are quantities having no fixed value. In the equation X + 7 = 10, X is the unknown or variable.
Constants (Knowns)
The parts of an equation that are given. In equations, the knowns are constants (numbers), which are quantities having a fixed value. In the equation X + 7 = 10, 7 and 10 are the knowns or constants.
Terms
The knowns (constants) and unknowns (variables) of an equation. In the equation X + 7 = 10, the terms are X, 7, and 10.
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4. Understanding Equations (cont’d)
Solve an Equation
The process of finding the numerical value of the unknown in an equation.
Solution (or Root)
The numerical value of the unknown that makes the equation true. Example: X + 7 = 10, 3 is the solution, because = 10.
Coefficient
A number or quantity placed before another quantity, indicating multiplication. For example, 4 is the coefficient in the expression 4C. This indicates 4 multiplied by C.
Transpose
To bring a term from one side of an equation to the other, with a corresponding change of sign.
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5. Solving Basic Equations
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6. Solving Basic Equations (cont’d)
Tips for Solving Equations
Whatever you do to one side of the equation, do to the other.
Use the opposite operation to get rid of (i.e., move) a term from one side of the equation to the other.
Isolate the unknown (variable).
Combine like terms.
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7. Example: Transposition+ 10 = 15 X
+ 10 = 15
– 10 5 5 + 10 = 15 T – 4 = 20 T
– 4 = 20
+ 4 24 24 – 4 = 20
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8. Example: Division 5R = 25 5R 5 = 25 5 R 5 5 (5) = 25 C6 = 6 C 6 = 6 ×
6 × 6 C 36
36 6 = 6
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9. Example: Multiple Operations– 5 = 20 5A –5 = 20 +5
5A 5
255 A 5
5(5) – 5 = 20
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10. Example: Multiple Operations (cont’d)+ 4 = 9 X + 4 = 9 –
– 4 5 4X
5(4) 20 20
+ 4 = 9 4
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11. Example: Multiple Operations with Parentheses= 24
3(2 × 2 + 4) = 24
3(8)
3(2X + 4) = 24
6X + 12
–12 6X 12 6 X 2
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12. Solving Basic Equations (cont’d)
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13. Solving Basic Equations (cont’d)
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14. Writing Expressions and Equations from Written Statements
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15. Creating Equations: Key Words and Phrases
Equal sign:
is, are, was, equals, gives, giving, leaves, leaving, makes, denotes
Addition:
and, added to, totals, the sum of, plus, more than, larger than, increased by, greater than, exceeds
Subtraction:
less, less than, smaller than, minus, difference between, decreased by, reduced by
Multiplication:
of, times, product of, multiplied by, twice, double, triple,
Division:
divide, divided by, the average of, divided into, the quotient of, the ratio of
Parentheses:
times the quantity of
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16. Writing Expressions A number increased by 12 20 less than A
The difference of R and 12
R-12
6 more than 4 times T
4T + 6
2 less than half of X
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17. A number increased by 24 is 35
Writing Equations
A number increased by 24 is 35
X + 24 = 35
A number totals 5 times B and C
X = 5B + C
12 less than 4G leaves 33
4G-12 = 33
The cost of R at $5.75 each is $28.75
5.75R = 28.75
Cost/persons is the total cost by the number of persons
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18. Using Equations to Solve Business-Related Word Problems
Step 1: Understand the situation
Step 2: Take inventory
Step 3: Make a plan—create an equation
Step 4: Work out the plan—solve the equation
Step 5: Check your solution
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19. Solving Business Equations Problem 1
Kathy and Karen work in a boutique. During a sale, Kathy sold eight less dresses than Karen. If they sold a total of 86 dresses, how many did each sell?
Karen = X Kathy = X–8
X + X – 8 = 862 2X = X = 94
X = 47X – 8 = 39
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20. Solving Business Equations Problem 2
One-fifth of the employees of Action Hydraulics Company work in the Midwest region. If the company employs 252 workers in that region, what is the total number of employees working for the company?
Total employees = X
X = 1,260
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21. Solving Business Equations Problem 3
Larry’s salary this year is $23,400. If this is $1,700 more than he made last year, what was his salary last year?
S = Larry’s salary S + 1,700 = 23, , ,700 S = 21,700
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22. Solving Business Equations
Price and Quantity Business Equations
Step 1: Determine which item gets the single variable
Hint: Let X equal the more expensive item to avoid working with negative numbers.
Step 2: Multiply the price times quantity
Step 3: Solve the equation
Step 4: Calculate the quantity of both items
Step 5: Calculate the dollar sales of each by multiplying the price per item by the quantity sold
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23. Solving Business Equations
Captain Cookie sells oatmeal cookies for $1.30 per pound and peanut butter cookies for $1.60 per pound. They sold 530 pounds valued at $755. Calculate how many pounds of each cookie was sold and the dollar amount of each type sold.
X = Peanut Butter 530–X = Oatmeal 1.60X (530–X) = 755
1.60X + 689–1.3X = 755
.30x = 755
.30X = X = –220 = 310
220(1.60) = 352
310(1.3) = 403
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24. Understanding And Solving Ratio And Proportion Problems
A fraction that describes a comparison of two numbers or quantities.
For example, five cats for every three dogs would be a ratio of 5 to 3, written as 5:3.
proportion
A mathematical statement showing that two ratios are equal.
For example, 9 is to 3 as 3 is to 1, written 9: 3 5 3: 1.
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25. Solving Proportion Problems
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26. Proportion and Ration Problem 1
If the interest on a $4,600 loan is $370, what would the interest on a loan of $9,660 be?
4,600X = 370(9,660)
4,600X = 3,574,000
X = 777
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27. Proportion and Ration Problem 2
At Rainbow Fruit Distributors, Inc., the ratio of fruits to vegetables sold is 5 to 3. If 1,848 pounds of vegetables are sold, how many pounds of fruit are sold?
3X = 5(1,848)
3X = 9,240
X = 3,080
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28. Chapter Review Problem 1
Solve the following:
B + 11 = 24
B = 24
C – 16 = 5
C = 21
50Y = 375
Y = 7.5
R/ = 84
R = 255
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29. Chapter Review Problem 2
Sylvia’s Bookstore makes four times as much money in paperback books as in hardcover books. If last month’s sales totaled $124,300, how much was sold of each type book?
X = Hardcover 4X = Paperback X + 4X = 124,300
5X = 124,300
X = 24,860
4X = 99,440
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30. Chapter Review Problem 3
Toy World placed a seasonal order for stuffed animals from a distributor. Large animals cost $20 and small ones cost $14. The total cost of the order was $7,320 for 450 pieces. Calculate the quantity of each ordered and the total cost of each size ordered.
X = Large 450–X = Small 20X + 14(450–X) = 7,320
20X + 6,300–14X = 7,320
6X = 1,020
X = 170
450 – X = × 20 = 3, × 14 = 3,920
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31. Chapter Review Problem 4
If auto insurance costs $6.52 per $1,000 of coverage, what is the cost to insure a car valued at $17,500?
1,000X = 6.52(17,500)
1,000X = 114,000
X =
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