Using Equations to Solve Business Problems CHAPTER 5 Using Equations to Solve Business Problems © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

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. © 2009 Cengage Learning. All rights reserved.

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 3 + 7 = 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. © 2009 Cengage Learning. All rights reserved.

Solving Basic Equations © 2009 Cengage Learning. All rights reserved.

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. © 2009 Cengage Learning. All rights reserved.

Example: Transposition + 10 = 15 X + 10 = 15 – 10 5 5 + 10 = 15 T – 4 = 20 T – 4 = 20 + 4 24 24 – 4 = 20 © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

Example: Multiple Operations – 5 = 20 5A –5 = 20 +5 5A 5 255 A 5 5(5) – 5 = 20 © 2009 Cengage Learning. All rights reserved.

Example: Multiple Operations (cont’d) + 4 = 9 X + 4 = 9 – – 4 5 4X 5(4) 20 20 + 4 = 9 4 © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

Solving Basic Equations (cont’d) © 2009 Cengage Learning. All rights reserved.

Solving Basic Equations (cont’d) © 2009 Cengage Learning. All rights reserved.

Writing Expressions and Equations from Written Statements © 2009 Cengage Learning. All rights reserved.

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, at, @ Division: divide, divided by, the average of, divided into, the quotient of, the ratio of Parentheses: times the quantity of © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

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 = 86 +8 +8 2X = 94 X = 47X – 8 = 39 © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

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 = 23,400 - 1,700 -1,700 S = 21,700 © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

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 + 1.30(530–X) = 755 1.60X + 689–1.3X = 755 .30x +689 = 755 -689 -689 .30X = 66 X = 220 530–220 = 310 220(1.60) = 352 310(1.3) = 403 © 2009 Cengage Learning. All rights reserved.

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. © 2009 Cengage Learning. All rights reserved.

Solving Proportion Problems © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

Chapter Review Problem 1 Solve the following: B + 11 = 24 B = 24 C – 16 = 5 C = 21 50Y = 375 Y = 7.5 R/5 + 33 = 84 R = 255 © 2009 Cengage Learning. All rights reserved.

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 © 2009 Cengage Learning. All rights reserved.

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 = 280 170 × 20 = 3,440 280 × 14 = 3,920 © 2009 Cengage Learning. All rights reserved.

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 = 114.10 © 2009 Cengage Learning. All rights reserved.