Using Equations to Solve Business Problems ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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 providing 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 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Understanding Equations continued 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. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For solving equations and providing the solution Step 1 Transpose all the unknowns to the left side of the equation and all the knowns to the right side of the equation by using the following “order of operations” for solving equations. • Parentheses, if any, must be cleared before any other operations are performed. To clear parentheses, multiply the coefficient by each term inside the parentheses. 3(5C + 4) = 2 3(5C) + 3(4) = 2 15C + 12 = 2 • To solve equations with more than one operation: ■ First, perform the additions and subtractions. ■ Then perform the multiplications and divisions. Step 2 Prove the solution by substituting your answer for the letter or letters in the original equation. If the left and right sides are equal, the equation is true and your answer is correct. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Solving Basic Equations 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. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Transposition Example + 13 = 22 X + 13 = 22 – 13 9 9 + 13 = 22 T – 6 = 32 T – 6 = 32 + 6 38 38 – 6 = 32 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Division Example 3R = 24 3R 3 = 24 3 R 8 3 (8) = 24 C8 = 8 C 8 = 8 × 8 × 8 C 64 64 8 = 8 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Multiple Operations Example – 5 = 30 7A – 5 = 30 +5 7A 7 357 A 5 7(5) – 5 = 30 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Multiple Operations Example continued X + 4 = 9 X + 4 = 9 – – 4 5 4X 5(4) 20 20 + 4 = 9 4 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Multiple Operations with Parentheses Example = 33 3(3 × 3 + 2) = 33 3(11) 3(3X + 2) = 33 9X + 6 –6 9X 27 9 X 3 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For Combining multiple unknowns Step 1 To combine unknowns, they must be on the same side of the equation. If they are not, move them to the same side. 5X = 12 + 2X 5X – 2X = 12 Step 2 Once the unknowns are on the same side of the equation, add or subtract their coefficients as indicated. 3X = 12 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Equations with Multiple Unknowns Example ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For Writing Expressions and Equations Step 1 Read the written statement carefully. Step 2 Using the list on the following slide, identify and underline the key words and phrases. Step 3 Convert the words to numbers and mathematical symbols. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Writing Expressions and Equations ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Writing Expressions Example A number increased by 18 Y + 18 13 less than P P – 13 The difference of R and 25 R - 25 6 more than 4 times T 4T + 6 2 less than half of X ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Writing Equations A number increased by 33 is 67 X + 33 = 67 A number totals 5 times B and C X = 5B + C 12 less than 4G leaves 36 4G - 12 = 36 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 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For Setting up and solving word problems STEP 1 Understand the situation. If the problem is written, read it carefully, perhaps a few times. If the problem is verbal, write down the facts of the situation. STEP 2 Take inventory. Identify all the parts of the situation. These parts can be any variables, such as dollars, people, boxes, tons, trucks, anything! Separate them into knowns and unknowns. STEP 3 Make a plan—create an equation. The object is to solve for the unknown. Ask yourself what math relationship exists between the knowns and the unknowns. Use the chart of key words and phrases on page 132 to help you write the equation. STEP 4 Work out the plan—solve the equation. To solve an equation, you must move the unknowns to one side of the equal sign and the knowns to the other. STEP 5 Check your solution. Does your answer make sense? Is it exactly correct? It is a good idea to estimate an approximate answer by using rounded numbers. This will let you know if your answer is in the correct range. If it is not, either the equation is set up incorrectly or the solution is wrong. If this occurs, you must go back and start again. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Solving Business Equations Problem 1 Gene and Fatima work in an electronics store. During a sale, Gene sold 8 less items than Fatima. If they sold a total of 86 items, how many did each sell? Fatima = X Gene = X – 8 X + X – 8 = 86 2X = 86 +8 +8 2X = 94 X = 47 – 8 = 39 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Solving Business Equations Problem 2 One-third of the products made by Vega Inc. are produced by the soap division. If the soap division makes 23 products, how many total products are made by Vega Inc.? Total employees = X X = 69 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Solving Business Equations Problem 3 Chris’s salary this year is $38,500. If this is $1,650 more than she made last year, what was her salary last year? S = Chris’s salary S + 1,650 = 38,500 - 1,650 -1,650 S = 36,850 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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 = 3:1. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

For solving proportion problems using cross-multiplication Step 1 Assign a letter to represent the unknown quantity. Step 2 Set up the proportion with one ratio (expressed as a fraction) on each side of the equal sign. Step 3 Multiply the numerator of the first ratio by the denominator of the second and place the product to the left of the equal sign. Step 4 Multiply the denominator of the first ratio by the numerator of the second and place the product to the right of the equal sign. Step 5 Solve for the unknown. ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Proportion and Ratio 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 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Proportion and Ratio Problem 2 At Kelley 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 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter Review Problem 1 Solve the following : B + 11 = 24 B = 13 C – 16 = 5 C = 21 50Y = 375 Y = 7.5 R/5 + 33 = 84 R = 255 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter Review Problem 2 Solve the following word problem: Emilio’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 of book? X = Hardcover 4X = Paperback X + 4X = 124,300 5X = 124,300 X = 24,860 4X = 99,440 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter Review Problem 3 Solve the following word problem: Kidsmart 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 = 3,400 280 × 14 = 3,920 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter Review Problem 4 Solve the following word problem: 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 = 114,100 X = 114.10 ©2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.