Solving Proportions 5.4

Transcription

1 . Solving Proportions. OBJECTIVE. Solve a proportion for an unknown value A proportion consists of four values. If three of the four values of a proportion are known, you can always find the missing or unknown value. a 0 In the proportion, the first value is unknown. We have chosen to represent the unknown value with the letter a. Using the proportion rule, we can proceed as follows.? 0 NOTE is a proportion in which the first value is unknown. Our work in this section will be learning how to find that unknown value. a 0 a 0 or a 0 The equals sign tells us that a and 0 are just different names for the same number. This type of statement is called an equation. Definitions: Equation An equation is a statement that two epressions are equal. One important property of an equation is that we can divide both sides by the same nonzero number. Here let s divide by. a 0 a 0 NOTE We will always divide by the number multiplying the variable. This is called the coefficient of the variable. a 0 a Divide by the coefficient of the variable. Do you see why we divided by? It leaves our unknown a by itself in the left term. You should always check your result. It is easy in this case. We found a value of for a. Replace the unknown a with that value. Then verify that the fractions are proportional. We a 0 started with and found a value of for a. So we write 00 McGraw-Hill Companies NOTE Replace a with and multiply The value of for a is correct. The procedure for solving a proportion is summarized as follows. 7

2 8 CHAPTER RATIOS AND PROPORTIONS Step by Step: To Solve a Proportion NOTE This gives us the unknown value. Now check the result. Step Use the proportion rule to write the equivalent equation a d b c. Step Divide both terms of the equation by the coefficient of the variable. Step Use the value found to replace the unknown in the original proportion. Check that the ratios or the rates are proportional. Solving Proportions for Unknown Values Find the unknown value. (a) Eample Step Using the proportion rule, we have the following. NOTE You are really using algebra to solve these proportions. In algebra, we write the product 6 as 6, omitting the dot. Multiplication of the number and the variable is understood and doesn t need to be written or 6 7 Step Locate the coefficient of the variable, 6, and divide both sides of the equation by that coefficient Step To check, replace with in the original proportion. 8 Multiply: The value of for checks. (b) 6 9 b Step Use the proportion rule. b or b 7 Step Locate the coefficient of the variable,, and divide both sides of the equation by that coefficient. b 7 b 00 McGraw-Hill Companies

3 SOLVING PROPORTIONS SECTION. 9 Step Multiply: To check, replace b with in the original proportion. 7 7 The value of checks for b. CHECK YOURSELF Solve the proportions for n. Check your result. (a) n (b) 7 9 n In solving for a missing term in a proportion, we may find an equation involving fractions or decimals. Eample involves finding the unknown value in such cases. Eample Solving Proportions for Unknown Values (a) Solve the proportion for. 00 McGraw-Hill Companies NOTE 8 8 To check, replace with 8 in the original proportion. 8 8 We divide by the coefficient of. In this case it is. Remember: is. Invert the divisor and multiply.

4 0 CHAPTER RATIOS AND PROPORTIONS NOTE Here we must divide 6 by 0. to find the unknown value. The steps of that division are shown below for review.. ^ ^ 0. B6.0 (b) Solve the proportion for a. 0. a 0.a 6 0.a a CHECK YOURSELF (a) Solve for a. (b) Solve for. a Divide by the coefficient, 0.. We will leave it to you to confirm that CHECK YOURSELF ANSWERS. (a) n 00 To check: (b) 7n 9 To check: n n n 0 0 n n (a) 0; (b) 6 00 McGraw-Hill Companies

5 Name. Eercises Section Date Solve for the unknown in each of the following proportions n 6 ANSWERS y 8 n 7 m a 6 8 p a 8 n n 8 9 a m p a 6... a n 00 0 p McGraw-Hill Companies b a p

6 ANSWERS..... m y n a a 0 n p y a. Answers. 9 8; n 80; n p ; p a 0; a McGraw-Hill Companies