Ratio, Proportion, and Percent

Transcription

1 Pre-Algebra Ratio, Proportion, and Percent Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. LAST REVISED June, 2007

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3 Statement of Prerequisite Skills Complete all previous TLM modules before beginning this module. Required Supporting Materials Access to the World Wide Web. Internet Explorer 5.5 or greater. Macromedia Flash Player. Rationale Why is it important for you to learn this material? Ratio, proportion, and percent are basic math skills that the student will encounter in many applied situations. These skills are also essential to a beginning algebra student. Learning Outcome When you complete this module you will be able to Solve problems using ratio, proportion, and percent. Learning Objectives. Determine equivalent ratios and solve. 2. Change percent to fractions. 3. Change fractions to percent. 4. Change percent to decimals. 5. Change decimals to percent. 6. Solve percent questions. 7. Solve percent error in measurement problems. Connection Activity Consider the many times you have encountered fractions or percentages in daily life: /3 off regular cost 7% gst Top 5% of the class What percentage of your paycheck do you spend in rent? Can you think of other applications of ratio, proportion, and percent?

4 OBJECTIVE ONE When you complete this objective you will be able to Determine equivalent ratios and solve. Exploration Activity A ratio is a comparison of two quantities. The ratio of one number to another is the first number divided by the second number. That is, the ratio of a to b is: a b Therefore, a ratio is a comparison of numbers by division. EXAMPLE NOTE: a) The ratio of 2 to 9 is b) The ratio of 7 to 3 is A PROPORTION is a statement of equality between two ratios; 2 4 i.e. is a proportion

5 EXAMPLE 2 If a car travels 80 km in 2 hours, the ratio of distance to time is: 80km 2h reducing this gives us; and 40km h 80km 40km 2h h The ratios are equal. CHECK: To see if the ratios are equal, perform the cross products If this is true, then: The cross products are equal, therefore the ratios are equal. The general statement for the equality of 2 ratios is: a c if b d then a d b c Notice the proportion has 4 components which are a, b, c, and d. We use ratios to solve problems when we are given 3 of these 4 components. 3

6 EXAMPLE 3 a x b d If we are given the values for a, b, d, then we could solve for x. x b a d a d x b EXAMPLE 4 The ratio of a given number to 3 is the same as the ratio of 6 to 6. Find the given number.. Maintain proper order; i.e. use given number to 3 and 6 to 6 given number Let x given number x If these ratios are equal then 6 ( x) 6( 3) 6 x 6 x 8 () 3 4. Check by using cross products in original proportion () 8 3( 6) The cross products are equal, therefore: x 8 is correct. 4

7 EXAMPLE 5 On a blueprint the scale is km to 25 cm. What is the actual distance between 2 points, if they are 5 cm apart?. Maintain order i.e. km to cm x let x actual distance and write ratios 5 25 x () () x x 5 25 km 5 x 0.2 km CHECK: () 25(0.2) 5 5 x 0.2 km is correct 5

8 EXAMPLE 6 A cedar board 8 m long is cut into two pieces that are in the ratio :4. Find the length of each piece. SOLUTION: Total number of units is Total 5 Therefore: 5 - total number of parts 8 - total length of board Therefore the ratio is either: smaller piece total or larger piece total or let x the length of the shorter piece. Therefore: x 5 8 x Shorter piece.6 m Longer piece 6.4 m 6

9 Experiential Activity One I. Solve the given proportions for x x x x 5 9 II. Solve the given problems by setting up the proper proportion. 4. The ratio of a number to 5 is the same as the ratio of 7 to 60. Find the number. 5. The ratio of a number to 40 is the same as the ratio of 7 to 6. Find the number g 2 lb; what weight in grams is 0 lbs? 7. Medication contains 2 substances, A and B, in the ratio of 3 to 5 respectively. If there is 200 mg of substance B, how many mg of substance A is there? 8. A 6 m length of pipe is cut into 2 parts that are in the ratio 8 to. Find the length of each part. Show Me. 9. A 5 m length of 2 by 0 planking is to be cut into 2 parts that are in the ratio of 4 to 3. Find the length of each part. Experiential Activity One Answers m, 5.33 m m, 2.4 m 7

10 OBJECTIVE TWO When you complete this objective you will be able to Change percent to fractions. Exploration Activity Percent To this point we have used fractions and decimals for representing parts of a unit or quantity. Now we will consider the concept of percent and shall find that percentages are useful in numerous applications. The word percent means by the hundred. Therefore, percent represents a decimal fraction with a denominator of 00. The symbol % is used to denote percent. EXAMPLE For 5% the denominator is 00 Write the fraction with a numerator 5 and get Reduce the fraction and get EXAMPLE 2 3 % : 4 the denominator is 00. Numerator is Write fraction Reduce and apply rules for dividing fractions

11 EXAMPLE 3 5 %: the denominator is Numerator is Write fraction 00 Reduce

12 Experiential Activity Two Change the following percent to fractions.. 50% 2. % % 4. 2% 4 5. % % % 8. 2 % Show Me % 0. 8 % Experiential Activity Two Answers

13 OBJECTIVE THREE When you complete this objective you will be able to Change fractions to percent. Exploration Activity Fractions EXAMPLE Change 3 5 to a percent. Use ratio and proportion. 3 is to 5 as a number is to 00 (% means per hundred) Let x a number Solve for x 3 x x x 5 60 so, 60% 00 and 3 60% 5

14 EXAMPLE 2 Change 5. So 5 is to 6 as a number is to Let x a number so we get, 5 x 6 00 Write ratios 6 x 5 00 Solve for x 5 00 x 6 x so, 83.3% 00 5 and 83.3% 6 2

15 Experiential Activity Three Change the following to percent.. 3/4 2. /00 3. /8 4. 4/5 Show Me. 5. /50 6. /4 Experiential Activity Three Answers. 75% 2. % % 4. 80% 5. 2% 6. 25% 3

16 OBJECTIVE FOUR When you complete this objective you will be able to Change percent to decimals. Exploration Activity EXAMPLE Change 25% to a decimal. Write it as a fraction with denominator 00 Divide by EXAMPLE 2 3 Write % as a decimal. Write it as a fraction with denominator Simplify the fraction

17 Experiential Activity Four Change the following percents to decimals.. 75% 3 2. % % % % % Show Me. 5 Experiential Activity Four Answers

18 OBJECTIVE FIVE When you complete this objective you will be able to Change decimals to percent. Exploration Activity EXAMPLE 0.5 % (multiply by 00) CHECK: % Change percent to decimal by dividing by % EXAMPLE 2. % (multiply by 00) CHECK:. 0% Change percent to decimal by dividing by %. 00 EXAMPLE % (multiply by 00) % CHECK: Change percent to decimal by dividing by %

19 Experiential Activity Five Change the following decimals to a percent Show Me Experiential Activity Five Answers. 0% 2. 80% 3. 25% % % 6. 5% 7

20 OBJECTIVE SIX When you complete this objective you will be able to Solve percent questions. Exploration Activity All problems using percent will be done using ratios. Therefore all percent problems can be grouped into 3 types. TYPE I: Calculate the percent of a quantity. Example: Find 30% of 75 TYPE II: Determine what percent one quantity is of another. Example: 25 is what percent of 60? TYPE III: Determine the quantity from percentage and percent. Example: 25 is 50% of what number? The following model will be used to solve all percentage problems: ( ) ( 2) () 3 ( 4) These are 4 positions; one of them is always taken up by the number 00 because percent is always based out of 00. ( ) () 3 ( 4) 00 Percent always goes over % 00 () 3 ( 4)

21 Finding the percent of a number the number always goes in position 4. % 00 of () 3 a number In position (3) we find the answer. % 00 of answer a number This is the model we will use to solve percent problems. TYPE I Problems EXAMPLE Find 30% of 75. One position is taken up by 00. Percent always goes over ? 00? We are finding 30% of the number 75. Answer in last position. 30 answer Replace the word answer with the variable x, 30 x Solve: ( x) ( 30)( 75) x x 00 x 22.5 Therefore: 30% of

22 EXAMPLE 2 Find 60% of 35. Place the 00. Then 60% goes over ? 00? 60% of the number 35 goes where? Answer x, goes where? 60 x Solve 00 x ( 60)( 35) ( 60)( 35) x 00 Therefore: 60% of 35 Fill in the blank. All problems finding the percent of a certain number are called TYPE. TYPE II Problems % 00 EXAMPLE answer of a number 36 is what percent of 48? Fill in the 00. Percent over 00. We do not know this value. Let it x. x (48)(x) (36)(00) (36)(00) x (48) x is 75% of 48.

23 EXAMPLE 2 8 is what percent of 72? Place 00. Percent over 00. x x ( 8)( 00) x 25 Fill in the blank. Therefore: 8 is % of 72. TYPE III Problems % 00 of answer a number EXAMPLE 30 is 50% of what number? Place the 00. Percent over % of a number. We do not know the number, therefore let x the number x ( 50 )( x) ( 30)( 00) x Fill in the blanks. Therefore: 30 is 50% of? 2

24 EXAMPLE 2 8 is 25% of what number? Place the 00. % over 00. Let x the number Fill in the blanks. ( 25 )( x) ( 8)( 00) x Therefore: 8 is 25% of? 22

25 Experiential Activity Six Solve for the following:. 52% of % of % of % of is what percent of 250? is what percent of 48? Show Me is what percent of 3.5? is what percent of 30? 9. 7 is 30% of what number? 0. 8 is 75% of what number? Show Me is 0% of what number? is 90% of what number? The following exercise is a review of the 3 types of percent problems just presented in this module. Complete the table as shown in number and solve for the indicated unknown. Problem Type I, II, III Model Solution. 4 is what % of 52 II x x? 2. 30% of is 30% of a number 4..75% of a number is % of is what % of 75? is 95% of what number? 8. 95% of is what % of 90? 0. 3% of a number is 7 23

26 Experiential Activity Six Answers % % 7. 4% % ANSWERS for review exercise % % %

27 OBJECTIVE SEVEN When you complete this objective you will be able to Solve percent error in measurement problems. Exploration Activity The percent error in a measurement is calculated from: measured value true value % error 00 true value EXAMPLE In a laboratory experiment a student determined the velocity of sound to be 352 m/s. The true value under the same conditions is 343 m/s. Determine the percent error in the measurement. Solution: True value 343 m/s Measured value 352 m/s Substituting into the above equation, we get % error % Notice the answer is positive. If the measurement were less than the true value the answer would have been negative. 25

28 Experiential Activity Seven. An airport runway is measured to be 5362 m in length. Its true value was supposed to be 5400 m. Find the percent error in the length of the runway. 2. A surveyor's tape reads 00 m. However on a particular day it is actually 00.2 m in length. Find the percent error in its length. 3. A grocer's scale reads 3 kg on an item that is actually 2.85 kg. Find the percent error in the measurement. Is the customer getting a deal? 4. If the present length of a steel rail is 3.0 m, what will be its length after a 0.5 percent expansion caused by heating? 5. The volume of a gas is measured to be 58.5 ml. If this is 6% lower than the true volume, what is the true volume? Show Me. Experiential Activity Seven Answers % % %; no m ml Practical Application Activity Complete the ratio, proportion, and percent module assignment in TLM. Summary This module introduced the student to the basic concepts of ratio, proportion, and percent. 26

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