1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 4.1 Ratios and Proportions

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Understand ratios. o Understand proportions.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Reciprocals A ratio is a comparison of two quantities by division. The ratio of a to b can be written as

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Compare the quantities 40 desks and 30 students as a ratio. Solution Example 1: Comparing Two Quantities as a Ratio The ratio is or in reduced form This means that there are 4 desks for every 3 students. (What is the ratio of desks to students in your classroom?)

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. During baseball season, major league players’ batting averages are published in the newspapers. If a player has a batting average of 0.250, what does this indicate? Solution A batting average is a ratio (or rate) of hits to times at bat. Thus a batting average of 0.250 means Example 2: Comparing Two Quantities as a Ratio

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Comparing Two Quantities as a Ratio (cont.) Reducing gives This means we can expect this player to get 1 hit for every 4 times at bat.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Comparing Two Quantities as a Ratio Write the comparison of 2 feet to 3 yards as a ratio. Solution a.We can write the ratio as b. Another procedure is to change to common units. Because 1 yd = 3 ft, we have 3 yd = 9 ft and we can write the ratio as an abstract number:

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Proportions A proportion is a statement that two ratios are equal. In symbols, is a proportion (b, d ≠ 0). A proportion is true if the cross products, and, are equal.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Therefore the cross products are equal and the proportion is true. Example 4: Verifying the Proportion Use the cross product technique to determine whether each proportion is true or false. Solution

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Verifying the Proportion (cont.) Solution Because the cross products are not equal (50 ≠ 56), the proportion is false.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Verifying the Proportion (cont.) Solution Because the cross products are equal, the proportion is true.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 5: Verifying the Proportion Determine whether the proportion is true or false. Solution Because the cross products are _________, the proportion is _______.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Write the following comparisons as ratios reduced to lowest terms. (Note: There may be more than one form for a correct answer.) 1.2 teachers to 24 students2.90 wins to 72 losses 3.147 miles to 3 hours 4.5 minutes to 200 seconds Determine whether the following proportions are true or false. 5. 6. 7.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.2. 3. 4.5. true6. false 7. true