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Lesson 4.3.2

Conversions and Proportions. Lesson 4.3.2. Lesson 4.3.2. Conversions and Proportions. California Standard: Algebra and Functions 2.1 Convert one unit of measurement to another (for example, from feet to miles, from centimeters to inches). What it means for you:

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Lesson 4.3.2

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  1. Conversions and Proportions Lesson 4.3.2

  2. Lesson 4.3.2 Conversions and Proportions California Standard: Algebra and Functions 2.1 Convert one unit of measurement to another(for example, from feet to miles, from centimeters to inches). What it means for you: You’ll convert measurements from one unit to another using proportions. • Key words: • proportion • conversion • ratio

  3. Lesson 4.3.2 Conversions and Proportions Last Lesson, you saw that to convert between different units you can multiply or divide by a conversion factor. But you can also think about conversion factors as ratios. And where there are ratios, proportions can’t be far behind. This Lesson is all about doing conversions using proportions.

  4. Lesson 4.3.2 Conversions and Proportions A Conversion Table Is a Set of Ratios A ratio is a way of comparing two quantities. But you’ve seen that ratios can also be used for converting quantities from one measuring system to another (think back to scale drawings, for example, where you saw things like “1 centimeter represents 10 meters”). In fact, you can think of the conversion tables you saw last Lesson as a table of ratios. For example, you can say the ratio of inches to feet is 12 : 1.

  5. Lesson 4.3.2 Conversions and Proportions Example 1 What is the ratio of: (i) feet to yards? (ii) yards to feet? Solution There are 3 feet in a yard. (i) This means the ratio of feet to yards is 3 : 1. (ii) Remember… the order of the quantities in a ratio is important. If the ratio of feet to yards is 3 : 1, then the ratio of yards to feet must be 1 : 3. Solution follows…

  6. Lesson 4.3.2 Conversions and Proportions Example 2 What is the ratio of: (i) meters to centimeters? (ii) centimeters to meters? Solution There are 100 centimeters in a meter. (i) The ratio of meters to centimeters is 1 : 100. (ii) The ratio of centimeters to meters is 100 : 1. Solution follows…

  7. Lesson 4.3.2 Conversions and Proportions Guided Practice What is the ratio of: 1. meters to kilometers? 2. kilometers to meters? 3. inches to yards? 4. yards to inches? 5. millimeters to centimeters? 6. centimeters to millimeters? 7. miles to feet? 8. feet to miles? 1000 : 1 1 : 1000 36 : 1 1 : 36 10 : 1 1 : 10 1 : 5280 5280 : 1 Solution follows…

  8. Lesson 4.3.2 Conversions and Proportions You Can Use Proportions to Convert Between Units You can use proportions to solve problems involving conversions. The method is exactly the same as the method you’ve seen in earlier Lessons. You find two equivalent ratios, write a proportion, then solve it using cross-multiplication.

  9. This is 1 : 10, or . 1 10 Then your second ratio is 8.5 : d, or . 8.5 d Lesson 4.3.2 Conversions and Proportions Example 3 The length of a bird is 8.5 cm. Use a proportion to find the length of the bird in millimeters. Solution You need two ratios to write a proportion. • The first ratio is the ratio of centimeters to millimeters. • The second ratio involves the length of the bird. The length in centimeters is 8.5 cm. Call its length in millimeters d. Solution follows… Solution continues…

  10. = 1 10 8.5 d Lesson 4.3.2 Conversions and Proportions Example 3 The length of a bird is 8.5 cm. Use a proportion to find the length of the bird in millimeters. Solution (continued) Now you can write and solve your proportion: d × 1 = 8.5 × 10 Cross-multiply d = 85 Simplify This means that the bird is 85 mm long.

  11. Lesson 4.3.2 Conversions and Proportions The last Example covered the method used to convert centimeters to millimeters: ● Find two ratios. ● Use them to write a proportion. ● Cross-multiply. ● Simplify. You use exactly the same method for converting millimeters to centimeters.

  12. The first ratio is the ratio of cm : mm, which is . Then your second ratio is d : 125.7, or . 1 10 d 125.7 Lesson 4.3.2 Conversions and Proportions Example 4 Convert 125.7 mm to centimeters. Solution As always, find two ratios. • The second ratio involves the length you’re converting. Call the length in centimeters d. Solution follows… Solution continues…

  13. = 1 10 d 125.7 Lesson 4.3.2 Conversions and Proportions Example 4 Convert 125.7 mm to centimeters. Solution (continued) Now write and solve a proportion: d × 10 = 1 × 125.7 Cross-multiply 10d = 125.7 Simplify Divide both sides of the equation by 10: d = 12.57 So 125.7 mm = 12.57 cm.

  14. 1 500 = 1 1 d d 10000 d = = 14 12 48 12 Lesson 4.3.2 Conversions and Proportions Guided Practice Use proportions to carry out the conversions in Exercises 9–11. 9. What is 48 inches in feet? d = 4 feet 10. What is 500 km in centimeters? d = 50,000,000 cm 11. Convert 14 inches into feet. d = 1.167 feet Solution follows…

  15. Lesson 4.3.2 Conversions and Proportions In Examples 3 and 4, the ratios were written without units. But if you prefer, you can include units in your ratios, just like you saw with scale drawings. The method works exactly the same.

  16. d d Then your second ratio is: 125.7 mm 125.7 mm 1 cm 1 cm 10 mm 10 mm This gives you a proportion: = Lesson 4.3.2 Conversions and Proportions Example 5 Convert 125.7 mm to centimeters. Solution Your first ratio is the ratio of centimeters to millimeters: Call the distance you need to find d. Solution follows… Solution continues…

  17. d 125.7 mm 1 cm 10 mm This gives you a proportion: = Lesson 4.3.2 Conversions and Proportions Example 5 Convert 125.7 mm to centimeters. Solution (continued) Solve by cross-multiplication in the usual way. d × 10 mm = 125.7 mm × 1 cm Cross-multiply d × 10 = 125.7 × 1 cm Divide both sides by “mm” Simplify 10d = 125.7 cm Divide both sides by 10 d = 125.7 cm

  18. 5280 9000 = d = 1.70 miles 1 d Lesson 4.3.2 Conversions and Proportions Guided Practice 12. What is the ratio of feet to miles? 5280 : 1 13. Convert 9000 feet into miles using your ratio from Exercise 12. Solution follows…

  19. Lesson 4.3.2 Conversions and Proportions Independent Practice 1. What is the ratio of yards to miles? 2. What is the ratio of miles to yards? 3. What is the ratio of centimeters to meters? 4. What is the ratio of meters to centimeters? 1760 : 1 1 : 1760 100 : 1 1 : 100 Solution follows…

  20. Lesson 4.3.2 Conversions and Proportions Independent Practice Use proportions to find the answers to Exercises 5–8. 5. Convert 7515 yards to miles. 6. Find 0.006 kilometers in millimeters. 7. Jonny needs 69 yards of fencing for his garden.What is this in feet? 8. An Egyptian camel trek is 8.75 km. How far is this in meters? 4.27 miles 6000 mm 207 ft 8750 m Solution follows…

  21. Lesson 4.3.2 Conversions and Proportions Round Up In this Lesson, you’ve learned to convert between units using proportions. You can use either method from the last two Lessons to solve conversion problems — you should get the same answer.

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