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Lesson 4.3.2Lesson 4.3.2

Conversions and Proportions

Conversions and Proportions

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Lesson

4.3.2Conversions and ProportionsConversions and Proportions

California Standard:Algebra and Functions 2.1Convert 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

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Lesson

4.3.2Conversions and ProportionsConversions 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.

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A Conversion Table Is a Set of Ratios

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

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.

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Example 1

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

What is the ratio of:

(i) feet to yards?

(ii) yards to feet?

Solution

(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.

There are 3 feet in a yard.

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Example 2

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

What is the ratio of:

(i) meters to centimeters?

(ii) centimeters to meters?

Solution

(i) The ratio of meters to centimeters is 1 : 100.

(ii) The ratio of centimeters to meters is 100 : 1.

There are 100 centimeters in a meter.

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Guided Practice

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

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?

10 : 1

1 : 10

1000 : 1

1 : 1000

36 : 1

1 : 36

1 : 5280

5280 : 1

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You Can Use Proportions to Convert Between Units

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

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.

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Example 3

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

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.

This is 1 : 10, or .110

• The second ratio involves the length of the bird. The length in centimeters is 8.5 cm. Call its length in millimeters d.

Then your second ratio is 8.5 : d, or .8.5d

Solution continues…

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Example 3

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

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:

110

8.5d=

Cross-multiplyd × 1 = 8.5 × 10

Simplifyd = 85

This means that the bird is 85 mm long.

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Lesson

4.3.2Conversions and ProportionsConversions 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.

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Example 4

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

Convert 125.7 mm to centimeters.

Solution

As always, find two ratios.

• The first ratio is the ratio of cm : mm, which is .1

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• The second ratio involves the length you’re converting.

Call the length in centimeters d.

Then your second ratio is d : 125.7, or .d

125.7

Solution continues…

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Example 4

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

Convert 125.7 mm to centimeters.

Solution (continued)

Now write and solve a proportion:

Cross-multiplyd × 10 = 1 × 125.7

110

d125.7=

Simplify10d = 125.7

Divide both sides of the equation by 10: d = 12.57

So 125.7 mm = 12.57 cm.

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Guided Practice

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

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?

11. Convert 14 inches into feet.

d = 50,000,000 cm

d = 1.167 feet

1

12

d

48=

1

10000

500

d=

1

12

d

14=

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Lesson

4.3.2Conversions and ProportionsConversions 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.

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Example 5

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

Convert 125.7 mm to centimeters.

Solution

Your first ratio is the ratio of centimeters to millimeters:

1 cm10 mm

Call the distance you need to find d.

Then your second ratio is: d125.7 mm

d125.7 mm

This gives you a proportion: 1 cm10 mm

=

Solution continues…

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Example 5

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

Convert 125.7 mm to centimeters.

Solution (continued)

Solve by cross-multiplication in the usual way.

Cross-multiplyd × 10 mm = 125.7 mm × 1 cm

Simplify10d = 125.7 cm

Divide both sides by “mm”d × 10 = 125.7 × 1 cm

Divide both sides by 10d = 125.7 cm

d125.7 mm

This gives you a proportion: 1 cm10 mm

=

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Guided Practice

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

12. What is the ratio of feet to miles?5280 : 1

d = 1.70 miles5280

1= 9000

d

13. Convert 9000 feet into miles using your ratio from Exercise 12.

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Independent Practice

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

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

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Independent Practice

Solution follows…

Lesson

4.3.2Conversions and ProportionsConversions and Proportions

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

Use proportions to find the answers to Exercises 5–8.

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Lesson

4.3.2Conversions and ProportionsConversions and Proportions

Round UpRound 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.