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Proportions and Unit Conversions
5-3: Solving Proportions
2 = 6 5 15
The tall stack of Jenga® blocks is 25.8 cm tall. How tall is the shorter stack of blocks? To find the answer you will need to solve a proportion.
5 · 6 = 30 2 · 15 = 30
CROSS PRODUCT RULE For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product. If the cross products of the ratios are equal, then the ratios form a proportion.
In the proportion a = c , the cross products, b d a · d and b · c are equal. You can use the cross product rule to solve proportions with variables.
Insert Lesson Title Here Additional Example 1: Solving Proportions Using Cross Products Use cross products to solve the proportion. 9 = m 15 5 15 · m = 9 · 5 15m = 45 15m = 45 15 15 m=3
The cross products are equal. Multiply. Divide each side by 15 to isolate the variable.
Try This: Example 1 Use cross products to solve the proportion. 6 = m 7 14 7 · m = 6 · 14
The cross products are equal.
7m = 84
Multiply.
7m = 84 7 7 m = 12
Divide each side by 7 to isolate the variable.
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Additional Example 2: Problem Solving Application When setting up a proportion to solve a problem, use a variable to represent the number you want to find. In proportions that include different units of measurement, either the units in the numerators must be the same and the units in the denominators must be the same or the units within each ratio must be the same. 16 mi = 8 mi 4 hr x hr
If 3 volumes of Jennifer’s encyclopedia takes up 4 inches of space on her shelf, how much space will she need for all 26 volumes? 1
Understand the Problem Rewrite the question as a statement. • Find the space needed for 26 volumes of the encyclopedia.
16 mi = 4 hr 8 mi x hr
List the important information: • 3 volumes of the encyclopedia take up 4 inches of space.
Additional Example 2 Continued 2
Make a Plan
Additional Example 2 Continued 3
3 volumes = 26 volumes 4 inches x
Let x be the unknown space.
3x = 104
Try This: Example 2 John filled his new radiator with 6 pints of coolant, which is the 10 inch mark. How many pints of coolant would be needed to fill the radiator to the 25 inch level?
Look Back
3 = 4
26 34 23
4 · 26 = 104 1 3·
34 23
= 104
The cross products are equal, so 3423 is the answer
Multiply.
Divide each side by 3 to isolate 3x = 104 the variable. 3 3 x = 34 2 3 She needs 34 2 inches for all 26 volumes. 3
Additional Example 2 Continued 4
Solve
3 = 26 Write the proportion. 4 x 3 · x = 4 · 26 The cross products are equal.
Set up a proportion using the given information.
Understand the Problem
Rewrite the question as a statement. • Find the number of pints of coolant required to raise the level to the 25 inch level. List the important information: • 6 pints is the 10 inch mark.
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Try This: Example 2 Continued 2
Try This: Example 2 Continued
Make a Plan
3
Set up a proportion using the given information.
6 = x Write the proportion. 10 25 10 · x = 6 · 25 The cross products are equal.
6 pints 10 inches
=
x 25 inches
Let x be the unknown amount.
Solve
10x = 150
Multiply.
10x = 150 10 10 x = 15
Divide each side by 10 to isolate the variable.
15 pints of coolant will fill the radiator to the 25 inch level.
Try This: Example 2 Continued 4
Look Back
6 = 10
15 25
10 · 15 = 150
5-4: Dimensional Analysis (unit conversions)
6 · 25 = 150 The cross products are equal, so 15 is the answer.
You can use a unit conversion factor to change, or convert, measurements from one unit to another. A unit conversion factor is a fraction in which the numerator and denominator represent the same quantity, but in different units. The fraction below is a unit conversion factor that can be used to convert miles to feet. Notice that it can be simplified to one. 5,280 ft = 5,280 ft 5,280 ft 1 mi
Multiplying a quantity by a unit conversion factor changes only its units, not its value. The process of choosing an appropriate conversion factor is called dimensional analysis.
=1
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Additional Example 1: Making Unit Conversions An oil drum holds 55 gallons. How many quarts of oil will fill the drum? Use a unit conversion factor to convert the units.
Helpful Hint When choosing a unit conversion factor, choose the one that cancels the units you want to change and replaces them with the units you want.
One gallon equals 4 quarts so use the conversion factor 1 gal or 4 qt . Choose the second one so the 4 qt
1 gal
gallon units will “cancel.” 55 gal · 4 qt = 55 · 4 qt 1 gal
1
Multiply.
= 220 qt 220 quarts of oil will fill the drum.
Dimensional Analysis Try This: Example 1 An ice cream recipe calls for 7 quarts of milk. How many pints of milk is this? Use a unit conversion factor to convert the units. One quart equals 2 pints so use the conversion factor 1 qt or 2 pt . Choose the second one so the 2 pt
1 qt
quarts units will “cancel.” 7 qt ·
7 · 2 pt 2 pt = 1 1 qt
Multiply.
= 14 pt 7 quarts of milk is 14 pints. Course 2
Additional Example 2A: Making Rate Conversions Use a unit conversion factor to convert the units within each rate. A. If orange juice sells for $1.28 per gallon, what is the cost per ounce? $1.28 · 1 gal · 1 qt = $1.28 · 1 · 1 gal 4 qt 32 oz 1 · 4 · 32 oz $1.28 = 128 oz $1.28 per gallon $1.28 ÷ 128 is $0.01 per = 128 oz ÷ 128 ounce. $0.01 = 1 oz
Multiply.
Course 2
Insert Lesson Title Here Additional Example 2B: Making Rate Conversions
Try This: Example 2A
Use a unit conversion factor to convert the units within each rate.
Use a unit conversion factor to convert the units within each rate.
B. Convert 80 miles per hour to miles per minute.
If milk sells for $2.24 per gallon, what is the cost per pint?
80 mi · 1 80 mi · 1 hr = 1 hr 60 min 1 · 60 min
Multiply.
80 mi ÷ 60 60 min ÷ 60 1.33 mi ≈ 1 min =
80 miles per hour is about 1.33 miles per minute.
$2.24 · 1 gal · 1 qt = $2.24 · 1 · 1 gal 4 qt 2 pt 1 · 4 · 2 pt $2.24 = 8 pt $2.24 per gallon $2.24 is $0.28 per pint. = 8 pt $0.28 = 1 pt
Multiply.
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Insert Lesson Title Here Try This: Example 2B Use a unit conversion factor to convert the units within each rate. B. Convert 50 miles per hour to miles per minute. 50 mi · 1 hr 50 mi · 1 = 1 hr 60 min 1 · 60 min
Multiply.
50 mi 60 min 0.833 mi ≈ 1 min
=
50 miles per hour is about 0.83 miles per minute.
Additional Example 3: Measurement Application The Mare Orientale crater on the Moon is more than 620 miles across. How many meters is this? Use unit conversion factors that convert miles to to kilometers, and then kilometers to meters. One kilometer is equivalent to 0.62 mile. 1 km 1,000 m 620 mi · · = 620 · 1 · 1,000 m 0.62 mi 1 km 0.62 = 620,000 m ÷ 0.62 0.62 ÷ 0.62 1,000,000 m = 620 miles is 1,000,000 meters.
Try This: Example 3 Mary went to the grocery store to buy 5 pounds of peaches. How many grams is this? Use unit conversion factors that convert pounds to to kilograms, and then kilograms to grams. One kilogram is equivalent to 2.2 pounds. 1 kg 2.2 lb
1,000 g = 5 · 1 · 1,000 g 1 kg 2.2 = 5,000 g ÷ 2.2 2.2 ÷ 2.2 ≈ 2,273 g 5 pounds is about 2,273 grams. 5 lb ·
·
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