holt ca course 1 5-4 solving proportions ns1.3 use proportions to solve problems (e.g., determine...
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Holt CA Course 1
5-4 Solving Proportions
NS1.3 Use proportions to solve problems (e.g., determine the value of N if = , find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Also covered: AF2.2, AF2.3
California Standards
4 7
N 21
Holt CA Course 1
5-4 Solving Proportions
Vocabulary
cross productThe product of the numerator in one ratio and the denominator in the other when comparing two ratios.
Holt CA Course 1
5-4 Solving Proportions
For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product.
If the two ratios form a proportion, then the cross products are equal.
5 · 6 = 30
2 · 15 = 30=2
56
15
Holt CA Course 1
5-4 Solving Proportions
You can use the cross product rule to solve proportions with variables.
Holt CA Course 1
5-4 Solving Proportions
Use cross products to solve the proportion.
Example 1: Solving Proportions Using Cross Products
15 · m = 9 · 5
15m = 45
15m15
= 4515
m = 3
The cross products are equal.
Multiply.
Divide each side by 15.
= m5
915
Holt CA Course 1
5-4 Solving Proportions
It is important to set up proportions correctly. Each ratio must compare corresponding quantities in the same order.
16 mi4 hr
= 8 mix hr
16 mi8 mi
= 4 hrx hr
Holt CA Course 1
5-4 Solving Proportions
The graph shows the time and distance Greg walked on his way to school. At this rate, how long would it take Greg to walk 7.5 miles?
Example 2: Sports Application
The labeled point on the graph shows that Greg walked 0.75 miles in 0.25 hr. Let t represent the time in hours it will take Greg to walk 7.5 miles. 0
0.1 0.2 0.3 0.4 0.5
0.25
0.5
0.75
Dis
tan
ce (
mi)
Walking Rate
Time (hr)
(0.25, 0.75)
Holt CA Course 1
5-4 Solving Proportions
The graph shows the time and distance Greg walked on his way to school. At this rate, how long would it take Greg to walk 7.5 miles?
Example 2 Continued
0.75 · t = 0.25 · 7.5
0.75t = 1.875
0.75t0.75
= 1.8750.75
t = 2.5 hr
DistanceTime
Divide each side by 0.75.
= 7.5 mi t hr
0.75 mi0.25 hr
Set up a proportion in which each ratio compares distance to the time needed to walk that distance.
The cross products are equal.
Multiply.
It will take Greg 2.5 hours to walk 7.5 miles.
Holt CA Course 1
5-4 Solving ProportionsExample 3: Problem Solving Application
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?
11 Understand the Problem
Rewrite the question as a statement.
• Find the space needed for 26 volumes of the encyclopedia.
List the important information:
• 3 volumes of the encyclopedia take up 4 inches of space.
Holt CA Course 1
5-4 Solving Proportions
22 Make a Plan
Set up a proportion using the given information. Let x represent the inches of space needed.
3 volumes4 inches
= 26 volumesx
volumesinches
Additional Example 3 Continued
Holt CA Course 1
5-4 Solving Proportions
Solve33
34
= 26x
3 · x = 4 · 26
3x = 104
3x3
= 1043
x = 34 23
She needs 34 23
inches for all 26 volumes.
Write the proportion.
The cross products are equal.
Multiply.
Divide each side by 3 to isolate the variable.
Additional Example 3 Continued
Holt CA Course 1
5-4 Solving Proportions
Look Back44
34
= 2634 2
3
4 · 26 = 104
3 · 34 23
= 104
The cross products are equal, so 34 23
is theanswer.
Additional Example 3 Continued
Holt CA Course 1
5-4 Solving Proportions
Check It Out! Example 1
Use cross products to solve the proportion.
7 · m = 6 · 14
7m = 84
7m7
= 847
m = 12
The cross products are equal.
Multiply.
Divide each side by 7 to isolate the variable.
67
= m14
Holt CA Course 1
5-4 Solving Proportions
The graph shows the time and distance Paige ran in one marathon. At this rate, how long would it take Paige to run 8.5 miles?
Check It Out! Example 2
The labeled point on the graph shows that a Paige ran 2.5 miles in 0.15 hr. Let t represent the time in hours it will take Paige to run 8.5 miles.
0 0.1 0.2 0.3 0.4 0.5
1.0
2.0
3.0
Dis
tan
ce (
mi)
Running Rate
Time (hr)
(0.15, 2.5)
Holt CA Course 1
5-4 Solving ProportionsCheck It Out! Example 2 Continued
2.5 · t = .15 · 8.5
2.5t = 1.275
2.5t2.5
= 1.2752.5
t = .51 hr
DistanceTime
Divide each side by 2.5.
= 8.5 mi t hr
2.5 mi.15 hr
Set up a proportion in which each ratio compares distance to the time needed to run that distance.
The cross products are equal.
Multiply.
The graph shows the time and distance Paige ran in one marathon. At this rate, how long would it take Paige to run 8.5 miles?
Holt CA Course 1
5-4 Solving Proportions
Check It Out! Example 3John 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?
11 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.
Holt CA Course 1
5-4 Solving Proportions
22 Make a Plan
Set up a proportion using the given information. Let p represent the pints of coolant.
6 pints10 inches
= p25 inches
pints inches
Check It Out! Example 3 Continued
Holt CA Course 1
5-4 Solving Proportions
Solve33
610
= p25
10 · p = 6 · 25
10p = 150
10p10
= 15010
p = 15
15 pints of coolant will fill the radiator to the 25 inch level.
Write the proportion.
The cross products are equal.
Multiply.
Divide each side by 10 to isolate the variable.
Check It Out! Example 3 Continued
Holt CA Course 1
5-4 Solving Proportions
Look Back44
610
= 1525
10 · 15 = 150
6 · 25 = 150
The cross products are equal, so 15 is the answer.
Check It Out! Example 3 Continued