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Proportion

A proportion is simply an equality between two ratios (fractions).
For example, the ratio of
x to y is equal to the ratio of 3 to 2 is translated as

or in ratio notation,
x : y :: 3 : 2

Two variables are
directly proportional if one is a constant multiple of the other:
y = kx
where k is a constant.
 
The above equation shows that as x increases (or decreases) so does y. This simple concept has numerous applications in mathematics. For example, in constant velocity problems, distance is directly proportional to time: d = vt, where v is a constant.
 
Note, sometimes the word directly is suppressed.
 
Example-2

If the ratio of y to x is equal to 3 and the sum of y and x is 80, what is the value of y?

A.  –10
B.  –2
C.  5
D.  20
E.  60


 Translating “the ratio of y to x is equal to 3” into an equation yields
Translating “the sum of y and x is 80” into an equation yields
y + x = 80

 

Solving the first equation for y gives y = 3x. Substituting this into the second equation yields
3x + x = 80
4x = 80
x = 20

 

Hence, y = 3x = 3(20) = 60. The answer is (E).
 

 

In many word problems, as one quantity increases (decreases), another quantity also increases (decreases). This type of problem can be solved by setting up a direct proportion.
 
Example-3

If Biff can shape 3 surfboards in 50 minutes, how many surfboards can he shape in 5 hours?

A.  16
B.  17
C.  18
D.  19
E.  20


As time increases so does the number of shaped surfboards. Hence, we set up a direct proportion. First, convert 5 hours into minutes: 5 hours = 5 × 60 minutes = 300 minutes. Next, let x be the number of surfboards shaped in 5 hours. Finally, forming the proportion yields

18 = x
The answer is (C).
 

 

Example-4

On a map, 1 inch represents 150 miles. What is the actual distance between two cities if they are 3½ inches apart on the map?

A.  225
B.  300
C.  450
D.  525
E.  600


As the distance on the map increases so does the actual distance. Hence, we set up a direct proportion. Let x be the actual distance between the cities. Forming the proportion yields

x = 525
The answer is (D).

Note, you need not worry about how you form the direct proportion so long as the order is the same on both sides of the equal sign. The proportion in Example 4 could have been written as . In this case, the order is inches to inches and miles to miles. However, the following is not a direct proportion because the order is not the same on both sides of the equal sign: . In this case, the order is inches to miles on the left side of the equal sign but miles to inches on the right side.

 

If one quantity increases (or decreases) while another quantity decreases (or increases), the quantities are said to be inversely proportional. The statement “y is inversely proportional to x” is written as

where k is a constant.
Multiplying both sides of  by x yields
yx = k

 

Hence, in an inverse proportion, the product of the two quantities is constant. Therefore, instead of setting ratios equal, we set products equal.
 

 

In many word problems, as one quantity increases (decreases), another quantity decreases (increases). This type of problem can be solved by setting up a product of terms.
 
Example-5

If 7 workers can assemble a car in 8 hours, how long would it take 12 workers to assemble the same car?

A.  3 hrs.
B.  3 1/2 hrs.
C.  4 2/3 hrs.
D.  5 hrs.
E.  6 1/3 hrs.


As the number of workers increases, the amount of time required to assemble the car decreases. Hence, we set the products of the terms equal. Let x be the time it takes the 12 workers to assemble the car. Forming the equation yields
7 — 8 = 12 — x

 

The answer is (C).
 

 

To summarize: if one quantity increases (decreases) as another quantity also increases (decreases), set ratios equal. If one quantity increases (decreases) as another quantity decreases (increases), set products equal.

 

The concept of proportion can be generalized to three or more ratios.  A, B, and C are in the ratio 3:4:5 means , and .
 
Example-6
In the figure, the angles A, B, C of the triangle are in the ratio 5:12:13. What is the measure of angle A?
A.   15
B.   27
C.   30
D.   34
E.    40

 

Since the angle sum of a triangle is 180˚, A + B + C = 180.  Forming two of the ratios yields
      
Solving the first equation for B yields
Solving the second equation for C yields
Hence, . Therefore, 180 = 6A, or A = 30. The answer is choice (C).
 





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