# 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

*x*:

*y*:: 3 : 2

Two variables are

*directly proportional*if one is a constant multiple of the other:

*y*=

*kx*

*k*is a constant.

*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.

*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?*

*–10**–2**5**20**6**0*

*“the sum of y and x is 80”*into an equation yields

*y*+

*x*= 80

*y*gives

*y*= 3

*x*.

Substituting this into the second equation yields

*x*+

*x*= 80

4

*x*= 80

*x*= 20

*y*= 3

*x*= 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.

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

*16**17**18**19**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).

*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?*

*225**300**450**525**6**00*

Hence, we set up a direct proportion.

Let

*x*be the actual distance between the cities.

Forming the proportion yields

*x*= 525

**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.

*inversely*proportional.

The statement “

*y*is inversely proportional to

*x*” is written as

*k*is a constant.

Multiplying both sides of by

*x*yields

*yx*=

*k*

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.

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

*3 hrs.**3 ½ hrs.**4 2/3 hrs.**5 hrs.**6**1/3 hrs.*

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

^{.}8 = 12

^{.}

*x*

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 .

*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?*

*15**27**30**34**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 = 6*A*, or *A* = 30.

The answer is choice (C).