Proportions

This lesson about proportions will teach you some very important concepts you need to know. You should review the lesson about ratio before studying this lesson.

Take a look at the image below. Try to measure the ratio of arm's length to height for 5 people that you know. What did you notice? More on this later, so keep reading!

Definition:

Simply put, whenever we put an equal sign between two ratios and the ratio on the left is equal to the ratio on the right, we say that they form a proportion.

The ratio of arm's length to height
Look at the following two ratios. Right now they are just ratios.

 
3 / 5
and   
6 / 12


Do they form a proportion? They will form a proportion if the ratios are equal, so first put an equal sign between them.

Then, you have two choices to check if the ratios are equal.

Choice #1: You can just convert both ratios into decimals and see if the decimals are equal.

 
3 / 5
=   
6 / 12


3 / 5
= 0.60
  
6 / 12
= 0.5


Since 0.60 ≠ 0.50, the ratios are not equal and therefore do not for a proportion.

Choice #2: You can also tell if two ratios are equal by comparing their cross products. If the cross products are equal, then they form a proportion.

Recall that a cross product is obtained when you multiply the numerator of one fraction by the denominator of another fraction.

The cross products for  
3 / 5
 and  
6 / 12
 are


3 × 12 = 36 and

5 × 6 = 30

36 is not equal to 30. Therefore,
3 / 5
  and  
6 / 12
  do not form a proportion


Another example: Do 
10 / 16
  and  
5 / 8
  form a proportion ?


10 × 8 = 80 and

16 × 5 = 80

80 is equal to 80, therefore, they form a proportion.

When ratios are equal, they are also called equivalent fractions.

In general, if two fractions are equivalent, they form a proportion.

If  
a / b
=    
c / d
  is a proportion


 
3 / 5
and   
6 / 12


Do they form a proportion? They will form a proportion if the ratios are equal, so first put an equal sign between them.

Then, you have two choices to check if the ratios are equal.

Choice #1: You can just convert both ratios into decimals and see if the decimals are equal.

 
3 / 5
=   
6 / 12


3 / 5
= 0.60
  
6 / 12
= 0.5


Since 0.60 ≠ 0.50, the ratios are not equal and therefore do not for a proportion.

Choice #2: You can also tell if two ratios are equal by comparing their cross products. If the cross products are equal, then they form a proportion.

Recall that a cross product is obtained when you multiply the numerator of one fraction by the denominator of another fraction.

The cross products for  
3 / 5
 and  
6 / 12
 are


3 × 12 = 36 and

5 × 6 = 30

36 is not equal to 30.

Therefore,

3 / 5
  and  
6 / 12
  do not form a proportion


Another example:

Do 
10 / 16
  and  
5 / 8
  form a proportion ?


10 × 8 = 80 and

16 × 5 = 80

80 is equal to 80, therefore, they form a proportion.

When ratios are equal, they are also called equivalent fractions.

In general, if two fractions are equivalent, they form a proportion.

If  
a / b
=    
c / d
  is a proportion


Then, the proportion can also be written as   a:b::c:d

We can read   a:b::c:d  as " a is to b as c is to d "

The format above make it easy also to identify the means and the extremes.

Means and extremes in a proportion
In any proportion the product of the extremes is equal to the product of the means. It is just a different way of wording the procedure of cross multiplication.



Fourth proportional:

Looking at a:b::c:d, the fourth term is d. We call d fourth proportional.

Equivalent proportions:

You can get an equivalent proportion by inverting each ratio:

a:b::c:d is the same as b:a::d:c

You can get others equivalent proportions by interchanging either the means, the extremes or both

a:b::c:d is the same as a:c::b:d, d:b::c:a and d:c::b:a

Thinking a little deeper about proportions


Wages:

Is your wage or the money you make per hour a proportion?

As long as the wage stays the same, it will form a proportion

Suppose you make 25 dollars per hour and work 8 hours a day

Money per hour is 25:1

Money at the end of the day is 200:8

Let us see if 25:1::200:8 form a proportion

Product of means is 1 × 200 = 200

Product of extremes is 25 × 8 = 200

Ratio of arm's length to height:

Let us now talk about the picture shown at the beginning of this lesson.

In adult males, arm's length is about 5 cm greater than the height.

Is it ok to say that the ratio of arm's length to height form a proportion for adult males?

If you are willing to neglect the tiny differences and use exactly 5 cm, the answer is yes.

Say your husband's height is 152.4 centimeter, the arm length is then 152.4 + 5 = 157.4

Now, if you are a male and your height is 165 centimeter, the arm length is then 165 + 5 = 170

157.4 / 152.4
= 1.03032


170 / 165
= 1.03030


Since the ratios are closely equal, you could make a claim and say that ratio of arm's length to height for adult males form a proportion.



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