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.
|
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
|
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.
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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