Ratios and Proportions

Definition:
A ratio is a comparison of two similar quantities obtained by dividing one quantity by the other. Ratios are written with the : symbol.


Example: The ratio of 6 to 3 is

6 ÷ 3 = 6/3 = 6 : 3 = 2

Example: The ratio of 3 to 6 is

3 ÷ 6 = 3/6 = 3 : 6 = 1/2



Notes about ratios:
Since a ratio is only a comparison or relation between quantities, it is an abstract number. For instance, the ratio of 6 miles to 3 miles is only 2, not 2 miles.

As you can see above, ratios can be written as fractions. They also have all the properties of fractions that you have learned in the previous part of this station.

The ratio of 6 to 3 should be stated as 2 to 1, but common usage has shortened the expression of ratios to be called simply 2.

If two quantities cannot be expressed in terms of the same unit, there cannot be a ratio between them.

Can you find these ratios?

1.2 quarts to 7 gallons

2.5 dollars to 25 cents

3.4 meters to 3 kilograms

Problem:
If two full time employees accomplish 20 tasks in a week, how many such tasks will 5 employees accomplish in a week?



2 : 5 = 20 : x
2 × x = 5 × 20
x = 50 tasks

This answer is obtained by knowing about proportions and how they are used. You can set up proportions by using ratios. Remember, ratios are comparing similar things. In the problem above, the first ratio is comparing employees and the second is comparing tasks.

Definition:
A proportion is a statement of the equality of two ratios.

Example: 6 : 3 = 2 : 1 or 6 / 3 = 2 / 1or
62
- = -
31
are ways to write the proportion expressed as: 6 is to 3 as 2 is to 1

Example: 2 : 8 = 1 : 4 or 2 / 8 = 1 / 4or
21
- = -
84
are ways to write the proportion expressed as: 2 is to 8 as 1 is to 4

Notes about proportions:

If any three terms in a proportion are given, the fourth may be found. Given the proportion:

a : b = c : d or a / b = c / d

and using the principals of manipulating equations , we observe that

a = (b × c) / d and c = (a × d) / b
b = (a × d) / c and d = (b × c) / a

An easy way to remember this is to say that in a proportion:

The product of the means is equal to the product of the extremes.

It is important to remember that to use the proportion, the ratios must be equal to each other and must remain constant.

Try a few proportion problems:

1.Find the value of x in 16 : 8 = x : 5

2.Find the value of x in 25 : 15 = 10 : x

3.A pipe transfers 236 gallons of fuel to the tank of a ship in 2 hours. How long will it take to fill the tank of the ship that holds 4543 gallons?

4.An I-beam 12 feet long weighs 52 pounds. How much does an I-beam of the same width weigh if it is 18 feet long?