Ratios

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**

- 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.

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 / 4 or*

**21
- = -
84**

*are ways to write the proportion expressed as: 2 is to 8 as 1 is to 4*

Proportions are built from ratios. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20.

Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, its number must come first. If the expression had been "the ratio of women to men", then the numbers would have been "20 to 15".

Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:

**odds notation: 15 : 20**

**fractional notation: 15/20**

You should be able to recognize all three notations; you will probably be expected to know them for your test.

Given a pair of numbers, you should be able to write down the ratios. For example:

**There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese in all three formats.**

**Consider the above park. Express the ratio of geese to ducks in all three formats.**

The numbers were the same in each of the above exercises, but the order in which they were listed differed, varying according to the order in which the elements of the ratio were expressed. In ratios, order is very important.

Let's return to the 15 men and 20 women in our original group. I had expressed the ratio as a fraction, namely, 15/20. This fraction reduces to 3/4. This means that you can also express the ratio of men to women as 3/4, 3 : 4, or "3 to 4".

This points out something important about ratios: the numbers used in the ratio might not be the absolute measured values. The ratio "15 to 20" refers to the absolute numbers of men and women, respectively, in the group of thirty-five people. The simplified or reduced ratio "3 to 4" tells you only that, for every three men, there are four women. The simplified ratio also tells you that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men. In other words, the men comprise 3/7 of the people in the group. These relationships and reasoning are what you use to solve many word problems:

**In a certain class, the ratio of passing grades to failing grades is 7 to 5. How many of the 36 students failed the course?**

The ratio, "7 to 5" (or 7 : 5 or 7/5 ), tells me that, of every 7 + 5 = 12 students, five failed. That is, 5/12 of the class flunked. Then ( 5/12 )(36) = **15 students failed.**

**In the park mentioned above, the ratio of ducks to geese is 16 to 9. How many of the 300 birds are geese?**

The ratio tells me that, of every 16 + 9 = 25 birds, 9 are geese. That is, 9/25 of the birds are geese. Then there are ( 9/25 )(300) = **108 geese.**

*Generally, ratio problems will just be a matter of stating ratios or simplifying them. For instance:*

Consider the following :

- Length of a room is 30 m and its breadth is 20 m. So, the ratio of length of the room to the breadth of the room = 30/20 = 3/2 = 3:2
- There are 24 girls and 16 boys going for a picnic. Ratio of the number of

girls to the number of boys 24/16 = 3/2 = 3:2

The ratio in both the examples is 3 : 2.

** Example : **Length and breadth of a rectangular field are 50 m and 15m respectively. Find the ratio of the length to the breadth of the field.

**Solution : **Length of the rectangular field = 50 m Breadth of the rectangular field = 15 m The ratio of the length to the breadth is 50 : 15

The ratio can be written as

Thus, the required ratio is 10 : 3.

**Example : **Divide Rs 60 in the ratio 1 : 2 between Kriti and Kiran.

**Solution :** The two parts are 1 and 2.

Therefore, sum of the parts = 1 + 2 = 3.

This means if there are Rs 3, Kriti will get Re 1 and Kiran will get Rs 2.

Or, we can say that Kriti gets 1 part and Kiran gets 2 parts out of every 3 parts.

Therefore, Kriti’s share =1/3 60 = Rs 20

And Kiran’s share =2/3 60 = Rs 40.

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