**Ratios:**

Ratio, a very basic arithmetic concept commonly tested on reasoning/aptitude tests in different levels and different formats is simple to understand and sometimes tricky to handle.

"Ratios" - On the simplest level, by definition, it is a quantitative relation between 2 amounts showing the number of times one value contains or is contained within the other.

For example, the number of boys and girls in a class are in the ratio of 2:3 or 2/3, means that the number of boys is 2/3 of the number of girls. (There could be 20 boys and 30 girls)

Now, even if the actual number of boys and girls are not 2 and 3 respectively the ratio may still be 2:3.

For example, if the number of boys in a class were 12 and the number of girls were 18, then the ratio between the number of boys and girls still remains 12:18 simplified to 2:3.

On a bit higher level, the ratio can be expressed between more than 2 amounts too.

For example, ‘the ratio of number of roses to lilies to tulips in a bouquet of flowers is 1:3:5’, means that the number of roses and lilies are in the ratio of 1:3 while the number of lilies to tulips is 3:5. Again, the actual number of roses, lilies and tulips in the bouquet need not be exactly 1, 3, 5 respectively only. They could be in multiples. This means the flowers could be such that they are a set of 1 rose, 3 lilies and 5 tulips or 2 roses, 6 lilies and 10 tulips or 3 roses, 15 lilies and 18 tulips or so on. Generalising, they could be x, 3x and 5x respectively where x represents a positive integer.

The following 2 examples illustrate questions in which ‘ratio’ as a concept is tested in 2 different, commonly tested formats

**Example 1**:

If the ratio of the number of shirts, trousers, belts that Sam has is 2:5:13 and he has 120 accessories (shirts, trousers and belts) altogether, how many more trousers than Shirts does Sam have?

**Solution:**

Given the ratio between the shirts, trousers, belts is 2:5:13. The actual number of shirts, trousers and belts that Sam has can be assumed to be 2x, 5x, 13x (as explained above) where x is a positive integer. The sum of all of these equals 20x (2x+3x+15x) which is given in the question equivalent to 120. Hence, equating 20x to 120 we get to find the value of x as 6. Now, to know how many trousers than shirts did sam have, we can calculate it by subtracting 2x from 5x. We get it as 3x, which equals 3 times 6 = 18.

**Example 2**:

If the ratio of red colour marbles to green colour marbles in a bag of marbles is 3:5 and the ratio of green colour marbles to yellow colour marbles in the bag is 2:7, then what is the ratio of the number of red colour marbles to the number of yellow colour marbles?

**Solution:**

Given, the ratio between red colour marbles to green colour marbles = 3:5

Assuming, the number of red colour marbles = 3x

and hence the number of green colour marbles = 5x

Also, the ratio of the number of green colour marbles to yellow ones = 2:7

Assuming, the number of green colour marbles to equal = 2y

and hence the number of yellow colour marbles = 7y

Notice that the number of green colour marbles is assumed to be 5x as well as 2y. Equating them, 5x = 2y

Implying, x = 2y/5

The question asks us to find the ratio of number of red colour marbles to yellow ones.

The red coloured ones are 3x in number which also equals, 3x = 3 (2y/5) = 6y/5 (Substituting the x value)

Now, the required ratio of the number of red coloured marbles to number of yellow coloured marbles will be (6y/5) / (7y) = 6/35

**You may try doing this now:**

If the ratio of a to b to c is 2:3:5 and the ratio of b to c to d is 6:10:15, then what is the ratio of a to d?