# Dividing a Quantity in a given Ratio

We will follow the rules of dividing a quantity in a given ratio (two or three) to solve different types of problems.

1. 20 apples are distributed between Aaron and Ben in the ratio 2 : 3. Find, how many does each get?

Solution:

Aaron and Ben get apples in the ratio 2 : 3 i.e., if Aaron gets 2 parts, B should get 3 parts.

In other words, if we make (2 + 3) = 5 equal parts, then Aaron should get 2 parts out of these 5 equal part

i.e. Aaron gets = $$\frac{2}{5}$$ of the total number of apples = $$\frac{2}{5}$$ of 20 = $$\frac{2}{5}$$ × 20 = 8 apples

Similarly, Ben gets 3 parts out of 5 equal parts

i.e. Ben gets = $$\frac{3}{5}$$ of the total number of apples = $$\frac{3}{5}$$ of 20 = $$\frac{3}{5}$$ × 20 = 12 apples

Therefore, Aaron gets 8 apples and Ben gets 12 apples.

In other way we can solve this by the direct method,

Since, the given ratio = 2 : 3 and 2 + 3 = 5

Therefore, Aaron gets = $$\frac{2}{5}$$ of the total number of apples

= $$\frac{2}{5}$$ × 20 apples = 8 apples

and, Ben gets = $$\frac{3}{5}$$ of the total number of apples

= $$\frac{3}{5}$$ × 20 apples = 12 apples

2. Divide $120 between David and Jack in the ratio 3 : 5. Solution: Ratio of David’s share to Jack’s share = 3 : 5 Sum of the ratio terms = 3 + 5 = 8 Thus we can say David gets 3 parts and Jack gets 5 parts out of every 8 parts. Therefore, David’s share = $$\frac{3}{8}$$ ×$ 120

= $$$\frac{3 × 120}{8}$$ =$ 45

And, Jack’s share = $$\frac{5}{8}$$ × $120 =$ $$\frac{5 × 120}{8}$$

= $75 Therefore, David get$ 45 and Jack gets $75. More solved problems on dividing a quantity in a given ratio: 3. Divide$ 260 among A, B and C in the ratio $$\frac{1}{2}$$ : $$\frac{1}{3}$$ : $$\frac{1}{4}$$.

Solution:

First of all convert the given ratio into its simple form.

Since, L.C.M. of denominators 2, 3 and 4 is 12.

Therefore, $$\frac{1}{2}$$ : $$\frac{1}{3}$$ : $$\frac{1}{4}$$ = $$\frac{1}{2}$$ × 12 : $$\frac{1}{3}$$ × 12 : $$\frac{1}{4}$$ × 12 = 6 : 4 : 3

And, 6 + 4 + 3 = 13

Therefore, A’ share = $$\frac{6}{13}$$ of $260 =$ $$\frac{6}{13}$$ × 260 = $120 B’ share = $$\frac{4}{13}$$ of$ 260 = $$$\frac{4}{13}$$ × 260 =$ 80

C’ share = $$\frac{3}{13}$$ of $260 =$ $$\frac{3}{13}$$ × 260 = $60 Therefore, A get$ 120, B gets $80 and C gets$ 60.

4. Two numbers are in the ratio 10 : 13. If the difference between the numbers is 48, find the numbers.

Solution:

Let the two numbers be 10 and 13

Therefore, the difference between these numbers = 13 – 10 = 3

Now applying unitary method we get,

When difference between the numbers = 3; 1st number = 10

⇒ when difference between the numbers = 1; 1st number = $$\frac{10}{3}$$

⇒ when difference between the numbers = 48; 1st number = $$\frac{10}{3}$$ × 48 = 160

Similarly, in the same way we get;

When difference between the numbers = 3; 1st number = 13

⇒ when difference between the numbers = 1; 1st number = $$\frac{13}{3}$$

⇒ when difference between the numbers = 48; 1st number = $$\frac{13}{3}$$ × 48 = 208

Therefore, the required numbers are 160 and 208.

The above examples on dividing a quantity in a given ratio will give us the idea to solve different types of problems on ratios.

5. Divide $40 in the ratio of 3 : 2 Solution: Sum of the terms of ratio 3 : 2 = 3 + 2 = 5 1st part of$ 40 = $$\frac{3}{5}$$ × $40 =$ 24

2nd part of $40 = $$\frac{2}{5}$$ ×$ 40

= \$ 16

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