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

1^st part of $ 40 = \(\frac{3}{5}\) × $ 40

                       = $ 24

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

                        = $ 16

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