# Dividing a Quantity into Three Parts in a Given Ratio

We will discuss here how to solve different types of word problems on dividing a quantity into three parts in a given ratio.

1. Divide $5405 among three children in the ratio 1$$\frac{1}{2}$$ : 2 : 1$$\frac{1}{5}$$. Solution: Given ratio = 1$$\frac{1}{2}$$ : 2 : 1$$\frac{1}{5}$$ = $$\frac{3}{2}$$ : 2 : $$\frac{6}{5}$$ Now multiply each term by the L.C.M. of the denominators = $$\frac{3}{2}$$ × 10 : 2 × 10 : $$\frac{6}{5}$$ × 10, [Since, L.C.M. of 2 and 5 = 10] = 15 : 20 : 12 So, the amount received by three children are 15x, 20x and 12x. 15x + 20x + 12x = 5405 ⟹ 47x = 5405 ⟹ x = $$\frac{5405}{47}$$ Therefore, x = 115 Now, 15x = 15 × 115 =$ 1725

20x = 20 × 115 = $2300 12x = 12 × 115 =$ 1380

Therefore, amount received by three children are $1725,$ 2300 and $1380. 2. A certain sum of money is divided into three parts in the ratio 2 : 5 : 7. If the third part is$224, find the total amount, the first part and second part.

Solution:

Let the amounts be 2x, 5x and 7x

According to the problem,

7x = 224

⟹ x = $$\frac{224}{7}$$

Hence, x = 32

Therefore, 2x = 2 × 32 = 64 and 5x = 5 × 32 =160.

So, the first amount = $64 and the second amount =$ 160

Hence, total amount = First amount + Second amount + Third amount

= $64 +$ 160 + $224 =$ 448



3.  A bag contains $60 of which some are 50 cent coins, some are$ 1 coins and the rest are $2 coins. The ratio of the number of respective coins is 8 : 6 : 5. Find the total number of coins in the bag. Solution: Let the number of coins be a, b and c respectively. Then, a : b : c is equal to 8 : 6 : 5 Therefore, a = 8x, b = 6x, c = 5x Therefore, the total sum = 8x × 50 cent + 6x ×$ 1 + 5x × $2 =$ (8x × $$\frac{1}{2}$$ + 6x ×  1 + 5x ×  2)

= $(4x + 6x + 10x) =$ 20x

Therefore, according to the problem,

$20x =$ 60

⟹ x = $$\frac{ 60}{ 20}$$

⟹ x = 3

Now, the number of 50 cent coins = 8x = 8 × 3 = 24

The number of $1 coins = 6x = 6 × 3 = 18 The number of$ 2 coins = 5x = 5 × 3 = 15

Therefore, the total number of coins =  24 + 18 + 15 = 57.

4. A bag contains $2,$ 5 and 50 cent coins in the ratio 8 : 7 : 9. The total amount is $555. Find the number of each denomination. Solution: Let the number of each denomination be 8x , 7x and 9x respectively. The amount of$ 2 coins = 8x × 200 cents = 1600x cents

The amount of $5 coins = 7x × 500 cents = 3500x cents The amount of 50 cent coins = 9x × 50 cents = 450x cents The total amount given = 555 × 100 cents = 55500 cents Therefore, 1600x + 3500x + 450x = 55500 ⟹ 5550x = 55500 ⟹ x = $$\frac{55500}{5550}$$ ⟹ x = 10 Therefore, the number of$ 2 coins = 8 × 10 = 80

The number of \$ 5 coins = 7 × 10 = 70

The number of 50 cent coins = 9 × 10 = 90

● Ratio and proportion