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
10th Grade Math
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