Shortcuts for Ratio and Proportion – 2

Shortcut 8: Finding number of coins in a bag

pAx+qBx+rCx=T

p : q : r = ratio of number of coins

A, B, C = value of respective coins.

T = total amount; x = common factor.

Question:

A bag contains 5 paise, 10 paise and 20 paise coins in the ratio 2:4:5. Total amount in the bag is Rs. 4.50. How many coins are there in 20 paise?

(2)(5)x + (4)(10)x + (5)(20)x = (4.50)(100)

To convert Rs. 4.50 into paise, multiply it by 100.

10x + 40x + 100x = 450

150x = 450 x = 3

Substitute x = 3 in 2x : 4x : 5x to get number of coins.

5x = number of 20 paise coins = 15

Shortcut 9: 1 : 2 to 2 : 1

Ans=Initial quantity

Question:

In a 45 liters mixture milk and water are in the ratio 1:2. After adding certain quantity of milk, the ratio becomes 2:1. What is the quantity of milk added?

45, since it is the initial quantity.

Proof:

Initial quantity of milk = (1/3) x 45 = 15 liters

Initial quantity of water = 30 liters

Final ratio of milk and water = 2 : 1

Final quantity of water = 30 = (1/3) x final quantity of solution

Final quantity of solution = 90 liters.

Added quantity = Final quantity – Initial quantity

= 90 – 45

Quantity of milk added = 45 liters.

Shortcut 10:  2 : 1 to 1 : 2

Ans=Final quantity/2

Question:

To a mixture of milk and water in the ratio 2:1 a certain quantity of water is added. The ratio becomes 1:2. If the quantity of mixture at present is 90, what was the quantity of water added to initial quantity?

Since final quantity is 90, answer = 90/2 = 45 liters

Proof:

Final quantity of water = (2/3) x 90 = 60

Final quantity of milk = 30

Initial quantity of milk = 30

Initial ratio = 2 : 1

30 : water = 2 : 1

Water = 15 liters.

Added quantity of water = final – initial quantity

= 60 – 15 = 45 liters

46