A farmer wants to divide Rs 1,22,000 between his sons , who are 18 and 20 years old respectively, in such a way that the sum divided at the rate of 20% per annum, compounded annually, will give the same amount to each of them when they attain the age of 22 years. How should he divide the sum?

Solution

Let the farmer give Rs x to the 18 years old son and the remaining Rs (1,22,000 - x) to his 20 years old son. Now,  [x(1+(20/100))]⁴ = (1,22,000 - x) (1+(20/100))² ⇒ [x(120/100)]² = (1,22,000 - x)   ⇒ [x(6/5)]² = (1,22,000 - x)   ⇒ x(36/25) = (1,22,000 - x) ⇒ ((36/25)+1) x = 1,22,000 ⇒ ((36 + 25)/25) x = 1,22,000 ⇒ x = (1,22,000 × 25)/61 = 50,000 ∴ x = Rs 50,000 For 18 years old son = Rs 50,000 For 20 years old son = Rs 72,000 Alternate shortcut method: They will get the sum in 2nd to 1st child in the ratio  of = (1+R/100)^{(difference between their age)} = (1+(20/100))^(20-18) = (6/5)² = 36/25 So for 18 years old(1st child)  , sum = [25/(36+25)] × 122000 = (25/61) × 122000 = 50000 & for 20 years old(2nd child)  , sum = [36/(36+25)] × 122000 = (36/61) × 122000 = 72000