Solve this question : Three friends whose ages form a G P divide a certain sum of money - Biology - The Living World

Question

Solve this question :

Three friends whose ages form a G P divide a certain sum of money
in proportion to their ages If they do that three years later, when
the youngest is half the age of the oldest, then he will receive
105 rupees more that he gets now and the middle friend will get 15
rupees more that he gets now, then ages of the friends are
(A) 12, 18, 27
(B) 6, 12, 24
(C) 8, 18, 36
(D) divisible by 3

Sneha Apar · Accepted Answer

Dear student,

Let the present  age of 3 friends be a, ar and ar2
Since a certain amount of money is distributed among them in the proportion of their ages, soshare of younger friend = aa+ar+ar2 = 11 + r + r2share of second friend = ara+ar+ar2 = r1 + r + r2share of elder friend = ar2a+ar+ar2 = r21 + r + r2After 3 years :age of younger friend = a+3 yrsage of second friend = ar + 3 yrsage of elder friend = ar2+3 yrsNow,share of younger friend = a+3a+3+ar+3+ar2+3 = a+3a1+r+r2 + 9share of second friend = ar + 3a+3+ar+3+ar2+3 = ar + 3a1+r+r2 + 9Now, a+3a1+r+r2 + 9 = 11 + r + r2 + 105    
1also, age of younger friend = 12age of elder friend⇒a + 3 = 12ar2 + 3     
2ar + 3a1+r+r2 + 9 = r1 + r + r2 + 15   
3put 1 + r + r2 = m, then from 1, we geta+3am + 9 - 1m = 105⇒ma + 3m - am - 9mam+9 = 105⇒3m - 9mam+9 = 105     
4

Now, from 3, we getar+3am + 9 - rm = 15⇒3m - 9rmam+9 = 15   
5dividing 4 by 5, we get3m - 93m - 9r = 7⇒2m - 7r + 1 = 0      ⇒21+r+r2 - 7r + 1 = 0⇒2r2 - 5r + 3 = 0⇒2r2 - 2r - 3r + 3 = 0⇒2rr - 1 - 3r - 1 = 0⇒2r-3r-1 = 0⇒2r-3=0  or  r-1=0⇒r = 32  or r = 1rejectedso, r = 32put r = 32 in 2, we geta + 3 = 129a4 + 3⇒a +3 = 9a8 + 32⇒9a8 - a = 3 - 32⇒a8 = 32⇒a = 12So, present ages of 3 friends are :12 , 12×32 and 12×94   i
e 12 yrs , 18 yrs and 27 yrs

Therefore option (A)

Regards