Numbers Question

[tarun'susername]

The sum of two numbers is 's' and their quotient is p'/q. The numbers are? :

a> ps/q,qs/p
b> s/p,s/q
c> s-p/q,s-q/p
d> ps/p+q,qs/p+q

Please answer the right option and explain

[sreak1089]

IMO Ans D

Solution below:


Let us say a & b are the two numbers:
a+b = s --> (1)
and a = (p/q)b

substituting for a in (1):

b(p/q) + b = s
b(p+q)/q = s
=> b = qs/(p+q)

substitute for b in (2):

a + qs/(p+q) = s
a = s - qs(p+q)
= (s(p+q) - qs)/(p+q)
= ps/(p+q)[spoiler][/spoiler][/spoiler]

[Ian Stewart]

Let's call the two numbers a and b. Three options here:

-choose numbers for a and b. Work out s, p and q, and plug these into each answer. If only one answer gives you your numbers a and b, it must be right. If two answers both give you the right numbers a and b, try again.

-we know the two numbers add to s. Well, try adding the numbers in each answer choice - the right answer has to give you s. Only D does.

-do the algebra:

a + b = s
a/b = p/q

So a = s - b, and a = pb/q, so s-b and pb/q must be equal:

pb/q = s - b
pb/q + b = s
b(p/q + 1) = s
b = s/[p/q + 1]
b = sq/(p + q)

from which D must be correct (no need to find a).