### CAT - Percentages

This
is an interesting question from Percentages. Simple framework, but finding the
answer is not that easy

Question

A is x% more than B and is x% of sum of A and B. What is the value of x? Give approximate answer.

Question

A is x% more than B and is x% of sum of A and B. What is the value of x? Give approximate answer.

Correct Answer

Roughly 62%

Explanatory Answer

a = b (1 + x) => a/b = 1 + x

a
= x (a + b) , dividing by a through out

1
= x (1 + b/a)

1
= x (1 + 1/(1 + x))

1
= x (x + 2) / (x + 1)

x
+ 1 = x

^{2}+ 2x
=>
x

^{2}+ x - 1 = 0
Now,
we need to solve this equation. Using the discriminant method, when we solve
this, x turns out to be {-1 + sqrt (5)}/2

x
has to lie between 0 and 1 and therefor cannot be { -1 - sqrt (5) }/2.

So,
the only solution is { -1 + sqrt (5) }/2. This is roughly 0.62.

Or,
x has to be 62% approximately. The ration 1.618 is also called the golden ratio, and is the
conjugate and reciprocal of 0.618.

The
golden ratio finds many mentions, from the Fibonacci series to Da Vinci. So, it
is a big favourite of mathematician.

Labels: CAT 2013, CAT 2013 questions, CAT Percentages, CAT questions