# CAT Practice : Percents, Profits

Golden ratio is a wonderful concept. Go on, google it.

## Percentages - Integers

Q.5: A is x% more than B and is x% of sum of A and B. What is the value of x?
1. 50%
2. 62%
3. 75%
4. 37%

Choice B. 62%

## Detailed Solution

a = b (1 + x) => ${a \over b}$ = 1 + x
a = x (a + b), dividing by a through out
1 = $x {({1} {+} {b \over a})}$
1 = $x {({1} {+} {1 \over 1 + x})}$
1 = $x {({x + 2 \over x + 1})}$
x + 1 = x2 + 2x
=> x2 + x - 1 = 0

Now, we need to solve this equation. Using the discriminant method, when we solve this, x turns out to be ${-1 + {\sqrt 5} \over 2}$.
x has to lie between 0 and 1 and there for cannot be ${-1 - {\sqrt 5} \over 2}$.
So, the only solution is ${-1 + {\sqrt 5} \over 2}$. 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.

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## More questions from Averages, Ratios, Mixtures

A number of beautiful questions from three allied topics. Most sportsmen will tell you - If you know the percentage play, you can profit well from it,