CAT Practice : Percents, Profits

When thieves share loot, things always get interesting.

Percentages - Variables

Q.6: A, B, C and D share a loot. A gets a% of the total. B gets b% of the remaining (after A has taken his share). C gets c% of the remaining and D gets the rest. D gets a% less than what A gets, B and C get equal amounts. b = 2a.
1.   What percentage of what A got did C get?
2.   If the total amount is equal to Rs. 1000, what is the difference between what A got and what D got?

1. 160%
2. 80%
3. 175%
4. 150%

Choice A. 160%

Detailed Solution

Let total amount = T.

A would have got T * a. After A takes his share, there would be T (1 - a) left
B would have got T (1 - a) * b
C would have got T (1 - a)(1 - b) * c
D would have got T (1 - a) (1 - b) (1 - c)

D gets a % less than what A gets, or, T *a (1 - a) = T(1 - a) (1 - b) (1 - c), or
a = (1 - b)(1 - c)   -------- Eqn 1

B and C get the same amounts, or (1 - a) * b = (1 - a) * (1 - b) * c, or b = (1 - b) c, or, b = c - cb, or
b = ${c \over c}$ + 1   ------------ Eqn 2
b = 2a   -----------------Eqn 3

Substituting, eqn 2 in and 3 in eqn 1, we get
${{c \over 2(c + 1)} {=} {1 - c \over c + 1}}$
Or, c = 2 -2c, or c = ${2 \over 3}$

b = ${{c \over c + 1} {=} {2 \over 5}}$, a = ${1 \over 5}$

a gets 20% of the loot, b gets 40% of the remaining 80%, or 32% of the loot. C gets 66.6% of 48%, or 32% of the overall loot. And d gets the final 16% of the overall amount (20% lesser than A's share).

Now, to the questions,

1. What percentage of what A got did C get?
A got 20% of overall, C got 32% of overall. Or, C got 160% of A

2. If the total amount is equal to Rs. 1000, what is the difference between what A got and what D got?
A would have got Rs. 200, D Rs. 160. A would have got Rs. 40 more than D.

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