CAT - Percents

Question: 

 

P is x% more than Q. Q is (x - 10)% less than R. If P > R, what is the range of values x can take?

A.  10% to 28%

B.  10% to 25%

C.  10% to 37%

D.  10% to 43% 

Correct Answer: (C)

Explanation:

P = Q

Q = R

R =

P > R

(100 + x) (110 – x) > 100 x 100

11,000 + 110x – 100x – x² > 10000

1000 + 10x – x² > 0

x² – 10x – 1000 < 0

x² – 10x + 25 < 1000 + 25

(x – 5)² < 1025

x – 5 < 32

x < 37

x could range from 10% to 37%

Answer Choice (C)

Difficulty level II

Percentages Question

Percentages Question

Question:

An alloy of copper and aluminum has 40% copper. An alloy of Copper and Zinc has Copper and Zinc in the ratio 2: 7. These two alloys are mixed in such a way that in the overall alloy, there is more aluminum than Zinc, and copper constitutes x% of this alloy. What is the range of values x can take?

A.  30% < x < 40%

B.  32.5% < x < 42%

C.  33.33% < x < 40%

D.  32.5% < x < 40%

Correct Answer : (A)

Explanation:

Alloy 1	Copper	Aluminium
	2x	3x
Alloy 2	Copper	Zinc
	2y	7y

If Aluminium and Zinc have to be equal, 3x = 7y. The simplest case occurs at the LCM, when both Aluminium and Zinc are 21kg.

Alloy 1	Copper	Aluminium
	14	21
Alloy 2	Copper	Zinc
	6	21

Even a gram of alloy 1 above this level would mean that there is more Aluminium than Zinc in the total alloy. So, this table is a limit on the percentages of different metals.

The percentage of Copper in the total alloy would be (14 + 6)* = 32.25%

On the other hand, if we want more Aluminium than Zinc, we can use tons and tons of alloy 1, and only a microgram of Alloy 2. The quantity of alloy 2 can be made so small that its presence can be almost neglected, as it will have as good as zero impact on the overall percentage of copper. This is another extreme case, in which Copper will have 40% weight in the alloy.

Thus, the percentage of Copper ranges from 32.25% to 40% in this alloy.

Answer Choice (A)

Level of difficulty 2

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.

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 = 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)}/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.