In a shop, 40% socks are white and rest are black. 60% socks are made

In such questions, it's very important to wisely choose your initial figures, here "the total number of socks". If you notice carefully, there are percentages involved in this question i.e. 40%, 60%, 25 %. Always remember the tip:

"Fractions are always better than Percentages"

This will help you in choosing your initial figures carefully.

Here, you have 40% which is nothing but 2/5 in terms of fractions. 60% is 3/5 & 25% is 1/4. If you notice the denominators of these fractions, you will come to the conclusion that the initial figure you assume i.e. the total number of socks, will be divided by 5 & 4 i.e. if you assume the total number of socks to be a multiple of 25, your calculation is going to be a lot easier.

What's the best number that you can assume which is a multiple of 25? 100, of course.

So, let's start by assuming the total number of socks to be "100S". Notice I have not taken the number of socks to be 100 but instead, I have taken it as "100S" where S is a variable. This will make sure that you are under the purview of all the mathematical rules & staying away from marking a possibly hidden wrong answer.

Why "S" though? "S" will remind me that it's Socks. Had it been Tshirts in place of socks, I would have assumed 100T.

Now, it's all about comprehending the problem statement one by one. Let's break it down.


1. 40% of all the socks are white -> "40S" socks are white.
2. The rest are black -> "60S" are black.
3. 60% socks are made of cotton -> "60S" are made of cotton.
4. The rest are made of wool -> "40S" are made of wool.
5. 25% white socks are made of cotton i.e. 1/4th of white socks are made of cotton -> 1/4th of "40S" are made of cotton i.e. 10S. Here, if 10S are made of cotton, it is implied that 30S are made of wool since there are only 2 kinds of socks i.e. cotton & wool.

Moreover, the total number of cotton socks are -> 60S (refer to 3rd point)

Out of the 60S, 10S are made white cotton socks (refer to 5th point). How many are black cotton socks then? 60S - 10S i.e. 50S & further, the rest i.e. 10S are black wool socks. This is what we are supposed to find out.

Do we have the answer yet?

No, since it depends on the value of "S". If S=1, our answer is 50, if S is 1/2, our answer is 25 & so on.

What now?

There is one statement that we have not made use of. It says "20 of the black socks are made of wool." This means:

10S = 20
=> S= 2.

BINGO.


How many black socks are made of cotton?

50S i.e. 50*2 = 100.


Try the same question by changing the figures. An example is as follows:

In a shop, 28.56% socks are white and rest are black. 16.66% socks are made of cotton and rest are made of wool. 10% white socks are made of cotton and 242 of the black socks are made of wool. How many black socks are made of cotton?

Here, you will understand the importance of wisely choose your initial figures.

I came about it a different way but let me be sure I got it
20 black wool socks
1/4 the white socks are cotton - 3/4 the white socks are wool
if 20 sock are 1/4 then 60 socks are 3/4 and total 80 socks are wool
if 60% are cotton and 40% are wool then Total socks are 200.
60% = 120 black total - 20 wool socks = 100 black cotton
Still took me way too long though

Hi
Welcome to the GMATCLUB, and thanks for the solution.
Just two points.
1) when you are dealing with such questions, the TIP is that take % as numbers such as 100x or so. In the solution exactly same thing has been done but after wasting some time. First convert % as fractions and then deal with denominators to get LCM and then finally get back to 100. In such question this could have been the first step itself.
2) secondly, if you are calling out Bunuel to give him a solution, he is a math expert on our club and way above such solutions. Rather he is here to help students and has been doing it for a very long time.

Let the total number of socks be 20x.
The number of White socks is 8x of which 2x is Cotton and 6x is Wool.
The number of Black socks is 12x of which 20 is Wool and (12x-20) is Cotton.
So the total number of Wool socks is 6x+20 which is equal to 8x (40% of the total number of socks 20x)
6x+20=8x....> x=10
The number of Black Cotton socks is 12x-20=12*10-20=100. ANS: A

2 by 2 MATRIX is the most suitable in such questions...

--------White--------Black--------Total
C--- --0.25*40x ----- a ---------60x ...................Thus, 0.25*40x+a=60x.....a=50x
W----------b---------20-----------40x
Total---40x-------60x------100x

So, from black socks, we get a+20=60x...50x+20=60x......10x=20....x=2
We are looking for a, which is 50x=50*2=100

A

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