Yesterday, the ratio of males to females working at Gigacorp

Question

Yesterday, the ratio of males to females working at Gigacorp was 2:3. Today, Gigacorp hired an additional 81 employees, and the ratio of males to females is now 6:5 (no employees left or were fired). What is the least number of males that could have been working at Gigacorp yesterday?

A) 16
B) 24
C) 39
D) 40
E) 42

*Kudos for all correct solutions

A) 16
B) 24
C) 39
D) 40
E) 42

*Kudos for all correct solutions

BrentGMATPrepNow · Accepted Answer

Yesterday, the ratio of males to females working at Gigacorp was 2:3  
There are many scenarios that meet this condition: 
- Gigacorp employs 2 males and 3 females (for a TOTAL of  5  employees) 
- Gigacorp employs 4 males and 6 females (for a TOTAL of  10  employees) 
- Gigacorp employs 6 males and 9 females (for a TOTAL of  15  employees) 
- Gigacorp employs 8 males and 23 females (for a TOTAL of  20  employees)
.
.
.
etc.

Notice that the  TOTAL number of employees yesterday is a  multiple of 5  . This is because our ratio of 2:3 tells us that, for every 5 employees, 2 are males and 3 are females. Since the number of males and females must be POSITIVE INTEGERS, the TOTAL number of employees yesterday must be a  multiple of 5 
 
  Today, Gigacorp hired an additional 81 employees, and the ratio of males to females is now 6:5  
By the same logic, the NEW TOTAL number of employees must be a multiple of 11, since this new information tells us that, for every 11 employees, 6 are males and 5 are females.

So, once we add 81 to  yesterday's TOTAL ,  the NEW TOTAL must be a multiple of 11 .

Let's see what happens if there was a TOTAL of  5  employees yesterday. 
NEW TOTAL =  5  + 81 = 86. 
Since 86 is NOT a multiple of 11, it  cannot  be the case that there was a TOTAL of  5  employees yesterday.

Let's see what happens if there was a TOTAL of  10  employees yesterday. 
NEW TOTAL =  10  + 81 = 91. 
Since 91 is NOT a multiple of 11, it  cannot  be the case that there was a TOTAL of  10  employees yesterday.

Let's see what happens if there was a TOTAL of  15  employees yesterday. 
NEW TOTAL =  15  + 81 = 96. 
Since 96 is NOT a multiple of 11, it  cannot  be the case that there was a TOTAL of  15  employees yesterday.

Notice that each NEW TOTAL follows the pattern: 86, 91, 96, 101, etc (the units digit of each number is either 1 or 6. 
So, let's continue the pattern until we get to the first  multiple of 11 
86, 91, 96, 101, 106, 111, 116,  121

BINGO! 
When the NEW TOTAL = 121, then the OLD TOTAL = 121 - 81 = 40
So, 40 is the smallest possible number of employees yesterday. 
Since the ratio of males to females working at Gigacorp was 2:3 yesterday, we can conclude that there were 16 males and 24 females.

Answer: A

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vitaliyGMAT · Answer

Hi

Nice question.

(1) 2x + 3x = N

5x = N

(2) 6y + 5y = N + 81

11y = N + 81

(2) - (1) 
 11y - 5x = 81

y_0 = 11, x_0 = 8

2x = 2*8 = 16

Answer A

pballijepalle · Answer

The answer should be a multiple of 2 as the ratio of male to female originally was 2:3 . Now eliminate ans choice 39. Remaining 16,24,40 and 42.

A. If we choose 16 as 2x , then 3x will be 24 and the total will be (16+24) 40. Adding 81 employees gives total of 121 employees. we know that the new ratio is 5:6. There fore 5x + 6x = 11x. This tells us,the answer has to be divided by 11. Clearly 121 is divisible by 11. 
Perform same logic for other options 24, 40 and 42. The end result is not divisible by 11. So going for Option A 16.

There must by easy way to solve this. But this came in handy as we have the options listed out.

elegantm · Answer

Logical Approach for this NICE question -

Original Ratio of Male to Female = 2x:3x
Total Original Population = 5x

After adding 81 employee New Ratio of Male to Female = 6y:5y
Total New Population = 11y

We May Write - 
5x + 81 = 11y
 Above equation conveys that LHS must be DIVISIBLE by 11 
5x + 4+ 77 = 11y

After Reducing
 5x + 4 = 11y
 
 Value of 2x is given in the options. Just Plug the value of x in above equation to get the answer.

Only option A would FIT.

Hence Answer A

doomedcat · Answer

Pls help m stuck on how to solve this problem

let initial ratio  \frac{Male}{Female}  =  \frac{2x}{3x} 
now let y males were hired of the 81 employees , then females = 81-y

new ratio  \frac{(2x+y)}{(3x+(81-y)}  =  \frac{6}{5}

which comes down to y= \frac{(8x+486)}{11}

i am lost here, what to do next; x has to INT den min value to y should be the ans

DebbieChats · Answer

Hi

there are multiple ways of solving a maths problem especially with Ratio - 1 variable 2 variable 3 variable etc. Though I agree with the above logic I am trying to find a hole in my logic - feel free to post it out to me please...
The first post of gmat prep CEO actually tried the 1 variable approach albeit picking numbers.

M/F = 2x/3x Total is 5x
Now add 81 so 5x + 81
Ratio turns to 6x/5x - total 11x

5x+81 = 11x implies 6x = 81
Initial male is 2x so 2x = 81/3 = 27

So the least from the choices given has to be 39?

2nd way: start from the answers and back calculate and try to arrive at answers
16 - x=2
24 - x=12
39 - x= 19.5
40 - x=20
42 -x= 21

Now the irony is as soon as I read the question, I did some mental calculations and without putting any pen to paper I arrived at the answer intuitively at 39. But obviously the answer is not 39 so then start doubting yourself, do multiple complicated x y variables and reach at another answer 16.
Request here is to find and discuss the two ways of thinking in ratio - using 1 variable vs multiple variables vs back calculating

Similar question solved with 1 variable
https://gmatclub.com/forum/the-ratio-of ... fl=similar

BrentGMATPrepNow · Answer

Once you have y= \frac{(8x+486)}{11} , you need to find the smallest value of x such that y= \frac{(8x+486)}{11}  is an integer.

Cheers,
Brent

Abhishek009 · Answer

(2x + 3x ) + 81 = ( 6y + 5y)

Or,  5x + 81 = 11y

Or,  x = \frac{11y - 81}{5}  {We are intentionally finding a solution for x for ease of calculation. as any no will be divisible by 5 only when the last digit is 0 Or 5 }

Now, The least possible value of y to make RHS divisible by 5 is 11, as  121 - 81 = 40  (Which is completely divisible by 5)

Now,  x = 8  , So the no of males yesterday is 16 , Answer must be (A) 16

leePNWBA · Answer

Although this is a 700 level question, it surprisingly simple if we are able to see the question in a different perspective. I'm not sure if this method works or it was just luck.

Since we are asked what the  LEAST  number of males are, I started with the least number in the answer choice.

2:3 ratio = 5X
2X = 16 (From answer choice)
X= 8
5X = 40

+81 Employees = 121

6:5 Ratio = 11X 
11X = 121
X= 11

16 works, and since it is the smallest number in the answer choice, (A) is correct.

Pritishd · Answer

Sharing an approach to solve this backwards from answer choices. In questions pertaining to least/smallest, always start from the least/smallest answer choice. In this question lets start with A i.e.  16 
We are told that the ratio of male to female was  2:3  yesterday, hence if there were  16  males then:
 2x = 16 
 x = 8

Using  x=8 , the number of females would be  3x = 3*8 = 24 
Total employees yesterday = males + females =  16 + 24 = 40

Today,  81  new employees were hired hence total would be  40 + 81 = 121  and the new ratio given to us is  6:5 . Using this ratio the total employees would be  11x .
 11x = 121 
 x = 11

We do not need to solve other options as the first option worked for us. In case you do try to solve then you will realize that the total employees today will not be divisible by  11  in case of the other answer choices.

Ans.  A

jabhatta2 · Answer

Experts --  chetan2u   Bunuel   Gladiator59   generis

I have a specific question on this one

If

Original Ratio of Male to Female = 2x:3x
Total Original Population = 5x

After adding 81 employee New Ratio of Male to Female = 6y:5y
Total New Population = 11y

I get that 5x + 81 = 11 y

Question : 
I always thought x and y do not have to be integers

Logic : 5x has to be an integer == thus x can still be a fraction ...like x can be 4.8 (as 4.8 x 5 = integer) or 4.6 (as 4.6 x 5 = integer) or 4.2 (as 4.2 x 5 = integer)

Similarly, Y does not have to be an integer

== y can be a fraction like ....like y can be 5/11 (as 5/11 x 11 = integer) or 20/11 (as 20/11 x 11 = integer) or 4.2 (as 26/11 x 11 = integer)

Pritishd · Answer

Hi,

We are discussing about men and women here. Imagine, can men and women be fractions in real life? Can there be 4.8 men, No right. The same rule applies in GMAT Quant as well. In questions pertaining to humans and animals we will always consider the quantities to be integers.

Hope this helps.

Warm Regards,
Pritishd

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Yesterday, the ratio of males to females working at Gigacorp

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