Problem Solving

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

Answer: A

Source: GMAT Prep Now
Difficulty level: 650 - 700

Brent@GMATPrepNow wrote: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

Answer: A

Source: GMAT Prep Now
Difficulty level: 650 - 700

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 2:5 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

Matt@VeritasPrep wrote:Now for a slightly better one.

Let's say that yesterday we had x men and y women. We know that x + y is a multiple of 5, since our initial ratio has five parts.

After we hire the 81 employees, we know that x + y + 81 is a multiple of 11, since our new ratio has eleven parts.

This gives us an equation like 5a + 81 = 11b, and we want to minimize a. With that in mind:

5a = 11b - 81

a = (11b - 81)/5

a = 2.2b - 16.2

a must be an integer, so we need 2.2b to end in .2. For that to happen, b must end in 1 or 6. We also need 2.2b to be greater than 16.2 (because a can't be negative), so b > 7. The first choice that satisfies both conditions is b = 11. Plugging that in, we find b = 11, a = 8.

Since a = 8, we had 5a = 5*8 = 40 employees yesterday. (2/5) of them were men, so we had 16 men.

Sweeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeet!!!!