Ratios - marching band

[Brent@GMATPrepNow]

Yesterday, the ratio of boys to girls in the marching band was 4:5. Today, 19 additional students joined the band and the ratio of boys to girls is now 5:6. What is the least number of students that could have been in the band yesterday?
A) 27
B) 36
C) 45
D) 54
E) 63

Please note that this is not an official GMAT question; it’s my attempt to create difficult (650+ level) GMAT-style questions for this forum.

[Fab]

I think there's an error on the answer choices.

4X + 5X = 9X
5X + 6X = 11X

THEN: 11X - 9X = 19
X = 9.5

Then at least there should be 9 * 9.5 = 85.5 = 86 students at least initially.

[krisraam]

Answer is B 36

[rajatmeh]

Should be 36.

Yesterday boys: girls = 4 : 5 i.e. 4y and 5y
Today boys : girls = 5: 6 i.e. 5t and 6t

4 y + 5y + 19 = 5t + 6t
i.e. 9y +19 = 11t

y and t can only be positive integers.

Clearly with y=4, t = 5

So minimum girls and boys yesterday would be 9*4 = 36

[Brent@GMATPrepNow]

Correct - the answer is B. Nice work.

Here's my solution, which is a little different.

Let 4x be the number of boys in the band yesterday.
This means that there were 5x girls in the band yesterday.
So, yesterday’s ratio of boys to girls can be viewed as 4x:5x

Of the 19 students who joined the band today, say there were k boys.
This means that the number of girls who joined the band is 19-k

So, once the boys and girls are added to yesterday’s boys and girls we get the equation: 4x+k : 5x + (19-k) = 5:6

Cross-multiply to get 24x+6k = 25x+95-5k
Simplify to get 11k-95 = x
Our goal here is to minimize x (while keeping x positive, since we can’t have a negative number of band members).
The smallest possible value for x is when k=9. This gives us x=4
If x=4, then yesterday there were 16 boys and 20 girls.
So, the total number of band members was 36.
Answer: B

[krisraam]

Initial ratio is 4:5 So total no of students are multiples of 9
Final ratio is 5:6 So total no of students are multiples of 11

Multiple of 9 when added by 19 should be a multiple of 11.

ie. Multiple of 9 when added by 1 should be a multiple of 11

so 9* 6 = 54

54 + 1 = 55

54 - 18 = 36

Thanks
Raama

[valdemas]

well i did it in a different way
the answer should be a number that when 19 is added... it should result in a multiple of 11...

[rolrol19]

agree. I used the same approach. Faster than using different unknowns

[GID09]

In order to satisfy the ratio of number of people in the band 5:6, yesterdays number + 19 should be divisible by 11 ( or 11x = 5x+6x) since there cannot be half people.

Now by simple addition, only 36 +19 =55 is divisible by 11. Hence B

[NavdeepG]

I did it the same way..

Started with 45 since its a multiple of 9..then tried 36..

x=4

x+19 should be a multiple of 11..hence answer is 36..