ratio problem

TeddyBonham wrote:The ratio of statues to paintings in Ann's villa was 3:10. After 30 paintings were stolen from Ann's house the ratio of statues to paintings became 12:25. What was the original number of statues in Ann's house?

(A) 18 (B) 24 (C) 30 (D) 48 (E) 80

Here's an algebraic solution.

Let S = original number of statues
Let P = original number of paintings

The [original ] ratio of statues to paintings in Ann's villa was 3:10
So, S/P = 3/10
Cross multiply to get 3P = 10S
Rearrange to get: 3P - 10S = 0

After 30 paintings were stolen from Ann's house the ratio of statues to paintings became 12:25
New number of paintings = P - 30
Number of statues still = S
So, S/(P-30) = 12/25
Cross multiply to get: 25S = 12(P - 30)
Expand to get: 25S = 12P - 360
Rearrange to get: 12P - 25S = 360

We now have the following system:
3P - 10S = 0
12P - 25S = 360

Multiply blue equation by 4 to get:
12P - 40S = 0
12P - 25S = 360

Subtract the red equation from the blue equation to get -15S = -360
Solve to get: S = 24

Answer = B

Cheers,
Brent

Hi TeddyBonham,

Since the question asks for the original number of statues, and the answer choices are NUMBERS, we can TEST THE ANSWERS.

We're told that the original ratio of statues to paintings was 3:10. This means that the number of statues MUST be a multiple of 3 and the number of paintings MUST be an equivalent multiple of 10. Next, we're told that after 30 paintings were stolen, the ratio become 12:25 (now, a multiple of 12 and an equivalent multiple of 25).

Since the original number of statues MUST be a multiple of 3, answer E can be eliminated (it's NOT a multiple of 3). We just need to do the necessary arithmetic to prove which of the remaining 4 answers matches the above information.

Answer A: 18 statues

With 18 statues (3x6), we have 60 paintings. Removing 30 paintings gives us a ratio of:
18: 30
This reduces to 3:5 which is NOT a match. Eliminate A.

Answer B: 24 statues

With 24 statues (3x8), we have 80 paintings. Removing 30 paintings gives us a ratio of:
24: 50
This reduces to 12:25 which IS a match. This MUST be the answer.

Final Answer: B

GMAT assassins aren't born, they're made,
Rich