GMAT Math: Step by Step Word Problem From Student

Step 1) Translate Sentence
“The number of stamps that Kaye and Alberto had were in the ratio 5:3 respectively”
(K/A) = (5/3)
Cross Multiply:
(Equation #1): 5A = 3K

Step 2) Translate sentence
” After Kaye gave alberto 10 of her stamps, the ratio of the number Kaye had to the number Alberto had was 7:5.”
(K-10) / (A+10) = (7/5)
Cross Multiply

(Equation #2): 7(A+10) = 5(K-10)

Step 3) Translate Question
“As a result of this gift, Kaye had how many more stamps than Alberto”
This translates to: What is (end result of Kaye) – (end result of Alberto)?
(K-10) – (A+10) = ?

Step 4) Do we have enough info?
Before we answer the question, first we need to figure out if we have enough info to answer the question.
Well, you see to variables K and A. You also see two equations. Since # of equations = # of variables, you have enough information.

OK, now just simplify the second equation
7(A+10) = 5(K-10)
7A + 70 = 5K – 50
7A – 5K = -120

Step 5) Combine Equations That We Have To Solve for one of the variables
From Equation #1: 5A = 3K
5A – 3K = 0

Combine
7A – 5K = -120 –>Match K’s –>Multiply by 3 —>21A – 15K = -360
5A – 3K = 0 —>Match K’s —->Multiply by 5—->25A – 15K = 0

21A – 15K = -360
25A – 15K = 0

(21A – 25A) + (-15K – (-15K) = (-360 – 0)

-4A = -360
A = 90

Step 6) Use what we found to find the other variable

Since 5A = 3K, then 5(90) = 450 = 3K –> K = 150

Step 7) Now what’s the original question asking?
It’s asking:
(K-10) – (A+10) = ?

So (150-10) – (90 + 10 = 140 – 100 = 40
So 40 is the answer.

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