GMAT Practice Question: Ada and Paul received their scores on three tests. On the first test, Ada's score was 10 points...
Question
Ada and Paul received their scores on three tests. On the first test, Ada's score was 10 points higher than Paul's score. On the second test, Ada's score was 4 points higher than Paul's score. If Paul's average (arithmetic mean) score on the three tests was 3 points higher than Ada's average score on the three tests, then Paul's score on the third test was how many points higher than Ada's score?
- 9
- 14
- 17
- 23
- 25
Topics: linear equations, word problems, translations, mean
Solution
Step 1
We create variables for Ada's and Paul's scores on each of the three tests.
A_1 = \text{Ada's first test score} A_2 = \text{Ada's second test score} A_3 = \text{Ada's third test score} P_1 = \text{Paul's first test score} P_2 = \text{Paul's second test score} P_3 = \text{Paul's third test score}
Step 2
We translate \textbf{\text{"On the first test, Ada's score was 10 points higher than Paul's score."}} into an equation.
A_1 = P_1 + 10
Step 3
We translate \textbf{\text{"On the second test, Ada's score was 4 points higher than Paul's score."}} into an equation.
A_2 = P_2 + 4
Step 4
We translate \textbf{\text{"Paul's average (arithmetic mean) score on the three tests was 3 points higher than Ada's average score on the three tests."}} into an equation.
\frac{P_1 + P_2 + P_3}{3} = \frac{A_1 + A_2 + A_3}{3} + 3
Step 5
We eliminate the denominators by multiplying both sides of the equation by 3.
P_1 + P_2 + P_3 = A_1 + A_2 + A_3 + 9
Step 6
We substitute the expressions for A_1 and A_2 from earlier steps into the equation.
P_1 + P_2 + P_3 = (P_1 + 10) + (P_2 + 4) + A_3 + 9
Step 7
We simplify and cancel common terms to isolate P_3.
P_1 + P_2 + P_3 = P_1 + P_2 + A_3 + 10 + 4 + 9 \cancel{P_1} + \cancel{P_2} + P_3 = \cancel{P_1} + \cancel{P_2} + A_3 + 23 P_3 = A_3 + 23
Answer
23 (Choice D)
