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?

GMAT Panda · Accepted Answer

23 (Choice D) We create variables for Ada's and Paul's scores on each of the three tests. A_1 = Ada's first test score A_2 = Ada's second test score A_3 = Ada's third test score P_1 = Paul's first test score P_2 = Paul's second test score P_3 = Paul's third test score We translate "On the first test, Ada's score was 10 points higher than Paul's score." into an equation. A_1 = P_1 + 10 We translate "On the second test, Ada's score was 4 points higher than Paul's score." into an equation. A_2 = P_2 + 4 We translate "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. P_1 + P_2 + P_3/3 = A_1 + A_2 + A_3/3 + 3 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 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 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 P_1 + P_2 + P_3 = P_1 + P_2 + A_3 + 23 P_3 = A_3 + 23

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$We create variables for Ada's and Paul's scores on each of the three tests.$	$A_{1} = Ada’s first test score A_{2} = Ada’s second test score A_{3} = Ada’s third test score P_{1} = Paul’s first test score P_{2} = Paul’s second test score P_{3} = Paul’s third test score$
$We translate "On the first test, Ada’s score was 10 points higher than Paul’s score." into an equation.$	$A_{1} = P_{1} + 10$
$We translate "On the second test, Ada’s score was 4 points higher than Paul’s score." into an equation.$	$A_{2} = P_{2} + 4$
$We translate "Paul’s average (arithmetic mean) score on the three tests was 3 points higher than Ada’s average score on the three tests." into an equation.$	$\frac{P _{1} + P _{2} + P _{3}}{3} = \frac{A _{1} + A _{2} + A _{3}}{3} + 3$
$We eliminate the denominators by multiplying both sides of the equation by 3.$	$P_{1} + P_{2} + P_{3} = A_{1} + A_{2} + A_{3} + 9$
$We substitute the expressions for A_{1} and A_{2} from earlier steps into the equation.$	$P_{1} + P_{2} + P_{3} = (P_{1} + 10) + (P_{2} + 4) + A_{3} + 9$
$We simplify and cancel common terms to isolate P_{3} .$	$P_{1} + P_{2} + P_{3} = P_{1} + P_{2} + A_{3} + 10 + 4 + 9 P_{1} + P_{2} + P_{3} = P_{1} + P_{2} + A_{3} + 23 P_{3} = A_{3} + 23$

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