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Question

On the first test, Rowena’s score was 6 points less than Spencer’s score. On the second test, Rowena’s score was 15 points more than Spencer’s score. If Spencer’s average score on the three tests was 4 points lower than Rowena’s average score, then Spencer’s score on the third test was how many points lower than Rowena’s score?

Answer Choices

A. $1$
B. $3$
C. $5$
D. $9$
E. $12$

Steps

Explanation	Calculations	Help
$We create variables for Rowena's and Spencer's scores on each of the three tests.$	$R_{1} = Rowena’s first test score R_{2} = Rowena’s second test score R_{3} = Rowena’s third test score S_{1} = Spencer’s first test score S_{2} = Spencer’s second test score S_{3} = Spencer’s third test score$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate "On the first test, Rowena's score was 6 points less than Spencer's score." into an equation.$	$R_{1} = S_{1} - 6$	Theory & Tactics Method Card TRANS1-D Click to view full details
$We translate "On the second test, Rowena's score was 15 points more than Spencer's score." into an equation.$	$R_{2} = S_{2} + 15$	Theory & Tactics Method Card TRANS1-C Click to view full details
$We translate "Spencer's average score on the three tests was 4 points lower than Rowena's average score." into an equation.$	$\frac{S _{1} + S _{2} + S _{3}}{3} = \frac{R _{1} + R _{2} + R _{3}}{3} - 4$	Theory & Tactics Method Card TRANS1-B Click to view full details Theory & Tactics Method Card TRANS1-D Click to view full details
$We eliminate the denominators by multiplying both sides of the equation by 3.$	$S_{1} + S_{2} + S_{3} = R_{1} + R_{2} + R_{3} - 12$
$We substitute the expressions for R_{1} and R_{2} from earlier steps into the equation.$	$S_{1} + S_{2} + S_{3} = (S_{1} - 6) + (S_{2} + 15) + R_{3} - 12$
$We simplify and cancel common terms to isolate S_{3} .$	$S_{1} + S_{2} + S_{3} = S_{1} + S_{2} + R_{3} - 3 S_{1} + S_{2} + S_{3} = S_{1} + S_{2} + R_{3} - 3 S_{3} = R_{3} - 3$

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Final Answer

B

Question

On the first test, Rowena’s score was 6 points less than Spencer’s score. On the second test, Rowena’s score was 15 points more than Spencer’s score. If Spencer’s average score on the three tests was 4 points lower than Rowena’s average score, then Spencer’s score on the third test was how many points lower than Rowena’s score?

Answer Choices

A. $1$
B. $3$
C. $5$
D. $9$
E. $12$

Steps

Explanation	Calculations	Help
$We create variables for Rowena's and Spencer's scores on each of the three tests.$	$R_{1} = Rowena’s first test score R_{2} = Rowena’s second test score R_{3} = Rowena’s third test score S_{1} = Spencer’s first test score S_{2} = Spencer’s second test score S_{3} = Spencer’s third test score$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate "On the first test, Rowena's score was 6 points less than Spencer's score." into an equation.$	$R_{1} = S_{1} - 6$	Theory & Tactics Method Card TRANS1-D Click to view full details
$We translate "On the second test, Rowena's score was 15 points more than Spencer's score." into an equation.$	$R_{2} = S_{2} + 15$	Theory & Tactics Method Card TRANS1-C Click to view full details
$We translate "Spencer's average score on the three tests was 4 points lower than Rowena's average score." into an equation.$	$\frac{S _{1} + S _{2} + S _{3}}{3} = \frac{R _{1} + R _{2} + R _{3}}{3} - 4$	Theory & Tactics Method Card TRANS1-B Click to view full details Theory & Tactics Method Card TRANS1-D Click to view full details
$We eliminate the denominators by multiplying both sides of the equation by 3.$	$S_{1} + S_{2} + S_{3} = R_{1} + R_{2} + R_{3} - 12$
$We substitute the expressions for R_{1} and R_{2} from earlier steps into the equation.$	$S_{1} + S_{2} + S_{3} = (S_{1} - 6) + (S_{2} + 15) + R_{3} - 12$
$We simplify and cancel common terms to isolate S_{3} .$	$S_{1} + S_{2} + S_{3} = S_{1} + S_{2} + R_{3} - 3 S_{1} + S_{2} + S_{3} = S_{1} + S_{2} + R_{3} - 3 S_{3} = R_{3} - 3$

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Final Answer

B