Problem Solving

Magoosh

Sam's car was fined when he gave Joe and Peter a ride, so they decided to help Sam pay the fine. Joe paid $3 more than 1/4 of the fine and Peter paid $3 less than 1/3 of the fine, leaving pay $4 less than 1/2 the fine to complete the payment. What fraction of the fine did Sam pay?

A. $13
B. $15
C. $20
D. $28
E. $48

OA C.

Sam's car was fined when he gave Joe and Peter a ride, so they decided to help Sam pay the fine. Joe paid $3 more than 1/4 of the fine and Peter paid $3 less than 1/3 of the fine, leaving SAM TO pay $4 less than 1/2 the fine to complete the payment. HOW MUCH of the fine did Sam pay?

A. $13
B. $15
C. $20
D. $28
E. $48

(We changed the question stem so that the alternative choices are focused-related.)
\[{\text{fine}} = 12\,k\,\,\,\,\left( {k > 0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}
\,J = 3k + 3 \hfill \\
\,P = 4k - 3 \hfill \\
\,{\text{S}} = 6k - 4 = ? \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\left[ \$ \right]\,\]
\[{\text{fine}} = J + P + {\text{S}}\,\,\,\, \Rightarrow \,\,\,12k = 13k - 4\,\,\,\, \Rightarrow \,\,\,k = 4\]
\[? = 6 \cdot 4 - 4 = 20\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.

Given that the answers choices are integers, I believe that the following reflects the problem's intent:

AAPL wrote:Magoosh

Sam's car was fined when he gave Joe and Peter a ride, so they decided to help Sam pay the fine. Joe paid $3 more than 1/4 of the fine and Peter paid $3 less than 1/3 of the fine, leaving pay $4 less than 1/2 the fine to complete the payment. How much of the fine did Sam pay?

A. $13
B. $15
C. $20
D. $28
E. $48

The three fractions in the prompt -- 1/4, 1/3 and 1/2 -- imply that the fine must be divisible by 4, 3 and 2.

Smallest possible case:
Total fine = 4*3*2 = 24.
Since Sam paid $4 less than 1/2 of the fine, Sam's share = (1/2)(24) - 4 = 8.
Not among the answer choices.

Next largest case:
Total fine = 2*24 = 48.
Since Sam paid $4 less than 1/2 of the fine, Sam's share = (1/2)(48) - 4 = 20.
Since Joe paid $3 more than 1/4 of the fine, Joe's share = (1/4)(48) + 3 = 15.
Since Peter paid $3 less than 1/3 of the fine, Peter's share = (1/3)(48) - 3 = 13.
Sum of the three shares = 20+15+13 = 48.
Success!
Thus, Sam's share = 20.

The correct answer is C.

Brent@GMATPrepNow wrote:Hey Mitch,
I think you answered the question "What was the cost of the fine?" (which IS, indeed, $48), but we're asked to determine the dollar amount that Sam paid (= $20).

Cheers,
Brent

Thanks for the heads up.
I've revised my post accordingly.