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Sam's car was fined when he gave Joe and Peter a ride, so

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Sam's car was fined when he gave Joe and Peter a ride, so

by AAPL » Sat Sep 08, 2018 5:08 am

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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.
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by fskilnik@GMATH » Sat Sep 08, 2018 2:45 pm
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.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by Brent@GMATPrepNow » Sat Sep 08, 2018 3:33 pm
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. What fraction of the fine did Sam pay?

A. $13
B. $15
C. $20
D. $28
E. $48
Let F = the total cost of the fine

Joe paid $3 more than 1/4 of the fine
So, F/4 + 3 = the amount Joe paid

Peter paid $3 less than 1/3 of the fine
So, F/3 - 3 = the amount Peter paid

Sam paid $4 less than 1/2 the fine
So, F/2 - 4 = the amount Sam paid

(amount Joe paid) + (amount Peter paid) + (amount Sam paid) = total cost of fine
We can write: (F/4 + 3) + (F/3 - 3 ) + (F/2 - 4) = F
To eliminate the fractions, we'll multiply both sides of the equation by 12 (the least common multiple of 3, 4 and 2)
When we do this, we get: 3F + 36 + 4F - 36 + 6F - 48 = 12F
Simplify to get: 13F - 48 = 12F
Solve: F = 48

What fraction of the fine did Sam pay?
F/2 - 4 = the amount Sam paid
So, 48/2 - 4 = the amount Sam paid
Evaluate to see that Sam paid $20

ASIDE: the question asks What fraction of the fine did Sam pay?
So, the answer SHOULD be 20/48

Cheers,
Brent
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fine

by GMATGuruNY » Sat Sep 08, 2018 6:32 pm
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.
Last edited by GMATGuruNY on Sun Sep 09, 2018 7:04 am, edited 1 time in total.
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by Brent@GMATPrepNow » Sun Sep 09, 2018 6:01 am
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,
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by GMATGuruNY » Sun Sep 09, 2018 7:05 am
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.
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by Scott@TargetTestPrep » Mon Sep 17, 2018 5:32 pm
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. What fraction of the fine did Sam pay?

A. $13
B. $15
C. $20
D. $28
E. $48
We can let n = the amount of the fine and create the equation:

3 + n/4 + n/3 - 3 + n/2 - 4 = n

Multiplying by 12, we have:

36 + 3n + 4n - 36 + 6n - 48 = 12n

n = 48

Since Same paid $4 less than 1/2 the fine, he paid 48/2 - 4 = 24 - 4 = $20.

Answer: C

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