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Last year Jeff received 24 paychecks. Each of his last six paychecks was $800; this was a $50 raise from each of his earlier paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of the paychecks for the year?
A. $748
B. $750
C. $763
D. $775
E. $776
A. $748
B. $750
C. $763
D. $775
E. $776
27 Dec 2017, 01:20
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Bunuel
Last year Jeff received 24 paychecks. Each of his last six paychecks was $800; this was a $50 raise from each of his earlier paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of the paychecks for the year?
A. $748
B. $750
C. $763
D. $775
E. $776
A. $748
B. $750
C. $763
D. $775
E. $776
We're given the ratio between paychecks and asked to find the average.
Before calculating, we'll estimate the answer. This is an Alternative approach.
Jeff's paychecks were all $750 or $800 so the average must be somewhere in the middle.
(A), (B) are eliminated.
Had exactly half of the paychecks been $750 and half been $800 the average would in the middle of the two - $775.
But - there are more $750 paychecks than $800 so the answer must be smaller than $775.
(D), (E) are eliminated.
(C) is our answer.
** Note that estimating like this requires much easier calculations than the full solution. Usually it can eliminate 2-3 of the answers and the rest can be tested directly.
27 Dec 2017, 10:36
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Bunuel
Last year Jeff received 24 paychecks. Each of his last six paychecks was $800; this was a $50 raise from each of his earlier paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of the paychecks for the year?
A. $748
B. $750
C. $763
D. $775
E. $776
Estimation is a shrewd strategy here, as DavidTutorexamPAL demonstrates above.A. $748
B. $750
C. $763
D. $775
E. $776
If you do not use that method, you can shorten this problem in a couple of ways.
Use the ratio of 6 : 18
David gets 6 paychecks for $800. 24 checks total.
The other 18 are $50 less than $800 = $750
Express the ratio of number of checks as a fraction: \(\frac{6}{18} = \frac{1}{3}\)
1 check for $800
3 checks for $750
4 checks total
Average check amount: \(A = \frac{S}{n}\)
\(S = (1)800 + 3(750) = (800 + 2250) = $3,050\), and \(n = 4\)
\(A = \frac{$3,050}{4} = 762.5 \approx{$763}\)
ANSWER C
Double and halve*
The long route: 6 checks for $800. 18 for $750
Average check amount= \(\frac{6($800) + 18($750)}{24}\)
For (18 * $750), double $750 and halve 18:
(9 * $1,500) = $13,500. And
(6 * $800) = $4,800
Sum = ($13,500 + 4,800) = $18,300
\(n= 24\),
A =\(\frac{$18,300}{24}\)
From the answers you know: 18,300/24 = 7xx
After that, the arithmetic is quick. (Though this method is the most time-consuming.)
\(\frac{$18,300}{24}\\
= 762.5 \approx{$763}\)
ANSWER C
See GMATclub expert mikemcgarry, the doubling and halving trick for GMAT math
