Problem Solving

Last year Manfred received 26 paychecks. Each of his first 6 paychecks was $750; each of his remaining paychecks was $30 more than each of his first 6 paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of his paychecks for the year?

(A) $752
(B) $755
(C) $765
(D) $773
(E) $775

OA [spoiler][D][/spoiler]

Here are a couple ways to look at this one.

First 6 are 750 each = 4500

Next 20 are 30 more than the 750. So they are all 780. 780 * 20 = 15600

15600+4500 = 20100 Total paid

20100/26 = ~ 773


Or you can say

6/26 of the payments were 750

6/26 = 3/13 * 750 = 2250/13

20/26 of the payments were 780

20/26 = 10/13 * 780 = 7800/13

(7800 + 2250)/13 = ~ 773

dmateer25 wrote:Here are a couple ways to look at this one.

First 6 are 750 each = 4500

Next 20 are 30 more than the 750. So they are all 780. 780 * 20 = 15600

15600+4500 = 20100 Total paid

20100/26 = ~ 773

Thanks Dmateer.

kanha81 wrote:Last year Manfred received 26 paychecks. Each of his first 6 paychecks was $750; each of his remaining paychecks was $30 more than each of his first 6 paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of his paychecks for the year?

(A) $752
(B) $755
(C) $765
(D) $773
(E) $775

Great algebraic solutions from dmateer! However, as with many GMAT questions, there are other approaches which may be much quicker.

If we were in a rush, we could give ourselves a 50/50 shot at this question almost immediately and then use some strategic guessing to answer with no big calculations at all.

In a weighted average, the total average will always be closer to the group with the most weight.

In this question, we have a lot more checks for $780 than for $750. Therefore, the overall average will be weighted toward $780: eliminate (a), (b) and (c).

In fact, in many weighted average questions the answer choice pattern will be:

a) closer to group 1
b) closer to group 1
c) right in the middle
d) closer to group 2
e) closer to group 2

so if you can figure out which group "weighs more" you can often give yourself a quick 50/50 shot (the answer right in the middle is almost always a sucker bet).

Going one step further:

$775 is 25/30 or 5/6 of the way between $750 and $780. For $775 to be correct, the weight of group 2 would have to be 5 times the weight of group 1.

We have 20 checks for $780 and 6 for $750. Is 20:6 = 5:1? Nope, so we can eliminate (e) as well: choose (d).

Stuart Kovinsky wrote:

kanha81 wrote:Last year Manfred received 26 paychecks. Each of his first 6 paychecks was $750; each of his remaining paychecks was $30 more than each of his first 6 paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of his paychecks for the year?

(A) $752
(B) $755
(C) $765
(D) $773
(E) $775

Great algebraic solutions from dmateer! However, as with many GMAT questions, there are other approaches which may be much quicker.

If we were in a rush, we could give ourselves a 50/50 shot at this question almost immediately and then use some strategic guessing to answer with no big calculations at all.

In a weighted average, the total average will always be closer to the group with the most weight.

In this question, we have a lot more checks for $780 than for $750. Therefore, the overall average will be weighted toward $780: eliminate (a), (b) and (c).

In fact, in many weighted average questions the answer choice pattern will be:

a) closer to group 1
b) closer to group 1
c) right in the middle
d) closer to group 2
e) closer to group 2

so if you can figure out which group "weighs more" you can often give yourself a quick 50/50 shot (the answer right in the middle is almost always a sucker bet).

Going one step further:

$775 is 25/30 or 5/6 of the way between $750 and $780. For $775 to be correct, the weight of group 2 would have to be 5 times the weight of group 1.

We have 20 checks for $780 and 6 for $750. Is 20:6 = 5:1? Nope, so we can eliminate (e) as well: choose (d).

Thanks Stuart! Thanks a lot. Until now, I was not aware of the weighted averages. Really an excellent way to narrow down the answer choices

His average salary for first six paychecks was $750.For next 20 checks he got $780 in each.He got $30 more in each check.So (30*20)=$600.Now if we divide $ 600 by 26(as there were all together 26 checks) we get an approximate average 23. If we add,then we get 750+23=773.

Hope it helps.

Question - How did you so quickly knew that 775 is 5/6 of the way down from 750 and 780? Thank you!

Stuart Kovinsky wrote:

kanha81 wrote:Last year Manfred received 26 paychecks. Each of his first 6 paychecks was $750; each of his remaining paychecks was $30 more than each of his first 6 paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of his paychecks for the year?

(A) $752
(B) $755
(C) $765
(D) $773
(E) $775

Great algebraic solutions from dmateer! However, as with many GMAT questions, there are other approaches which may be much quicker.

If we were in a rush, we could give ourselves a 50/50 shot at this question almost immediately and then use some strategic guessing to answer with no big calculations at all.

In a weighted average, the total average will always be closer to the group with the most weight.

In this question, we have a lot more checks for $780 than for $750. Therefore, the overall average will be weighted toward $780: eliminate (a), (b) and (c).

In fact, in many weighted average questions the answer choice pattern will be:

a) closer to group 1
b) closer to group 1
c) right in the middle
d) closer to group 2
e) closer to group 2

so if you can figure out which group "weighs more" you can often give yourself a quick 50/50 shot (the answer right in the middle is almost always a sucker bet).

Going one step further:

$775 is 25/30 or 5/6 of the way between $750 and $780. For $775 to be correct, the weight of group 2 would have to be 5 times the weight of group 1.

We have 20 checks for $780 and 6 for $750. Is 20:6 = 5:1? Nope, so we can eliminate (e) as well: choose (d).

kanha81 wrote:Last year Manfred received 26 paychecks. Each of his first 6 paychecks was $750; each of his remaining paychecks was $30 more than each of his first 6 paychecks. To the nearest dollar, what was the average (arithmetic mean) amount of his paychecks for the year?

(A) $752
(B) $755
(C) $765
(D) $773
(E) $775

OA [spoiler][D][/spoiler]

The value of each paycheck is equal to one of the following:
700+80 or 700+50.
Since $700 is common to all of the paychecks, the average depends only on the LAST TWO DIGITS.
Thus, we can substitute $80 for each $780 paycheck and $50 for each $750 paycheck.

Of the 26 paychecks, there are 20 paychecks for $80 each and 6 paychecks for $50 each.
Average amount per paycheck = (20*80 + 6*50)/26 = 1900/26 = 950/13 ≈ 73.

The correct answer is D.