SOLUTION

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

The average amount of the paychecks =

\(\frac{750*6+ (750 +30)*20}{26}=\frac{750*26+ 30*20}{26}=750+\frac{30*20}{26}=750+\frac{30*10}{13}\approx{750+23}=773\).

Answer: D.
_________________

The first 6 paychecks are for 750 each and then next 20 paychecks are for 780 each. The average will be closer to 780. Eliminate options A, B and C.

How do we guesstimate the right option between D and E. Upon solving, we get 773 but the calculations are tedious. Any good way to eliminate 775?

the solutions posted thus far are good but lots of computation, an area where i tend to make mistakes. here is how i broke it down.

6 checks are at 750. 20 checks are at 780.

difference between these checks is 30.

20 checks * 30 = 600 (easy math)
when removing 600 from the latter checks, all checks are now 'evenly filled' at 750, as if they were glasses filled with 750ml of water, for example.

what's an even way to distribute the 600 among all of the 26 checks? ==> 600/26, but this is also a tedious calculation
however, 600/30 is 20, since our actual denominator is smaller (and 600 numerator the same) we know that the resulting number will be just a little above 770 (750 + 20)

773 is more 'little' above 750 than 775.

IMO, the simplest way to deal with this type of questions is by calculating the weighted average.

750..................780 (diff is 30)
6......................20(total=26)
So wted avg= 750+ (30/26)*20= 773
_________________

Hi,
We are given that there are 26 paychecks in all. 6 paychecks each pay $750, and rest of 20 paychecks each pay $30 more than $750.

New avg= 750+(20*30)/26.

Now lets assume there were 30 paychecks in total. 6 paychecks each pay $750, and rest of 24 paychecks each pay $30 more than $750. ( i just picked a number that would cancel the 30 in the numerator)


By 20th paychecks the average would have increased by 21. How {(14*30)/20}=21
By 30th paychecks the average would have increased by 24. How {(24*30)/20}=24

So the new avg would be between 771 to 774
We have 773 as option
Hence Choice D .

Probus
_________________