Bunuel wrote:

Over the past 7 weeks, the Smith family had weekly grocery bills of $74, $69, $64, $79, $64, $84, and $77. What was the Smiths' average (arithmetic mean) weekly grocery bill over the 7-week period?

A. $64

B. $70

C. $73

D. $74

E. $85

NEW question from GMAT® Official Guide 2019

(PS07369)

I took the approach described by

Iakovos1995 (second method) +1.

I do not want to attempt crunching these numbers.

Finding the mean as a balance point with numbers this large is faster.

The arithmetic mean can be viewed as a balancing point of a scale or seesaw.

The mean is the point at which the total distance of numbers below the mean is equal to the total distance above the mean

VeritasPrepKarishma explains the theory and method in

The Meaning of Arithmetic Mean, here,

here, and

here (important!)

This "balancing" method makes finding the average of a lot of large numbers much easier.

We have 64, 64, 69, 74, 77, 79, 84

74 is the median. The lowest and highest data values (64 and 84) are both 10 away from the median, so the list is not heavily skewed.

We make an educated guess about the mean: 74 is the middle number, and the center of the range.

Now measure distance from (difference from) the middle value. At the mean, the totals on each side will be equal. [74] is NOT the mean -- it is a number that is probably very close to the mean.

64, 64, 69, [74] 77, 79, 84

64 is 10 less than 74

64 is 10 less than 74

69 is 5 less than 74

Total: 25 below the middle

VeritasPrepKarishma calls this total the "overall shortfall."

77 is 3 more than 74

79 is 5 more than 74

84 is 10 more than 74

Total: 18 above the middle

This is called "overall excess."

Shortfall must balance out excess. Not true here, so the scale is not balanced. The overall shortfall is greater than the overall excess by 7: (25 - 18) = 7

This shortfall of 7 needs to be distributed evenly. There are a total of 7 numbers. So the mean must decrease (because we have a shortfall) by \(\frac{7}{7}=1\).

(74 - 1) = 73

Answer C

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"