Over the past 7 weeks, the Smith family had weekly grocery bills of $7

I do not want to crunch these numbers. With numbers this large, find the mean as if it were a balancing point of a scale or seesaw.*

The mean is the point at which the total distance of numbers below the mean (the total "weight" on a "scale") is equal to the total distance ("weight") above the mean.

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

Just make an educated guess about a number that is probably very close to the mean: 74 is good. It's the median and exactly halfway between 64 and 84.

Now measure distance from (difference from) the chosen number. At the mean, the totals on each side will be equal.

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

Overall shortfall?
-- Find the difference between 74 and each number to the left of 74. Sum the answers.
64 is 10 less than 74
64 is 10 less than 74
69 is 5 less than 74

-- Overall shortfall is (10+10+5) =
25 total on the left of (below) the middle

Overall excess?

Same process, but to the right of the chosen number:
(77-74) + (79-74) + (84-74) =
(3+5+10) =
18 total above the middle

Find the balance point, which is the mean

Overall shortfall and overall excess must be equal at the balancing point. There is too much shortfall. Redistribute that extra amount evenly.

\(\frac{(Shortfall - Excess)}{7}=\frac{(25-18)}{7}=\frac{7}{7}=1\)

Because the extra is shortfall, each list number must decrease by 1.

To find the exact mean, we need to adjust only that rough number that we chose.

The mean is (74 - 1) = 73

Answer C

*An explanation can be found in The Meaning of Arithmetic Mean, here, here, and here.