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Tue Feb 16 2021 Question:

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

Method 1: The regular Approach

It’s quite well known that average of a set of numbers is the sum of those numbers divided by the quantity of those numbers. In our case we have:

Average = (74 + 69 + 64 + 79 + 64 + 84 + 77)÷7

While this is a straight-forward problem, performing 74 + 69 + 64 + 79 + 64 + 84 + 77 without a calculator will be time-consuming job during which distractions could most likely happen!

At any rate, since: 74 + 69 + 64 + 79 + 64 + 84 + 77=511

And 511 ÷ 7 = 73

Method 2: Estimations Method

An alternative approach to decrease the amount of calculations with “big numbers” is to estimate (guess) what the average would be and add it to the average of the differences of every number

Let’s say that, from glancing at the numbers in the question, one would estimate their average to be 75.

Let’s calculate the differences between each number and 75

74 – 75 = -1

69 – 75 = -6

64 – 75 = -11

79 – 75 = 4

64 – 75 = -11

84 – 75 = 9

77 – 75 = 2

Since the sum of -1, -6, -11, 4, -11, 9 and 2 will be -14 and -14 ÷ 7 = -2, it comes out that the average of the differences is -2.

Therefore 75 - 2 = 73  – the correct answer is C!

These questions have been written by Cara Skikne and Shimon Goldchmit.

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