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

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Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 25 Jun 2018, 05:02
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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



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Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 27 Jun 2018, 18:28
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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


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(PS07369)

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.
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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 29 Jun 2018, 06:58
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First, choose a nice, round number, such as 70, within the range of values.

Then calculate the average with the following formula:

Average = Nice number + Average of differences from the nice number

Average = 70 + (4 - 1 - 6 + 9 - 6 + 14 + 7)/7 = 70 + 21/7 =73

Answer: C
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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 28 Jun 2018, 18:07
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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


We can determine the average using the formula: average = sum / number:

average = (74 + 69 + 64 + 79 + 64 + 84 + 77)/7 = 511/7 = 73

Answer: C
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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 19 Nov 2018, 16:13
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Admittedly I made a stupid mistake and I organised the numbers in ascending order then selected the median (74) instead of solving Sum/# Terms

The method I used is only for consecutive integers and the punishment answer D is there for suckers like me.
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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 25 Jun 2018, 05:10

Solution



Given:
    • Over the past 7 weeks, the Smith family had weekly grocery bills of $74, $69, $64, $79, $64, $84, and $77

To find:
    • Smith’s average weekly grocery bill over the 7-week period

Approach and Working:
    • Smith’s total bill amount over the 7-week period = (74 + 69 + 64 + 79 + 64 + 84 + 77) = 511
    • Therefore, Smith’s average bill amount over the same period = \(\frac{511}{7}\) = 73

Hence, the correct answer is option C.

Answer: C
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Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 25 Jun 2018, 05:28
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The first way to find the average weekly grossery bill is to find the sum of the bills and to divide it with seven.
In this case we have that: Average=[64+64+69+74+77+79+84][/7]=[511][/7]=73.

An another quickest way is to arrange the seven values in ascending order.
The arrengement is: 64 64 69 74 77 79 84
The median value is 74 but the three smaller values have a total difference of 25 from the median (74-64=10, 74-64=10, 74-69=5 so 10+10+5=25). The three bigger values have a total difference of 18 from the median (77-74=3, 79-74=5, 84-74=10 so 3+5+10=18). So clearly, the average weekly grossery bill has to be a little smaller from the median. The next smaller value from the median is 73.

In my opinion, if it is possible to make simple calculations without wasting a lot of time (as is the case in this problem), the first way is preferable. But if we have many different values and especially if the given numbers are difficult to handle, the second way can be really helpful.

In both cases, the correct answer is C.
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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 28 Aug 2018, 03:42
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



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Easy way out is consider avg 70 -
so diff for all terms is 4, -1 , -6, 9 , -6,14, 7 = sum is 21 = divided by 7 = equal to 3 so 70 +3 = 73 avg
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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 10 Sep 2018, 07:34
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


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(PS07369)


\(\frac{( 70 + 4 ) + ( 70 - 1 ) + ( 60 + 4 ) + ( 80 - 1 ) + ( 60 + 4 ) + ( 80 + 4 ) + ( 70 + 7 )}{7}\)

= \(\frac{( 70 + 70 + 60 + 80 + 60 + 80 + 70 ) + ( 4 - 1 + 4 - 1 + 4 + 4 + 7 )}{7}\)

= \(\frac{490 + 21}{7}\)

= \(\frac{490}{7} + \frac{21}{7}\)

= \(70 + 3\)

= \(73\), Answer must be (C)
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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 10 Sep 2018, 07:46
We can rule out
64, 74 and 85 as they are the extreme numbers.
That leaves just two choices to choose from 70 and 73.

Only 3 values in the 60s and therefore less likely for 70.

73 is the most likely choice. Confirm 73 by taking difference and summing it up
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Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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New post 02 Oct 2018, 00:09
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This solution might be too blunt for some people. Sum the unit digits and you will have 41.
Now there are 7 numbers, which mean if x multiply by 7 must will result a number with unit digit of 1, so the x is 3.
The only answer with 3 as its unit digit is C.
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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7  [#permalink]

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Re: Over the past 7 weeks, the Smith family had weekly grocery bills of $7   [#permalink] 02 Dec 2019, 16:30
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