Problem Solving

During the first 11 months of a recent year, a certain charitable organization received an average (arithmetic mean) of $20,600 per month in donations. How much did the organization receive in donations during December of that year if the average amount of donations per month for the entire year was $21,000?

a) 21,400
b) 21,600
c) 24,000
d) 25,400
e) 25,800

OA: D

nataras wrote:During the first 11 months of a recent year, a certain charitable organization received an average (arithmetic mean) of $20,600 per month in donations. How much did the organization receive in donations during December of that year if the average amount of donations per month for the entire year was $21,000?

a) 21,400
b) 21,600
c) 24,000
d) 25,400
e) 25,800

OA: D

Total Donation in 11 Months = 11 x $20,600 = $2,26,600
Total Desired Donation in 12 Months = 12 x $21,000 = $2,50,200

Required Donation in 12th Month December = $252,000 - $226,600 = $25,400

Answer: option D

nataras wrote:During the first 11 months of a recent year, a certain charitable organization received an average (arithmetic mean) of $20,600 per month in donations. How much did the organization receive in donations during December of that year if the average amount of donations per month for the entire year was $21,000?

a) 21,400
b) 21,600
c) 24,000
d) 25,400
e) 25,800

OA: D

ALTERNATIVE SOLUTION:

The difference between averages = 21000 - 20600 = 400

which has to be bridged for all 11 months = 400 x 11 = 4400

In 12th Month(December)
The amount contributed should be 21000 + the gap of past 11 months to be bridged= 4400+21000 = $25400

Answer: option D

nataras wrote:During the first 11 months of a recent year, a certain charitable organization received an average (arithmetic mean) of $20,600 per month in donations. How much did the organization receive in donations during December of that year if the average amount of donations per month for the entire year was $21,000?

a) 21,400
b) 21,600
c) 24,000
d) 25,400
e) 25,800

OA: D

To make the math easier, use the following values:
Average monthly donation for the first 11 months = 206.
Average monthly donation for December = D.
Average monthly donation for the MIXTURE of all the months = 210.

To determine the value of D, use ALLIGATION -- a great way to handle mixture problems.

Plot the 3 averages on a number line, as follows.
206-------x---------210-------11x---------D

The values in red represent the DISTANCES between the averages.
Here is the reasoning behind these values:
There are 11 MONTHS with an average of 206.
There is ONLY 1 MONTH with an average of D.
As a result, the value of D must be 11 TIMES FURTHER from the average for the whole year -- 210 -- than is average for the first 11 months.

The number line implies that x = 210-206 = 4.
Thus:
D = 210 + 11x = 210 + 11*4 = 254.

The correct answer is D.

One more:

During the first 11 months of a recent year, a certain charitable organization received an average (arithmetic mean) of $200 per month in donations. How much did the organization receive in donations during December of that year if the average amount of donations per month for the entire year was $300?

Here, we get the following number line:
200-------x---------300-------11x---------D

The number line indicates that x = 300-200 = 100.
Thus:
D = 300 + 11x = 300 + 11*100 = 1400.