How to solve this basis Formula

Brent@GMATPrepNow wrote:

[email protected] wrote:A machinist's salary at a factory increases by $2,000 at the end of each full year the machinist works. If the machinist's salary for the fifth year is $39,000, what is the machinist's average annual salary for his first 21 years at the factory?

There's a nice rule that says, "In a set where the numbers are equally spaced, the mean (average) will equal the median."

Here's an article I wrote about this property: https://www.beatthegmat.com/mba/2012/05/ ... d-the-mean

In this question, we have 21 years of salaries. Each year, the salary increases by $2000, so the yearly salaries might look something like this: $30000, $32000, $34000, $36000, etc
Since these salaries are equally spaced, the MEAN salary over the 21 year period will EQUAL the MEDIAN salary over the 21 year period.

The median salary will occur on the 11th year (the middlemost year).

We're told that in the 5th year, the salary $39,000
To find the salary in the 11th year, we'll add six increases of $2000 to get a salary of $51,000 in the 11th year

So, the mean salary = median salary = [spoiler]$51,000[/spoiler]

Cheers,
Brent

Hi Brent,

Does the given solution account for the fact that the first year counts as no increase? Hence 21 complete years of service includes 20 pay rises of 2000 each.

Total increase = 0(2000) + 1(2000) + 2(2000) + 3(2000) + 4(2000)+ ....+ 20(2000)

[email protected] wrote:A machinist's salary at a factory increases by $2,000 at the end of each full year the machinist works. If the machinist's salary for the fifth year is $39,000, what is the machinist's average annual salary for his first 21 years at the factory?

Nice Question.

[email protected] wrote:A machinist's salary at a factory increases by $2,000 at the end of each full year the machinist works. If the machinist's salary for the fifth year is $39,000, what is the machinist's average annual salary for his first 21 years at the factory?

It’s a very popular kind of series which appears on GMAT frequently. In the series here, the difference between two consecutive terms is a constant equal to 2000 (=2. In calculations, we can ignore 000) We don’t know the first term of this series, but we know that its fifth term is 39000 (=39. In calculations, we can ignore 000). We only need to find its 1st and 21st terms, because the average of an evenly spaced sequence is simply the average of its 1st and last terms.

Fifth term = 39. It means that we need to take 2 out four times from it to get to the first term, which is 39 – 4 (2) = 31, and we further need to add 2 to the fifth term sixteen times to get to the 21st term, which will be 39 + 16 (2) = 71.

Required average is [spoiler]½ (31 + 71) = ½ (102) = 51, Give those $ and 000 back here, answer is $51000.[/spoiler]

Raise Year Annual Salary
0 1 31
2 2 33
2 3 35
2 4 37
2 5 39
2 6 41
2 7 43
2 8 45
2 9 47
2 10 49
2 11 51
2 12 53
2 13 55
2 14 57
2 15 59
2 16 61
2 17 63
2 18 65
2 19 67
2 20 69
2 21 71
Sum 1071
Mean 51