Question: Total expenses of a boarding house are partly fixed and partly varying linearly with tile number of boarders. The average expense per boarder is $700 when there are 25 boarders and $600 when there are 50 boarders. What is the average expense per boarder when there are 100 boarders?

1. 540
2. 550
3. 560
4. 560
5. 570

Correct Answer: B
Solution and Explanation:
Approach Solution 1:
Let fixed expense be k and varying be x

Then we have (k+25x)/25=700
(k+50x)/50=600

simplifying
k/25 + x =700
k/50 + x =600

we need to find k/100+x =?
easily solve and get 550

Approach Solution 2:
Average expense [when there are 25 borders] = $700
Average expense [when there are 50 boarders] = $600
Average expense [when there are 100 boarders] = ?

It is given that Total expenses of a boarding house are partly fixed and partly varying linearly with the tile number of borders. So, let's assume that:

Fixed expense = k
Varying expense = x

The formulae being used are:
Average expense = total expense / no. of borders
Where, total expense = fixed expense [k] + varying expense [x] * no. of borders

Thus, we have two equations here
[k + 25x] / 25 = 700
k + 25x = 700 * 25
k + 25x = 17500……. (1)

[k + 50x] /50 = 600
k + 50x = 600 * 50
k + 50x = 30000 ……… (2)

What we need to know is the value of: [k + 100x] /100……. (3)

Subtract Equation 1 from Equation 2 to get the value of x
[k + 50x] – [k +25x] = 30000 – 17500
k + 50x – k – 25x = 12500
25x = 12500
x = 12500 / 25
x = 500

In order to get the value of k, we put the value of x into Equation 1
k + 25 [500] = 17500
k + 12500 = 17500
k = 17500 – 12500
k = 5000

So, we put all the values into Equation 3
[k + 100x] /100 = [5000 + 100 * 500] / 100
= [5000 + 50000] /100
= [55000] /100
= 550

Average expense [for 100 boarders] = $550

Approach Solution 3:
Average expense [when there are 25 borders] = $700
Average expense [when there are 50 boarders] = $600
Average expense [when there are 100 boarders] = ?

Average expense = total expense / no. of borders

Case no.#1: for 25 borders
700 = total expense / 25

Total expense = 700 * 25 = 17500 ……. (a)

Case no.#2: for 50 boarders
600 = total expense / 50

Total expense = 600 * 50 = 30000 ……… (b)

From a quick perusal of Equation (a) and Equation (b) it can be concluded that for an increment of 25 borders the total expense has increased by 12500.

Now, going from 50 boarders to 100 borders would mean 2 increments of 25 boarders or 2 increments of 12500 along with the total expense of 50 boarders [given in equation (b)]

Total expense [for 100 boarders] = [12500 * 2] + 30000
Total expense [for 100 boarders] = 25000 + 30000
Total expense [for 100 boarders] = 55000

Again, using the formula for average expense

Average expense [for 100 boarders] = Total expense [for 100 boarders] / 100
Average expense [for 100 boarders] = 55000 / 100
Average expense [for 100 boarders] = 550

“Total expenses of a boarding house are partly fixed and partly varying”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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