Total expenses of a boarding house are partly fixed and partly varying

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?

A. 540
B. 550
C. 560
D. 560
E. 570

A. 540
B. 550
C. 560
D. 560
E. 570

Shahalikhan · Answer

let fixed expense be k and variying 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

option c

ArunSharma12 · Answer

Total expenses = Fixed + Variable*number of boarders 
 TE(25) = F + V*25 ;  700*25 = F + V*25  --a
 TE(50) = F + V*50 ;  600*50 = F + V*50  --b
subtracting the above equations a and b,
 V*25=12500 ;  V=500 , put this value in one of the equations and we get F = 5000
thus,  TE(100) = 5000 + 500*100 
Avg Expense(100) = 50 + 500 = 550.
Ans: B

amak9114 · Answer

Hey! I know my logic is incorrect because I got the question wrong, but not sure why. I read that it was linear in the question so automatically through y=mx+b… then I used the two points (25,700) and (50,600) to get a slope and y-intercept and then plugged 100 for x. It didn't work, but I'm confused on why it's not the correct method. Any color on this would be great, thanks!

Abhishek009 · Answer

∑_25 = 17500
∑_50 = 30000

Thus, Fixed cost is 5000 , SO variable cost per boarder is $ 500

So, Total Cost of 100 boarder is  \frac{5000 + 5*100}{100} = 550 , Thus Answer must be (B)

tarasovk · Answer

25 boarders x 700 = 17500
50 boarders x 600 = 30000
Increment of 25 costs 12500
100 boarders = 12500x2 + 30000 =55000 
answer is 550

jcerdae · Answer

Thank you for the smart approaches!!!!

I would like to know what is wrong with my approach. Since the decrease per 25 boarders is 100 ($700 for 25 boarders and $600 for 50 boarders) then for me would be logical to assume that the average cost for 75 passengers would be $500 and for 100 passengers $400, in other words, $100 "discount" per 25 passengers.

Could you please help me?!

Thank you for the great answers

ScottTargetTestPrep · Answer

Since the total expenses is linear, we let it be y = mx + b where y is the total expense, m is the expense per boarder, x is the number of boarders, and b is the fixed cost. Using the information from the problem, we see that the total expenses are 25 x 700 = $17,500 when there are 25 boarders and 50 x 600 = $30,000 when there are 50 boarders. So we can create the equations:

25m + b = 17,500

and

50m + b = 30,000

Subtracting the first equation from the second, we have:

25m = 12,500

m = 500

Since m = 500, we have:

50(500) + b = 30,000

25,000 + b = 30,000

b = 5,000

Therefore, the total expenses equation is y = 500x + 5,000, and for 100 boarders, the total expenses are:

y = 500(100) + 5,000 = 50,000 + 5,000 = 55,000

Therefore, the average expense per boarder is 55,000/100 = $550 when there are 100 boarders.

Answer: B

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Total expenses of a boarding house are partly fixed and partly varying

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