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

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!

∑_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)

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

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

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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