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A manufacturer makes umbrellas at the cost of c dollars per umbrella,

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A manufacturer makes umbrellas at the cost of c dollars per umbrella, [#permalink] New post   24 May 2021, 10:18
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A manufacturer makes umbrellas at the cost of c dollars per umbrella, and sells them at the retail price of r dollars each. How many umbrellas can it afford to sell at the below-cost rate of b dollars per umbrella and still make a 100% profit on its lot of x total umbrellas?

A. b(2c−r)(x−r)

B. 2x(c−r)(b−r)

C. x(2c−r)(b−r)

D. 2b(c−r)(x−r)

E. 2(xc−r)(x−r)
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Re: A manufacturer makes umbrellas at the cost of c dollars per umbrella, [#permalink] New post   26 May 2021, 13:48
This is a tough one to use the INPUT-OUTPUT approach, but here is goes:

Let c = $2 (it cost $2 to make each umbrella)
Let x = 10 (we make 10 umbrellas)
Let r = $5 (the retail price is $5 per umbrella)
Let b = $0 (the below-cost sale price is $0 per umbrella)

So, the manufacturer made 10 umbrellas at the cost of $2 per umbrella. So the total cost = $20
We need a 100% profit. So, we must earn $40 in revenue. In other words, we must sell 8 umbrellas at $5 each.
This means we can "sell" 2 umbrellas at the below-cost sale price of $0 each.

At this point, we must plug c = 2, x = 10, r = 5 and b = 0 into each expression and see which one yields an OUTPUT of 2

A. b(2c−r)(x−r) = 0 ELIMINATE A

B. 2x(c−r)(b−r) = 12 ELIMINATE B

C. x(2c−r)(b−r) = 2 KEEP C

D. 2b(c−r)(x−r) = 0 ELIMINATE D

E. 2(xc−r)(x−r) = 6 ELIMINATE E

Answer : C
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Re: A manufacturer makes umbrellas at the cost of c dollars per umbrella, [#permalink] New post   26 May 2021, 00:40
option plz,
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Re: A manufacturer makes umbrellas at the cost of c dollars per umbrella, [#permalink] New post   26 May 2021, 13:50
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GreenlightTestPrep wrote:
This is a tough one to use the INPUT-OUTPUT approach, but here is goes:

Let c = $2 (it cost $2 to make each umbrella)
Let x = 10 (we make 10 umbrellas)
Let r = $5 (the retail price is $5 per umbrella)
Let b = $0 (the below-cost sale price is $0 per umbrella)

So, the manufacturer made 10 umbrellas at the cost of $2 per umbrella. So the total cost = $20
We need a 100% profit. So, we must earn $40 in revenue. In other words, we must sell 8 umbrellas at $5 each.
This means we can "sell" 2 umbrellas at the below-cost sale price of $0 each.

At this point, we must plug c = 2, x = 10, r = 5 and b = 0 into each expression and see which one yields an OUTPUT of 2

A. b(2c−r)(x−r) = 0 ELIMINATE A

B. 2x(c−r)(b−r) = 12 ELIMINATE B

C. x(2c−r)(b−r) = 2 KEEP C

D. 2b(c−r)(x−r) = 0 ELIMINATE D

E. 2(xc−r)(x−r) = 6 ELIMINATE E

Answer : C


Thank you sir for the explanation

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Re: A manufacturer makes umbrellas at the cost of c dollars per umbrella, [#permalink] New post   26 May 2021, 23:40
i solved it later, but it requires lot of time.
any simle approach?
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A manufacturer makes umbrellas at the cost of c dollars per umbrella, [#permalink] New post   08 Jun 2021, 15:09
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This one took me a while unfortunately so I probably would have skipped it on the actual test. I apologize in advance for being unfamiliar on the formatting for this forum.

Cost of manufacturing = c

Retail price = r

Below price = b

total quantity sold = x

Total Profit = 100%

The key to this question is starting with some solid variables and a fundamental definition what profit is.

Lets say the quantity sold for retail price is q1 and the quantity sold for below price is q2. The costs for both remain the same.

q1 + q2 must equal the total quantity x. I.e. x = q1 + q2

100% profit = You make 200% return whatever you spent, but lose 100% due to cost (doubling your investment).

100% profit = 2 x (Cost) - 1 x (Cost) = Cost = cx

Total Profit = Total Revenue - Total Cost
= (r-c)(q1) + (b-c)(q2)

Set Total profit = 100% profit

cx = (r-c)(q1) + (b-c)(q2)

Using substitution, q1 = x - q2. When you plug into this formula and set equal to the cost, you get the final equation

(r-c)(x-q2) + (b-c)(q2) = cx

solving for q2, the number of umbrellas sold at below cost, gives you x(2c-r)/(b-r)
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A manufacturer makes umbrellas at the cost of c dollars per umbrella, [#permalink]
08 Jun 2021, 15:09
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