A manufacturer makes umbrellas at the cost of c dollars pe

Question

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?

$\frac{b(2c-r)}{(x-r)}$

$\frac{2x(c-r)}{(b-r)}$

$\frac{x(2c-r)}{(b-r)}$

$\frac{2b(c-r)}{(x-r)}$

$\frac{2(xc-r)}{(x-r)}$

A.  \frac{b(2c-r)}{(x-r)}

B.  \frac{2x(c-r)}{(b-r)}

C.  \frac{x(2c-r)}{(b-r)}

D.  \frac{2b(c-r)}{(x-r)}

E.  \frac{2(xc-r)}{(x-r)}

MacFauz · Accepted Answer

Let "N" be the required answer.,

(x - N)r + Nb = 2xc

xr - Nr + Nb = 2xc

N(b - r) = x(2c - r)

N =  \frac{x(2c-r)}{(b-r)}

Answer is C

BrentGMATPrepNow · Answer

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.  \frac{b(2c-r)}{(x-r)}  =  0  ELIMINATE A

B.  \frac{2x(c-r)}{(b-r)}  =  12  ELIMINATE B

C.  \frac{x(2c-r)}{(b-r)}  =  2  KEEP C

D.  \frac{2b(c-r)}{(x-r)}  =  0  ELIMINATE D

E.  \frac{2(xc-r)}{(x-r)}  =  6  ELIMINATE E

Answer : C

Cheers, 
Brent

leekay · Answer

process of elimination + intuition

x and r are not of the same nature (cost in dollar and total umbrellas) => eliminate A,D,E now, with some intuition, if retail price is less than cost, B would be negative => eliminate B Answer is C

jlgdr · Answer

Could anybody try smart numbers on this one? I'm having a tough time coming up with the solution Thanks! Cheers! J

Maxirosario2012 · Answer

Good approach, but still I don´t understand how to choose between B and C.

BrentGMATPrepNow · Answer

MacFauz and I have demonstrated the two methods (Algebraic and Input-Output) for solving a question type I call Variables in the Answer Choices.

If you'd like more information on these approaches, we have some free videos:
- Variables in the Answer Choices - https://www.gmatprepnow.com/module/gmat- ... /video/933
- Tips for the Algebraic Approach - https://www.gmatprepnow.com/module/gmat- ... /video/934
- Tips for the Input-Output Approach - https://www.gmatprepnow.com/module/gmat- ... /video/935

Cheers,
Brent

BillyZ · Answer

Official solution from Veritas Prep.

Correct Answer: C

For this problem, it is important to note that  Profit = Revenue - Cost . The profit needs to be equal the cost in order to make a 100% profit, so essentially the equation should then be  Cost = Revenue - Cost . The cost to the manufacturer is the number of units it makes, x, multiplied by the cost per unit, c, for a product of  xc . The revenue that the manufacturer will bring in has two components: the number it sells at the sale price of b multiplied by that price, b, and the rest of the umbrellas, which it sells at r, times r. If we call the number it is allowed to sell at a discount y, then we can solve for y to get our answer. That would make the revenue  yb + (x-y)r .

Therefore, the equation Cost = Revenue - Cost will be:

xc = yb + (x-y)r - xc .

Algebraically, we need to manipulate this equation to solve for y:
 xc = yb + (x-y)r - xc 
 xc = yb + xr - yr - xc  distribute the multiplication to get y out of the parentheses
 2xc = yb + xr - yr  add xc to both sides
 2xc - xr = yb - yr  subtract xr from both sides to get the y terms on their own
 2xc - xr = y(b - r)  factor the y to get it on its own

\frac{(2xc−xr)}{(b−r)} = y  divide both sides by  (b - r)  to get y on its own

x \frac{(2c−r)}{(b−r)} = y  factor the x to make this equation look like the answer choices.

Because the solution for y matches answer choice C, the answer is C.

KarishmaB · Answer

Responding to a pm:

Cost of x umbrellas = c*x
Profit he intends to make is 100% i.e. = c*x 
So revenue must be = 2c*x

Say, he sells N umbrellas at below cost rate of b and rest of (x - N) umbrellas at retail price of r dollars.

2c*x = N*b + (x - N)*r

N = \frac{(2cx - rx)}{(b - r)}

Answer (C)

T1101 · Answer

Had a very hard time understanding the above solutions... 
So I tried to put it into simpler terms:

Profit = Revenue-Costs

Costs: c*x

Profit = Revenue - cx

Question stem tells us that the manufacturer needs to make a 100% profit. How do we get a 100% profit? Only when the profit is equal to the costs and the revenue is equal to 2 times the costs.

cx = Revenue - cx 
 cx = 2cx - cx

Therefore the revenue the manufacturer needs to make is equal to  2cx ! Now setting up the real equation:

Revenue (In terms of cost) = Revenue (In terms of below cost rate and normal cost rate)

--> 2cx=  Revenue below costs + Revenue normal costs

Lets use N for the #of terms that sell below cost price.

-->  2cx= N*b + (x-N)*r

--> N = \frac{(2cx - rx)}{(b - r)}

GMATGuruNY · Answer

How many umbrellas can it afford to sell at the below-cost rate of b dollars per umbrella and still make a 100% profit? 
If c=1 and r=2 -- implying that the cost per umbrella = $1 and that the retail price per umbrella = $2 -- 100% profit will be yielded only if ALL of the umbrellas are sold at the retail price.
Since this case requires that NONE of the umbrellas be sold at a discount, the correct answer must yield a value of 0 when c=1 and r=2.
Only A and C are guaranteed to yield a value of 0 when c=1 and r=2.

Given c=1 and r=2, it is possible that x=2 umbrellas are sold at the retail price.
In A, x=2 and r=2 will yield a denominator of 0, rendering A invalid.

C .

ScottTargetTestPrep · Answer

Let y = the number of umbrellas the manufacturer sold below cost for b dollars each. Thus, by is the revenue from the below-cost part of the sale.

Since x is the total number of umbrellas sold, then he sold x - y umbrellas for r dollars each. Thus, r(x - y) is the revenue from the above-cost part of the sale. Since we want to have 100% profit, the revenue R has to be twice the cost:

R = 2c

by + r(x - y) = 2cx

by + rx - ry = 2cx

by - ry = 2cx - rx

y(b - r) = 2cx - rx

y = (2cx - rx)/(b - r)

y = x(2c - r)/(b - r)

Answer: C

Bambi2021 · Answer

Maybe I complicated things but I used fractions of x to get the equation:

2cx = rx(1-y) + bxy

2cx = rx - rxy + bxy

2cx - rx = xy(b-r)

x(2c-r)/(b-r) = xy

xy = the fraction of the total number of umbrellas that can be sold for b dollars.

AnkithaSrinivas · Answer

Say, he sells N umbrellas at below cost rate of b and rest of (x - N) umbrellas at retail price of r dollars. 
how do we know this?

bumpbot · Answer

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A manufacturer makes umbrellas at the cost of c dollars pe

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