Marta plans to buy a certain number of toothbrushes for a total cost o

Question

Marta plans to buy a certain number of toothbrushes for a total cost of $60, excluding sales tax. If she buys 5 toothbrushes more than originally planned, she will receive a discount of $1 on each toothbrush and still pay $60 in total. If each toothbrush costs the same amount, what is the price of a single toothbrush before the discount?
A) $1
B) $3
C) $4
D) $5
E) $6

KarishmaB · Accepted Answer

\frac{60}{n} - 1 = \frac{60}{(n+5)}

Now check the options. They give us the value of 60/n.

If 60/n = 3, n = 20 - Doesn't satisfy.

If 60/n = 4, n = 15 - Satisfies

Answer (C)

"excluding sales tax" has no relevance here.

yashikaaggarwal · Answer

Let the total brush marta was buying earlier be X.
And y be the price/brush
60 is the amount she spent.
Therefore,
60/x=y ------(1)

Now she bought 5 more than the expected no. Of brush., So X+5
The price/brush reduced to 1 per brush.
Therefore equation will be,
60/(x+5)=(y-1)
60/(x+5)+1=y -----(2)

Equating both equations.
60/x = 60/(x+5)+1
60/x = 60+x+5/(x+5)
60(x+5) = (65+x)x
60x+300 = 65x+x^2
X^2+5X-300=0
X^2+20X-15X-300=0
(X+20)(X-15)
X=15,-20
Brush can't be negative.
So Brush no. Is 15
Putting value in equation 1
60/15=4
The earlier price of one brush is 4.
Answer is C

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TheUltimateWinner · Answer

Hello experts,
 EMPOWERgmatRichC ,  VeritasKarishma ,  IanStewart ,  Bunuel ,  chetan2u 
What if the highlighted parts are removed? Does it make any difference? 
Thanks__

BlackSky · Answer

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ScottTargetTestPrep · Answer

Solution:

To solve this problem, let’s look at the factors of 60 first (since the number of toothbrushes must be a whole number and assume that will be the case for the price of a toothbrush also):

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

We see that 15 is 5 more than 10. So let’s let them be the number of toothbrushes with and without the discount. If she bought 10 toothbrushes for $60 without discount, then each toothbrush costs $6. If she bought 15 toothbrushes for $60 with discount, then each toothbrush costs $4. However, $4 is NOT $1 less than $6.

Next, we see that 20 is also 5 more than 15. So let’s try these two numbers. If she bought 15 toothbrushes for $60 without discount, then each toothbrush costs $4. If she bought 20 toothbrushes for $60 with discount, then each toothbrush costs $3. We see that $3 is exactly $1 less than $4. Therefore, the price of a toothbrush must be $4 before the discount.

Alternate Solution:

Let n be the number of toothbrushes Martha planned to buy. Then, each toothbrush costs 60/n dollars before the discount. With the $1 discount, each toothbrush costs 60/n - 1 dollars and we are told that n + 5 toothbrushes at this price cost $60. We can create the following equation:

(60/n - 1)(n + 5) = 60

60 - n + 300/n - 5 = 60

n + 5 - 300/n = 0

Since n is not zero, we can multiply each side of this equation by n:

n^2 + 5n - 300 = 0

(n - 15)(n + 20) = 0

n = 15 or n = -20

Since the number of toothbrushes cannot be negative, n must be 15. Thus, the price of each toothbrush before the discount is 60/15 = $4.

Answer: C

armthan · Answer

x = original toothbrush price
N = number of toothbrushes bought

xN = 60

The question says  Marta bought 5 more than she intended      +  got a discount $1 on each toothbrush      +  ending up paying the $60

x ----- N ----- N+5 ----- x(N+5) ----- x(N+5) - (N+5) 
2-----30-----35--------70--------------35
3-----20-----25--------75--------------50
 4----15----20------80-----------60 
5-----12-----17--------85--------------68
6-----10-----15--------90--------------75

Ans:  C

Prip · Answer

Let x be the price of a single toothbrush and N be the no. of toothbrushes bought
Given, (xN) = 60 -------> Eq. 1
 and (x-1) (N-5) = 60 ----> Eq. 2

Substitute Eq. 1 in Eq.2 
xN-5x-N+5 = xN
N = 5 - 5x

From Eq.1 ------> x(5-5x) = 60
5x-5x^2 = 60
x-x^2 = 12
x^2-x+12 = 0
(x+3) (x-4) = 0

Since the price of a toothbrush can't be negative, ans would be x = $4

Marta plans to buy a certain number of toothbrushes for a total cost o

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