Problem Solving

A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel?
(A) $1
(B) $2
(C) $3
(D) $4
(E) $12

The answer choices all divide nicely into $120, so it might be fastest to PLUG in the answer choices.

A) $1
If towels are $1 each, $120 will buy 120 towels
Increase by $1 so that towels are $2 each. At this price, we can buy 60 towels
Difference = 120 - 60 = 60 fewer towels
We want 10 fewer
ELIMINATE A


B) $2
If towels are $2 each, $120 will buy 60 towels
Increase by $1 so that towels are $3 each. At this price, we can buy 40 towels
Difference = 60 - 40 = 20 fewer towels
We want 10 fewer
ELIMINATE B


c) $3
If towels are $3 each, $120 will buy 40 towels
Increase by $1 so that towels are $4 each. At this price, we can buy 30 towels
Difference = 40 - 30 = 10 fewer towels
PERFECT!

Answer: C

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John and Mary were each paid x dollars in advance to do a certain job together. John worked on the job for 10 hours and Mary worked 2 hours less than John. If Mary gave John y dollars of her payment so that they would have received the same hourly wage, what was the dollar amount, in terms of y, that John was paid in advance?
(A) 4y
(B) 5y
(C) 6y
(D) 8y
(E) 9y

For VIACs (Variables in the Answer Choices), we can use Algebra or the Input-Output approach,
Here's the algebraic approach:

Salary
Mary's NET salary was x - y dollars (because Mary gave John y dollars)
John's NET salary was x + y dollars

Hours worked
Mary worked 8 hours
John worked 10 hours


In the end, John and Mary received the SAME hourly wage.
So, John's hourly wage = Mary's hourly wage
Hourly wage = (total salary)/(hours worked)
So, (x + y)/10 = (x - y)/8

In terms of y, that John was paid in advance?
In other words, what is the value of x (in terms of y)
So, we'll solve our equation for x.

Take (x + y)/10 = (x - y)/8 and cross multiply to get:
10(x - y) = 8(x + y)
Expand: 10x - 10y = 8x + 8y
Rearrange: 2x = 18y
Divide by 2: x = 9y
So, John's advance payment = [spoiler]x = 9y = E[/spoiler]

Cheers,
Brent

Kayode wrote:I want to know whether their are simpler solution for these 2 questions apart from all the time consuming calcs involving quadratic equation.

2. John and Mary were each paid x dollars in advance to do a certain job together. John work on the job for 10 hrs and Mary worked 2 less than John. If Mary gave John y dollars of her payment so that they would have received the same hourly wage, what was the dollar amount, in terms f y that John was paid in advance.

a) 4y
b) 5y
c) 6y
d) 8y
e) 9y

Solution:

We are first given that John worked for 10 hours and that Mary worked for 2 hours less than John. It follows that:

John's hours = 10

Mary's hours = 8

We are also given that John and Mary were each given x dollars in advance. We can use this info to determine the hourly wage for both Mary and John.

Because (hourly wage)(# of hours) = total paid, we can say that:

hourly wage = (total paid)/(# of hours)

John's hourly wage = x/10

Mary's hourly wage = x/8

We are also told that Mary gave John y dollars of her payment so that they would have an equal hourly wage. It follows that Mary actually made x - y dollars. Since John received y dollars, he now made x + y dollars. Using this information, the hourly wages of John and Mary are:

Mary's wage = (x - y)/8

John's wage = (x + y)/10

Since we are told that the two hourly wages are the same, we can set the hourly wages of John and Mary equal to each other.

(x + y)/10 = (x - y)/8

We can cross multiply and solve:

8x + 8y = 10x - 10y

-2x = -18y

x = 9y

Answer is E