Problem Solving

Hello,

Can you please help with this. This is from MGMAT:

For a particular model of moving truck, rental agency A charges a daily fee of m dollars, plus n cents per mile. For the same model of truck, rental agency B charges a daily fee of p dollars, plus q cents per mile. If a driver plans to rent this model of truck for two days, which of the following expressions gives the number of miles this driver must drive for the two rental agencies' total charges to be equal?


Sorry, I don't have the official answer.

My answer: [spoiler]2( m - p )/( q - n )[/spoiler]


Thanks for your help,
Sri

For a particular model of moving truck, rental agency A charges a daily fee of m dollars, plus n cents per mile. For the same model of truck, rental agency B charges a daily fee of p dollars, plus q cents per mile. If a driver plans to rent this model of truck for two days, which of the following expressions gives the number of miles this driver must drive for the two rental agencies' total charges to be equal?

[100(m-p)]/(q-n)

[200(p-m)]/(n-q)

[50(m-p)]/(q-n)

[2(p-m)]/(n-q)

(m-p)/[2(q-n)]

Let the distance = 10 miles.
Each agency must charge the same amount to travel 10 miles over 2 days.

Rate agency A:
Let the fee per mile = n cents = 200 cents = $2.
Total fee for 10 miles = 10*2 = $20.
Let the daily fee = m = 5.
Total charge to drive 10 miles over 2 days = 20 + 2*5 = $30.

Rental agency B:
Let the fee per mile = q cents = 300 cents = $3.
Total fee for 10 miles = 10*3 = 30.
Since B must charge the same total amount as A -- $30 -- the daily fee for B = p = 0.

The question stem ask for the number of miles: 10.
This is our target.
Now we plug m=5, n=200, p=0, and q=300 into the answers to see which yield our target of 10.

Only B works:
200(p-m) / (n-q) = 200(0-5) / (200-300) = 200(-5) / (-100) = 10.

The correct answer is B.

Brent@GMATPrepNow wrote:

For a particular model of moving truck, rental agency A charges a daily fee of m dollars, plus n cents per mile. For the same model of truck, rental agency B charges a daily fee of p dollars, plus q cents per mile. If a driver plans to rent this model of truck for two days, which of the following expressions gives the number of miles this driver must drive for the two rental agencies' total charges to be equal?

[100(m-p)]/(q-n)

[200(p-m)]/(n-q)

[50(m-p)]/(q-n)

[2(p-m)]/(n-q)

(m-p)/[2(q-n)]

Here's the algebraic approach:

Rental agency A charges a daily fee of m dollars, plus n cents per mile.
IMPORTANT: a fee of n cents per mile driven is the same as n/100 dollars per mile driven
Let x = total number of miles driven
If the truck is rented for 2 days, the total cost (in dollars) = 2m + (n/100)x
= 2m + nx/100

Rental agency B charges a daily fee of p dollars, plus q cents per mile.
Let x = total number of miles driven
If the truck is rented for 2 days, the total cost (in dollars) = 2p + (q/100)x
= 2p + qx/100

Which of the following expressions gives the number of miles this driver must drive for the two rental agencies' total charges to be equal?
So, 2m + nx/100 = 2p + qx/100 [solve for x]
Rearrange: nx/100 - qx/100 = 2p - 2m
Multiply both sides by 100: nx - qx = 200p - 200m
Factor both sides: x(n - q) = 200(p - m)
Divide both sides by (n-q): x = 200(p - m)/(n - q)
= B

Cheers,
Brent

Hi Brent

Why do I get answer as A if I convert dollars into cents? Should give the same answer right?

Brent@GMATPrepNow wrote:

For a particular model of moving truck, rental agency A charges a daily fee of m dollars, plus n cents per mile. For the same model of truck, rental agency B charges a daily fee of p dollars, plus q cents per mile. If a driver plans to rent this model of truck for two days, which of the following expressions gives the number of miles this driver must drive for the two rental agencies' total charges to be equal?

[100(m-p)]/(q-n)

[200(p-m)]/(n-q)

[50(m-p)]/(q-n)

[2(p-m)]/(n-q)

(m-p)/[2(q-n)]

Here's the algebraic approach:

Rental agency A charges a daily fee of m dollars, plus n cents per mile.
IMPORTANT: a fee of n cents per mile driven is the same as n/100 dollars per mile driven
Let x = total number of miles driven
If the truck is rented for 2 days, the total cost (in dollars) = 2m + (n/100)x
= 2m + nx/100

Rental agency B charges a daily fee of p dollars, plus q cents per mile.
Let x = total number of miles driven
If the truck is rented for 2 days, the total cost (in dollars) = 2p + (q/100)x
= 2p + qx/100

Which of the following expressions gives the number of miles this driver must drive for the two rental agencies' total charges to be equal?
So, 2m + nx/100 = 2p + qx/100 [solve for x]
Rearrange: nx/100 - qx/100 = 2p - 2m
Multiply both sides by 100: nx - qx = 200p - 200m
Factor both sides: x(n - q) = 200(p - m)
Divide both sides by (n-q): x = 200(p - m)/(n - q)
= B

Cheers,
Brent

Hi Brent

Why do I get answer as A if I convert dollars into cents? Should give the same answer right?

sinhap07 wrote:Why do I get answer as A if I convert dollars into cents? Should give the same answer right?

I think you had it right, but you forgot the TWO DAY rental. Just to be sure, let's walk through the steps.

If you're working with cents, you should have the following.

A charges: (2 days)*100m¢ + (n¢ per mile)
B charges: (2 days)*100p¢ + (q¢ per mile)

x will be the number of miles we seek. We then have 2*100m¢ + nx¢ = 2*100p¢ + qx¢, or

200m - 200p = qx - nx, or

200(m - p) = x(q - n), or

200(m - p)/(q - n) = x

This is the same as answer B, since (m - p)/(q - n) = (p - m)/(n - q) for all n ≠ q.

Hi Experts ,

Please advise whats wrong with my solution.

Let M = 2$ and N = 100 cents

Let P= 1$ and q = 200 cents

Let the no. of miles be x

So the equation will become

2(2)+100x = 1(2)+200x

So the x = 1/50

Now this is our target , if I put the value in option B, then I do not find the answer.

Option B : 200(1-2)/100-200

this will give 2 ?

Please advise and correct me.

Many thanks in advance.

SJ

jain2016 wrote:Hi Experts ,

Please advise whats wrong with my solution.

Let M = 2$ and N = 100 cents

Let P= 1$ and q = 200 cents

Let the no. of miles be x

So the equation will become

2(2)+100x = 1(2)+200x

In the equation above, the values in blue represent DOLLARS, while the values in red represent CENTS.
The equation must be in terms of dollars OR cents but NOT BOTH:
(2*2 dollars) + (100x cents) = (4 dollars) + (x dollars) = (4 + x) dollars.
(1*2 dollars) + (200x cents) = (2 dollars) + (2x dollars) = (2 + 2x) dollars.

Since the dollar amounts are equal, we get:
4 + x = 2 + 2x
x = 2.
Since the question stem asks for the distance -- the value of x -- our target is 2.

if I put the value in option B, then I do not find the answer.

Option B : 200(1-2)/100-200

this will give 2

The correct answer is B.