Question

A car service charges customers a flat fee per ride (which is higher during rush hour traffic) plus charges for each

minute and each mile. Suppose that, in a certain metropolitan area during rush hour, the flat fee is $4, the cost per
minute is $0.20, and the cost per mile is $1.50. Let x be the number of minutes and y the number of miles. At the end
of a ride, the driver said that the passenger owed $18.70 and remarked that the number of minutes was three times
the number of miles. Find the number of minutes and the number of miles for this trip.

Answer 1

Explanation:

Let's use the given information to set up two equations involving the number of minutes and the number of miles, and then solve for those variables.

First, we know that the total cost of the ride is the sum of the flat fee, the cost per minute times the number of minutes, and the cost per mile times the number of miles. So we have:

$18.70 = 4 + 0.20x + 1.50y$

We also know that the number of minutes is three times the number of miles, so we have:

$x = 3y$

Now we can substitute the second equation into the first equation and solve for y:

$18.70 = 4 + 0.20(3y) + 1.50y$

$18.70 = 4 + 0.60y + 1.50y$

$18.70 = 4 + 2.10y$

$14.70 = 2.10y$

$y = 7$

So the number of miles for this trip is 7. Using the second equation, we can find the number of minutes:

$x = 3y = 3(7) = 21$

Therefore, the number of minutes for this trip is 21.

Answer 2

• 7 miles
• 21 minutes

Explanation:

You want the minutes and miles for a trip that had a total cost of $18.70, which included a $4 flat fee, and $1.50 per mile plus $0.20 per minute. The number of minutes was 3 times the number of miles.

Effective rate

If each mile had a charge of $1.50 plus a charge for 3 minutes at $0.20 per minute, the effective travel charge per mile was ...

$1.50 +3(0.20) = $2.10 per mile

Travel charge

The total cost of the trip included a fixed $4 fee plus the travel charge, so the travel charge was ...

travel charge = total - fixed fee = $18.70 -4.00 = $14.70

Miles

The number of miles that result in this travel charge is ...

$14.70 / ($2.10/mile) = 7 miles

The number of minutes is 3 times this, or ...

(7 miles)·(3 minutes/mile) = 21 minutes

The charge was for 21 minutes and 7 miles.

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Additional comment

Often, when there is a fixed relationship between elements of a problem, it is convenient to combine them in a group. The equation for the number of groups is often much simpler, even to the point of allowing you to solve the problem mentally. Here, our "group" was 1 mile and 3 minutes.

This can be a useful strategy even when there is an excess. For example, if there are 3 more nickels than quarters, we know those 3 nickels add 15¢ to the value of the 30¢ groups that consist of 1 quarter and 1 nickel. When you find the number of groups, you immediately know the number of quarters. You have to add 3 to get the number of nickels.