June 2023
Problem of the Month

How far did Jonah drive?
Submitted by K. Sengupta

If Jonah had driven for 20 minutes less than the time he would have driven if he had driven 20 miles less than he did drive but at two-thirds the speed at which he drove, he would have driven 10 miles less than he did.

However, if he had driven 20 minutes longer than the time he would have driven if he had driven 10 miles less than he did drive but at three-quarters the speed at which he drove, he would have driven 20 miles further.

How far, in miles, did Jonah drive?

Solution:

Jonah had driven for a distance of 64 miles.

Jonah had driven for a distance of 64 miles.

Let D = original mileage.
Let S = original speed

Then you can write equations for the two statements:


Now solve the two equations simultaneously:


Here is K. Sengupta's solution:

Let d be the actual distance (miles).
Let s be the actual speed (miles/hr).

To get all units in miles and hours, we must use
20 min = 1/3 hr.

From the two given sentences, we can write two equations, keeping in mind that Distance = Rate x Time and Time = Distance / Rate

((d-20)/(2s/3) - 1/3) s = d - 10

and

((d-10)/(3s/4) + 1/3) s = d + 20

From the first:

3s(d-20)/(2s) - s/3 = d - 10
3d/2 - 30 - s/3 = d - 10
9d - 180 - 2s = 6d - 60
3d = 2s + 120

From the second

4s(d - 10)/(3s) + s/3 = d + 20
4d/3 - 40/3 + s/3 = d + 20
4d - 40 + s = 3d + 60
d = 100 - s
3d = 300 - 3s

2s + 120 = 300 - 3s
5s = 180

s = 36
d = 64

Consequently, Jonah had driven for a distance of 64 miles.

Correctly solved by:

1. Seth Cohen	Concord, New Hampshire
2. Kelly Stubblefield	Mobile, Alabama
3. Rob Miles	Northbrook, Illinois

Send any comments or questions to: David Pleacher