# How Many Minutes?

#### magentarita

Skipper can mow his lawn in 20 minutes less times with the power mower than with his hand mower. One day the power mower broke down 15 minutes after he started mowing, and he took 25 minutes more time to complete the job with the hand mower. How many minutes does Skipper take to mow the lawn with the power mower?

I wrestled with this question for more than 60 minutes and could not come up with the correct equation. Can someone help me? I dislike the way the question is worded. Do you agree?

#### mr fantastic

MHF Hall of Fame
Skipper can mow his lawn in 20 minutes less times with the power mower than with his hand mower. One day the power mower broke down 15 minutes after he started mowing, and he took 25 minutes more time to complete the job with the hand mower. How many minutes does Skipper take to mow the lawn with the power mower?

I wrestled with this question for more than 60 minutes and could not come up with the correct equation. Can someone help me? I dislike the way the question is worded. Do you agree?
Let x be the time (in minutes) it takes using the power mower, let y be the time (in minutes) it takes using the hand mower.

y = x + 20 .... (1)

After 15 minutes the fraction of time remaining to do the line and hence the fraction of lawn still to be mowed is $$\displaystyle \frac{x - 15}{x}$$. So the time it will take to finish the job using the hand mower is $$\displaystyle \frac{x - 15}{x} \cdot y$$.

But taking 25 minutes more to complete the job means that the lawn got mowed in 15 + 25 = 40 minutes. Hence:

$$\displaystyle 15 + \frac{x - 15}{x} \cdot y = 40 \Rightarrow 15x + (x - 15)y = 40x \(\displaystyle \Rightarrow (x - 15)y = 25x$$ .... (2)

Substitute equation (1) into equation (2):

$$\displaystyle (x - 15) (x + 20) = 25x$$.

You want the positive solution to this equation. I get x = 30.\)

• magentarita

#### magentarita

ok..

Let x be the time (in minutes) it takes using the power mower, let y be the time (in minutes) it takes using the hand mower.

y = x + 20 .... (1)

After 15 minutes the fraction of time remaining to do the line and hence the fraction of lawn still to be mowed is $$\displaystyle \frac{x - 15}{x}$$. So the time it will take to finish the job using the hand mower is $$\displaystyle \frac{x - 15}{x} \cdot y$$.

But taking 25 minutes more to complete the job means that the lawn got mowed in 15 + 25 = 40 minutes. Hence:

$$\displaystyle 15 + \frac{x - 15}{x} \cdot y = 40 \Rightarrow 15x + (x - 15)y = 40x \(\displaystyle \Rightarrow (x - 15)y = 25x$$ .... (2)

Substitute equation (1) into equation (2):

$$\displaystyle (x - 15) (x + 20) = 25x$$.

You want the positive solution to this equation. I get x = 30.\)
$$\displaystyle I would have never come up with those two equations. First of all, I had no clue that this question is one involving a system of equations in two unknowns. Secondly, the wording is absolutely horrible, to say the least.$$

#### mr fantastic

MHF Hall of Fame
I would have never come up with those two equations. First of all, I had no clue that this question is one involving a system of equations in two unknowns. Secondly, the wording is absolutely horrible, to say the least.
I agree it could have been worded better. In particular:

One day the power mower broke down 15 minutes after he started mowing, and he took 25 minutes more time to complete the job with the hand mower

is ambiguous and could be interpretted several different ways. But the way I interpretted it is the only way that gives a sensible answer.