Solving Rational Equations

Question

John takes 2 hours longer than Andrew to peel 600 pounds of apples. If together they can peel 600 pounds of apples in 8 hours, then how long would it take John to peel the apples working alone?

Answer 1

Let t be the time it takes John to peel 600 hours.  This means he peels the 600 apples at the rate of 600 apples/ t hours.

 

Andrew takes (t-2) hours to peel 600 apples.  Andrew's rate is 600 apples/ (t-2) hours.

 

Together it takes them 8 hours, so their combined rate is 600 apples/ 8 hours.

 

Their combined rate is also their individual rates combined: 600/t + 600/(t-2)

 

Their two rates are equal: 600/t + 600/(t-2) = 600/8 => 1/t + 1/(t-2) = 1/8

 

Solving this equation, multiple sides of it by 8 * t*(t-2) => 8*(t-2) + 8*t = t * (t -2)

                                                                          =>  16*t - 16 = t² - 2t

                                                                          => t² - 18*t + 16 = 0

 

t = (18 ± √(18² - 4 * 16))/2 = (18 ± √(260))/2 = 9 ± √(65)

 

This also means that Andrew takes t -2 hours to peel the 600 apples, which would be 7 ± √(65)

The value of 7 - √(65) is negative which is not possible.

 

∴ The only possible value is it took John 9 + √(65) ≈ 17 hours to peel the 600 pounds of apples by himself.

 

Answer 2

Let us assume that John takes t hours to peel 600 pounds. Thus, Andrew should take (t-2) hours to peel the same amount. Let's convert these quantities in terms of how much apple they can peel in unit time (1 hour in this case):

 

In 1 hour, John can peel 600/t pounds of apple.

In 1 hour, Andrew can peel 600/(t-2) pounds of apple.

 

Thus, while working together:

 

In 1 hour, John and Andrew can peel (600/t + 600/(t-2)) pounds of apple.

 

Since we have some information about the amount of apple they peeled together in 8 hours, let's convert this in 8 hours instead of just 1 hour:

 

In 8 hours, John and Andrew can peel 8(600/t + 600/(t-2)) pounds of apple.

 

According to the problem statement, the actual amount of apple was 600 pounds. Thus, the equation should be:

 

8(600/t + 600/(t-2)) = 600

8/t + 8/(t-2) = 1

8t-16+8t = t^2-2t

t^2 - 18t + 16 = 0

t = 17 (approximately)

 

Thus, it takes 17 hours for John to peel 600 pounds of apple.

 

* The most important part in this solution is to identify that we need to convert the calculation in terms of unit time. Staying within 600 pounds-based calculation, or even changing the amount of apple to unit (e.g. finding out the time required to peel 1 pound of apple) won't help much since we cannot come up with something useful with this when they are working together.

 

** Please let me know if it helps you (or something still seems confusing). Thanks.