. How fast could the new belt do the job by itself?

Question

an old conveyor belt takes 21 h to move one days coal output, working with a new conveyor belt, they finish the job in 9 hours. 

Answer 1

Hi Shaw, the old conveyor belt takes 21 hours to complete the job, or it can complete 1/21 part of the job each hour. We know that the old and new conveyor belts can complete the job in 9 hours, or they can complete 1/9 part of of the job each hour.

 

we will let "x" represent the time it takes the new conveyor belt to complete one days coal output, or it can complete 1/x part of the job each hour.

 

Create an equation using what we know:

 

1/21 + 1/x = 1/9

 

Subtract 1/21 from both sides of the equation giving:

 

1/x = 1/9 - 1/21

 

We need to find a common denominator for x, 9, and 21. A simple way to eliminate all of the fractions at the same time is to multiply each term by x * 9 * 21.

 

This gives:

 

(x * 9 * 21) / x = (x * 9 * 21) / 9 - (x * 9 * 21) / 21

 

The x cancels out in the first term, the 9 cancels out in the second term, and the 21 cancels out in the last term giving:

 

9 * 21 = 21x - 9x

 

189 = 12x or x = 15.75 hours.

 

Therefore, the new conveyor belt can move one days coal output in 15.75 hours.

 

Answer 2

This is a "classic", for better or worse.

 

The key is to think about how much of the job each belt does in an hour.  Say the old belt takes 21 hours, the new belt takes x hours, and the two together take 9 hours.  This means that in an hour, the old belt does 1/21 of the work, the new belt does 1/x of the belt, and the two together do 1/9 of the job.

 

This gives us the equation 1/21 + 1/x = 1/9.  Clear fractions and solve.

 

The "trick" is to think about rates, which DO combine by addition, instead of the times, which don't.