SOLUTION: In 8 minutes, a conveyor belt can move 1,000 bushels of corn into a storage bin. A smaller belt can move 1,000 bushels to the storage bin in 14 minutes. If both belts are used, how

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Question 1206862: In 8 minutes, a conveyor belt can move 1,000 bushels of corn into a storage bin. A smaller belt can move 1,000 bushels to the storage bin in 14 minutes. If both belts are used, how long will it take to move 1,000 bushels to the storage bin?
Found 3 solutions by greenestamps, math_tutor2020, MathTherapy:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


If this problem is on a competitive timed test, then you can use the standard shortcut for solving problems like this involving two workers: If the times for the two workers to perform the job individually are A and B, then the time required for them to do the job together is %28A%2AB%29%2F%28A%2BB%29

For this problem: %2814%2A8%29%2F%2814%2B8%29=112%2F22=56%2F11

ANSWER: 56/11 hours, or 5 1/11 hours

If this problem is on a test in a classroom where you are supposed to show a formal algebraic solution, then the solution might look something like this.

Let x be the number of hours it takes the two of them together to do the job.

1/8 = fraction of job the first belt does in 1 minute
1/14 = fraction of job the second belt does in 1 minute
1/x = fraction of job the two belts together do in 1 minute

The fraction of the job they do together is the sum of the fractions of the job each does alone in 1 minute:

1%2F8%2B1%2F14=1%2Fx

Multiply the equation by the least common multiple of the denominators, which is 56x:

7x%2B4x=56
11x=56
x=56%2F11

And here is a different formal method for solving the problem and any similar problem.

8 times 14 is 112.

In 112 minutes, the first belt could do the job 112/8 = 14 times; in 112 minutes the second belt could do the job 112/14 = 8 times. So together in 112 minutes the two could do the job 14+8 = 22 times.

And so the number of minutes it will take the two belts together to do the one job is 112/22 = 56/11 = 5 1/11.

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 5.090909 minutes approximately
The "09" decimal portion repeats forever.
This is equivalent to the mixed number 5 & 1/11 minutes.

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Explanation

The LCM of 8 and 14 is 56 because:
8*14/2 = 56
where the "divide by 2" portion is the GCF of 8 and 14.
LCM = x*y/GCF

Conveyor A moves 1000 bushels of corn in 8 minutes.
If it keeps this same rate then it moves 7*1000 = 7000 bushels in 7*8 = 56 minutes.

Conveyor B moves 1000 bushels of corn in 14 minutes.
If it keeps this same rate then it moves 4*1000 = 4000 bushels in 4*14 = 56 minutes.

The key is to get both conveyors moving some amount of corn in the same amount of time.
Combine those amounts:
A+B = 7000+4000 = 11000

11,000 bushels of corn are moved in 56 minutes when the conveyor belts work together.
This assumes neither conveyor belt hinders the other.
The combined unit rate would be 11000/56 = 1375/7 = 196.42857 bushels per minute approximately.

Then,
rate*time = amountDone
time = amountDone/rate
time = (1000 bushels)/( 1375/7 bushels per min)
time = (1000*7/1375) minutes
time = (56/11) minutes
time = 5.090909090909 minutes approximately
The "09" part repeats forever.

If you want the answer as a mixed number then,
5.090909090909 = 5 & 1/11

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
In 8 minutes, a conveyor belt can move 1,000 bushels of corn into a storage bin. A smaller belt can move 1,000 bushels to the storage bin in 14 minutes. If both belts are used, how long will it take to move 1,000 bushels to the storage bin?

As the 1,000 bushels is considered a SINGLE job, you CERTAINLY don't need to include the 1,000 bushels
in your calculations as the other person did.  


Let the time it takes both belts to do the 1 job (move 1,000 bushels into the storage bin), be T. 
Since the faster belt takes 8 minutes to do the job, it does %281%2F8%29%5Eth of the job in 1 minute
Since the slower belt takes 14 minutes to do the job, it does %281%2F14%29%5Eth of the job in 1 minute
We then get: matrix%281%2C3%2C+1%2F8+%2B+1%2F14%2C+%22=%22%2C+1%2FT%29
            7T + 4T = 56 ----- Multiplying by LCD, 56T
                11T = 56
Time it takes both belts to do the job (move 1,000 bushels into the storage bin), or