Question 1198343: A student must leave for campus in half an hour or he will be late for class. Unfortunately, he is snowed in. He can shovel the driveway in 35 minutes, and his brother claims to be able to do it in 14 minutes. If they shovel together, how long will it take to clear the driveway? Will this give enough time for the student to leave for campus?
Found 3 solutions by josgarithmetic, math_tutor2020, ikleyn:
Answer by josgarithmetic(39625)   (Show Source):
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
Person A can do the job in 35 minutes if working alone.
Person B (the brother) can do the same job in 14 minutes if working alone.
35 = 5*7
14 = 2*7
LCM = 2*5*7 = 10*7 = 70
or
LCM(x,y) = x*y/GCF
LCM(14,35) = 14*35/7
LCM(14,35) = 70
In short, the LCM of 14 and 35 is 70.
I'll use this value in the example problem below.
Consider the task is to shovel 70 cubic feet of snow.
Person A has a unit rate of 70/35 = 2 cubic feet per minute.
Person B has a unit rate of 70/14 = 5 cubic feet per minute.
If working together, without hindering one another, then their combined rate is 2+5 = 7 cubic feet per minute.
Therefore, the total time taken if working together is 70/7 = 10 minutes.
Relevant formulas
(rate)*(time) = amount done
rate = (amount done)/(time)
time = (amount done)/(rate)
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Algebraic method:
x = total time taken if working together
time is in minutes
Person A has a unit rate of 1/35 of a job per minute.
i.e. every 35 minutes, person A does 1 full job. "Job" in this case refers to shoveling the entire driveway.
I'm using the informal idea that
rate = (amount done)/time
rate = (1 job)/time
rate = 1/time
Person B has a unit rate of 1/14 of a job per minute using similar logic as earlier discussed.
The combined unit rate is
1/35 + 1/14
2/70 + 5/70
(2+5)/70
7/70
1/10
Therefore,
(unit rate)*(time) = amount done
(unit rate)*(time) = 1 job
(1/10 of a job per min)*(x minutes) = 1 job
(1/10)*x = 1
x = 10 minutes
Note:
the equation
(1/35 + 1/14)*x = 1
is equivalent to
1/35 + 1/14 = 1/x
We can multiply both sides by the LCD 70x to clear out the fractions and we'd get
2x+5x = 70
Solving that equation leads to x = 10
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Answer: 10 minutes
This gives plenty of time to make it to campus (since the value is less than 30 minutes, aka half an hour).
Answer by ikleyn(52854)  (Show Source):
You can put this solution on YOUR website! .
Hello,
it seems to me that the "problem" behind this message (given in perfect English) is a joke.
Why do I think so ?
Because the brother alone can complete the job in 14 minutes,
and it answers the question if they will be able to do the job faster than in half an hour, working together.
If I am not right, then please let me know and explain
for what reason the question about the half an hour is included to the problem.
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