SOLUTION: if a takes an old sweeper 30 hour to clean a street and takes a new and old sweepers together only 7.5 hours to clean the same street, then how long would it take for new sweeper

Algebra ->  Rate-of-work-word-problems -> SOLUTION: if a takes an old sweeper 30 hour to clean a street and takes a new and old sweepers together only 7.5 hours to clean the same street, then how long would it take for new sweeper       Log On


   

Question 734376: if a takes an old sweeper 30 hour to clean a street and takes a new and old sweepers together only 7.5 hours to clean the same street, then how long would it take for new sweeper to cleanse the same street?
Found 6 solutions by lynnlo, josmiceli, ikleyn, greenestamps, josgarithmetic, math_tutor2020:
Answer by lynnlo(4176) About Me  (Show Source):
Answer by josmiceli(19441) About Me  (Show Source):
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Let +t+ = time in hours for new sweeper to clean street
The new sweeper's rate of sweeping is ( 1 street cleaned ) / ( t hrs )
The old sweeper's rate of sweeping is ( 1 street cleaned / 30 hrs )
Together, their rate is ( 1 street cleaned ) / ( 7.5 hrs )
-------------
Add their rates of sweeping to get their rate sweeping together
+1+%2F+30+%2B+1+%2F+t+=+1%2F+7.5+
Multiply both sides by +30t+
+t+%2B+30+=+4t+
+3t+=+30+
+t+=+10+
It will take the new sweeper 10 hrs to clean street
check:
+1%2F30+%2B+3%2F30+=+4%2F30+
+4%2F30+=+4%2F30+
OK

Answer by ikleyn(53427) About Me  (Show Source):
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.
if highlight%28cross%28a%29%29 it takes an old sweeper 30 hour to clean a street and takes a new and old sweepers together
only 7.5 hours to clean the same street, then how long would it take for new sweeper
to highlight%28cross%28cleanse%29%29 clean the same street?
~~~~~~~~~~~~~~~~~~~~~~~ 

From the problem, the combined rate of the old and the new sweepers is  1%2F7.5  of the job
per hour, when they work together,

while the rate of the old sweeper is  1%2F30  of the job per hour.


So, the rate of the new sweeper, when it works alone, is the difference

    1%2F7.5 - 1%2F30 = %2830-7.5%29%2F%287.5%2A30%29 = 21.5%2F%287.5%2A30%29 = 3%2F30 = 1%2F10

of the job per hour.


Hence, it will take 10 hours for the new sweeper to clean the street working alone.    ANSWER

Solved.

This method analyzing rate of work via fractions is very powerful and effective way solving such problems.



Answer by greenestamps(13258) About Me  (Show Source):
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An informal solution, using logical reasoning instead of the formal algebra shown in the response from the other tutor....

When the new sweeper joins the old one, the job is completed in 1/4 as much time (7.5 hours instead of 30), so it is as if there are now 4 of the old sweepers working.

That means the new sweeper does as much work as 3 of the old sweepers.

And since the old sweeper can do the job alone in 30 hours, the new sweeper can do the job alone is 1/3 as much time, which his 10 hours.

ANSWER: 10 hours

Answer by josgarithmetic(39702) About Me  (Show Source):
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n hours just the new sweeper alone
                RATE        TIME          JOB

OLD             1/30         30            1

NEW             1/n           n            1

BOTH           1/7.5          7.5          1

Rates of the two working together are simply additive.
highlight%281%2F30%2B1%2Fn=1%2F7.5%29
Solve.

Answer by math_tutor2020(3828) About Me  (Show Source):
You can put this solution on YOUR website!

There are many great answers by the other tutors.

Here's yet another approach.
Let's say the street length is 3000 feet in total.
The value 3000 doesn't matter and can be changed to any other number you want, since the final answer at the end will be the same.

The old sweeper does the full job in 30 hours when working alone.
The old sweeper's rate is (3000 ft)/(30 hr) = 100 feet per hour.

The new sweeper takes x hours to do the job when working alone.
The new sweeper's rate is 3000/x feet per hour.

Their combined rate is 100 + (3000/x) feet per hour.
This assumes that neither sweeper hinders the other.
Multiplying this combined rate by 7.5 hours should lead to the total 3000 feet needed to be cleaned.
7.5*( 100 + (3000/x) ) = 3000
which solves to x = 10

Therefore the new sweeper needs 10 hours to clean the entire street by itself.