Question 162774This question is from textbook
: Lourdes can trim the hedges around her property in 8 hours by using an electric hedge trimmer. Rafael can do the same job in 15 hours by using a manual trimmer. How long would it take them to trim the hedge working together?
This question is from textbook
Found 2 solutions by user_dude2008, gonzo:
Answer by user_dude2008(1862)   (Show Source):
Answer by gonzo(654) (Show Source):
You can put this solution on YOUR website! this looks like the answer.
let H = the hedge.
i suspect this oould be anything, like 100 yards, or 75 acres. the size will affect the rate but the rate between lourdes and Rafael will be proportional and so the absolute rates will be adjusted accordingly.
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lourdes does it in 8 hours, so 8 * R1 = H
where 8 is the number of hours it took, and R1 is the amount of H that was done in one hour.
since 8 * R1 = H, then
R1 = H/8
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rafael does it in 15 hours, so 15 * R2 = H
where 15 is the number of hours it took, and R2 is the amount of H that was done on one hour.
since 15 * R2 = H, then
R2 = H/15
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working together, their rates are added together to get a combined rate, and they can do it in X hours (unknown at this time).
formula is
X * (R1 + R2) = H
where X is the number of hours it took, and (R1+R2) is the amount of H that was done in one hour.
since R1 = H/8 and R2 = H/15, then
X * (H/8 + H/15) = H
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multiply both sides of the equation by 120 to get rid of the denominators.
120*X * (H/8 + H/15) = 120*H
expanding, we get
120*X*H/8 + 120*X*H/15 = 120*H
this becomes
15*X*H + 8*X*H = 120*H
this becomes
23*X*H = 120*H
dividing both sides of the equation by 23*H and it becomes
X = 120*H/23*H
which becomes
X = 120/23 = 5.217391304
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the combined rate takes 5.217391304 hours to trim the hedge.
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to prove this formula correct, pick any value for H.
take 1000 feet of hedge.
Lourdes rate is 1000/8 = 125 feet of hedge per hour.
rafaels rate is 1000/15 = 66.6666667 feet of hedge per hour.
their combined rate is 191.6666667 feet of hedge per hour.
this takes them 1000/191.6666667 hours to complete which equals 5.217391304 hours to complete.
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any value for H will work.
try 75 acres.
R1 = 75/8 = 9.375 acres per hour.
R2 = 75/15 = 5 acres per hour.
combined rate is 14.375 acres per hour.
75/14.375 = 5.217391304 hours to complete.
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