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Question 1098582: If 6 men can harvest a field in 60 working hours, and a man works three times as fast as a boy:
(a) How long will 10 boys take to harvest the field?
(b) How many boys will be needed to help the 6 men finish the job in 40 hours?
(c) How many boys are needed with 8 men to finish the job in 20 hours?
(d) 12 boys help some men to harvest the field, but after only 15 working hours they tired and quit the job. The men worked on. Had the boys worked till the job was complete, they would have taken 45 hours.
(i) How many men are working with the boys initially?
(ii) How long did it take the men to complete the harvest after the boys quit?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 6 men can harvest the field in 60 working hours.
each man works 3 times as fast as each boy.
therefore it would take 18 boys to harvest the same field in 60 hours.
looked at another way:
6 men can harvest a field in 60 hours.
rate of each person * number of persons * time = quantity
this formula becomes rate * 6 * 60 = 1
rate = what you want to find.
number of persons is equal to 6 men
number of hours is 60.
quantity is 1 harvested field.
solve for rate of each person to get rate of each person = 1 / 360.
each man can harvest 1 / 360 of the field in 1 hour.
rate of each person * number of persons * time = quantity.
1/360 * 6 * 60 = 1 which becomes 1 = 1, so the value of rate of each person are correct.
each man works 3 times as fast as each boy, on the average.
the rate of each boy must therefore be equal to 1/3 * 1/360 = 1/1080 of the field in each hour.
you would require 18 boys to harvest the same field in 60 hours.
1/1080 * 18 * 60 = 1 becomes 1 = 1, confirming the rate of each boy is correct.
if the rate of each boy is 1/3 * the rate of each man, then it should take 3 times as many boys to harvest the field in 60 hours, and it does.
so, you have.
the rate of each man is 1/360 of the field in 1 hour.
the rate of each boy is 1/1080 of the field in 1 hour.
now to the questions.
(a) How long will 10 boys take to harvest the field?
1/1080 * 10 * T = 1
10/1080 * T = 1
T= 1 / (10/1080) = 108 hours.
(b) How many boys will be needed to help the 6 men finish the job in 40 hours?
when they work together, their rates are additive.
(1/1080 * x + 1/360 * 6) * 40 = 1
distribute the multiplication to get:
40/1080 * x + 240/360 = 1
multiply both sides of the equation by 1080 to get:
40 * x + 720 = 1080
subtract 720 from both sides of the equation to get:
40x = 360
solve for x to get x = 360/40 = 9
it would take 9 boys to assist 6 men to enable the job to be completed in 40 hours.
(1/1080 * 9) + (1/360 * 6) * 40 = 1 becomes 1 = 1, confirming the solution is correct.
(c) How many boys are needed with 8 men to finish the job in 20 hours?
(1/1080) * x + 1/360 * 8) * 20 = 1.
distribute the multiplication to get:
20/1080 * x + 160/360 = 1
multiply both sides of the equation by 1080 to get:
20 * x + 480 = 1080
subtract 480 from both sides of the equation to get:
20*x = 600
solve for x to get x = 30
it would take 30 boys to help 8 men to allow the harvesting of the field to be completed in 20 hours.
(1/1080 * 30 + 1/360 * 8) * 20 = 1 becomes 1 = 1, confirming the solution is correct.
(d) 12 boys help some men to harvest the field, but after only 15 working hours they tired and quit the job. The men worked on. Had the boys worked till the job was complete, they would have taken 45 hours.
if the boys had worked until the job was completed, then:
(1/1080 * 12 + 1/360 * x) * 45 = 1
distribute the multiplication to get:
540/1080 + 45/360 * x = 1
multiply both sides of the equation by 1080 to get:
540 + 135*x = 1080
subtract 540 from both sides of the equation to get:
135*x = 540
divide both sides of the equation by 135 to get:
x = 4
if the boys had worked until the job was completed, then 12 boys + 4 men would have completed the job in 45 hours.
(1/1080 * 12 + 1/360 * 4) * 45 = 1 becomes 1 = 1, confirming this solution is correct.
(i) How many men are working with the boys initially?
the answer to this was calculated above and is equal to 4 men.
(ii) How long did it take the men to complete the harvest after the boys quit?
the boys quit after 15 hours.
the 12 boys and the 4 men would have finished the following portion of the job in 15 hours.
(1/1080 * 12 + 1/360 * 4) * 15 = .3333333333 which is equal to 1/3 of the job.
the 4 men would need (1/360 * 4) * T = 2/3 in order to finish the job.
simplify to get:
4/360 * T = 2/3
divide both sides of this equation bhy (4/360) to get:
T = (2/3) / (4/360)
solve for T to get T = 60
once the boys stopped working, it would take the 4 men an additional 60 hours to complete the job.
the total time it took to complete the job would be 75 hours.
15 hours when the boys and men were working together and 60 more hours when the men men were working alone.
note that the 4 men working alone from the beginning would have taken 1/360 * 4 * T = 1 which would result in T = 1 / (4/360) = 90 days, so the boys did help, even though they quit after 15 hours.
the general formula applies, and is:
rate of each person * number of persons * time = quantity
this is an expansion of the general formula of rate * time = quantity, taking into account the number of people involved and the rate of each person.
the quantity in this case is 1, because we are talking about 1 harvested field.
if you have any problems or questions about this, let me know and i'll answer as best i can.
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