This document discusses direct and indirect proportions as they relate to time and work. It provides examples of how quantities that are directly proportional increase together, while quantities that are indirectly proportional decrease together. The document then introduces the man-hour formula for calculating work done based on the number of men, time worked, and hours worked per day. Several example problems are provided to demonstrate how to use the man-hour formula and other proportional reasoning to solve work and time word problems.

1. Time and Work

2. Chain Rule
• Direct Proportion: Any two quantities are said to be directly proportional, if on
the increase of one quantity, the other quantity increases and vice-versa.
• Example: Cost is directly proportional to number of objects
• Cost ∝ Number of objects
• Number of objects increases (↑) Cost (↑)
• Example: Work done is directly proportional to number of working men
• Work done ∝ Number of working men
• Number of men increase (↑) Work done (↑)

3. Chain Rule
• Indirect Proportion: Any two quantities are said to be indirectly
proportional, if on the increase of one quantity, the other quantity
decreases and vice-versa.
• Example: If speed of car is increases, then the time required to cover
the distance decreases.
• Speed of car (↑) Time required decreases (↓)
• Example: Time taken to finish work increases, if number of men
decrease.
• Time (↑) Number of men (↓)

4. 1) If the wages of 6 men for 15 days be Rs. 2100, then find the wages of
9 men for 12 days.
2) If 15 men, working 9 hours a day, can reap a field in 16 days, in how
many days will 18 men reap the field, working 8 hours a day?
3) If 9 engines consumes 24 metric tonnes of coal, when each is
working 8 hours a day, how much coal will be required for 8 engines,
each running 13 hours a day, it being given that 3 engines of former
type consume as much as 4 engines of latter type?
Examples

5. 4) A contract is to be completed in 46 days and 117 men were set to
work, each working 8 hours a day. After 33 days, 4/7 of the work is
completed. How many additional men may be employed so that the
work may be completed in time, each men working 9 hours a day?
Examples

6. • More men can do more work.
• More work means more time required to do work.
• More men can do more work in less time.
• M men can do a piece of work in T hours, then Total effort or work
=MT man hours.
• Rate of work * Time = Work Done
• If A can do a piece of work in D days, then A's 1 day's work = 1/D.
• Part of work done by A for t days = t/D.
• If A's 1 day's work = 1/D, then A can finish the work in D days.
Man - Work - Hour Formula:

7. • Part of work done by A for t days = t/D.
• If A's 1 day's work = 1/D, then A can finish the work in D days.
Man - Work - Hour Formula:

8. • If M1 men can do W1 work in D1 days working H1 hours per day and
M2 men can do W2 work in D2 days working H2 hours per day, then
Man - Work - Hour Formula:

9. Tips and Tricks:
• If A can do a piece of work is x days and B can do a piece of work in
y days, then both of them working together will do the same work
in:
• xy/(x+y) days
• Two persons A & B, working together, can complete a piece of work in
x days. If A, working alone, can complete the work in y days, then B,
working alone, will complete the work in
• ⇒xy/(y-x)

10. Tips and Tricks:
• If A & B working together, can finish a piece of work is x days, B & C in
y days, C & A in z days. Then, A + B + C working together will finish the
job is
• ⇒2xyz/(xy+yz+zx)
• If A working Alone takes a days more than A & B, & B working Alone
takes b days more than A & B. Then,
Number of days, taken by A & B working together to finish a job is =
√ab

11. 1) A and B together can complete a piece of work in 4 days. If A alone
can complete the same work in 12 days, in how many days can B alone
complete that work?
2) A and B can do a piece of work in 18 days, B and C can do it in 24
days, A and C can do it in 36 days. In how many days will A, B and C
finish it, working together.
3) A can do a piece of work in 7 days of 9 hours each and B can do it in
6 days of 7 hours each. How long will they take to do it, working
together 8(2/5) hours a day?
Examples

12. 4) A is twice as good as workman B. And together they finish a piece of
work in 18 days. In how many days will A alone finish the work?
5) A can do a certain job in 12 days. B is 60% more efficient than A.
How many days does B alone take to do the same job?
6) A and B working separately can do a piece of work in 9 and 12 days
respectively. If they work for a day alternately, A beginning, in how
many days, the work will be completed.
Examples

13. 7) 45 men can complete a work in 16 days. Six days after they started
working, 30 more men joined them. How many days will they now take to
complete the remaining work?
8) 2 men and 3 boys can do a piece of work in 10 days while 3 men and 2
boys can do the same work in 8 days. In how many days can 2 men and 1 boy
do the work?
9) A garrison of 3300 men had provisions for 32 days, when given at the rate
of 850 gms per head. At the end of 7 days, a reinforcement arrives and it was
found that the provisions will last 17 days more, when given at the rate of
825 gms per head. What is the strength of the reinforcement?
Examples