Shortcut -Time and Work
Work is directly proportional to time.
$\dfrac{W_{1}}{W_{2}}=\dfrac{T_{1}}{T_{2}}$
$W_{1} $ – Work done in first case.
$T_{1}$ – Time taken in first case.
$W_{2} $ – Work done in second case.
$T_{2}$ – Time taken in second case.
Question:
It took 40 days to build 10 buildings. How long will it take to build 50 buildings using same number of employees?
Answer:$W_{1} $= 10; $W_{2} $ = 50
$T_{1}$ = 40; $T_{2}$= ?
Substitute the values in the above formula, we get
10/50 = 40/$T_{2}$
$T_{2} $= 200 days
Shortcut -Time and Work
Work is directly proportional to Resource.
$\dfrac{W_{1}}{W_{2}}=\dfrac{R_{1}}{R_{2}}$
Question:
40 men are required to complete 20 buildings. How many men are required to build 50 buildings in the same time? 
$R_{1}$ = 40;
$R_{2} $= ?
$W_{1}$= 20;
$W_{2}$ = 50
Substitute the values in the above formula, we get
$40/R_{2}$ = 20/50
$R_{2}$ = 100
Shortcut -Time and Work
Time is inversely proportional to Resource. $\dfrac{T_{1}}{T_{2}}=\dfrac{R_{2}}{R_{1}}$
Question:
If 20 men can do the work in 40 days, how many men are required to do the same work in 10 days?
Answer: $R_{1}$ = 20;
$ R_{2}$ = ?
$T_{1}$= 40;
$T_{2}$ = 10
Substitute the values in the above formula, we get
$20/R_{2}$= 10/40
$R_{2}$ = 80
Shortcut -Time and Work
Comparison of Time, Work and Resource:
$\dfrac{W_{1}}{W_{2}}=\dfrac{R_{1}T_{1}}{R_{2}T_{2}}$
Question:
If 10 men can cut 40 trees in 4 days, how many trees can be cut by 40 men 6 days? 
$R_{1}$ = 10;
$R_{2}$ = 40
$T_{1}$ = 4;
$T_{2}$= 6
$W_{1}$ = 40;
$W_{2}$ = ?
Substitute the values in the above formula, we get
$40/W_{2}$ = (10 x 4)/(40 x 6)
$W_{2}$= 240 trees
Shortcut -Time and Work
When the number of working hours per day is included in the question. $\dfrac{W_{1}}{W_{2}}=\dfrac{R_{1}T_{1}H_{1}}{R_{2}T_{2}H_{2}}$
Question:
30 men working 6 hours day for 50 days can make 2000 toys. How many hours per day should 40 men work for 60 days to make 3000 toys? 
$R_{1}$ = 30; $R_{2}$ = 40, (Resources in each case)
$T_{1}$= 50;$T_{2}$ = 60, (Days in each case)
$W_{1}$ = 2000; $W_{2}$ = 3000, (Work in each case)
$H_{1}$= 6; $H_{2}$= ?, (Hours per day in each case)
Substitute the values in the above formula, we get
2000/3000 = (30 x 50 x 6)/(40 x 60 x H2)
$H_{2}$= 5(5/8) hours = 5 hours 32 minutes 30 seconds
Shortcut -Time and Work
Two resources working together:
$T=\dfrac{AB}{A+B}$ or $\dfrac{1}{T}=\dfrac{1}{A}+\dfrac{1}{B}$
Question:
A can complete a work in 40 days. B can complete the same work in 60 days. How long will they take to complete the work if they are working together?
Answer: A = 40
B = 60
Substitute the values in the above formula, we get
Time taken together = (40 x 60)/(40 + 60)
= 24 days
Shortcut -Time and Work
Three resources working together:
$T=\dfrac{ABC}{AB+BC+AC} $ or $\dfrac{1}{T}= \dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1} {C}$
Question:
A can do a work in 40 days, B in 50 days and C in 60 days. If they work together, how long will they take to complete the work?
Answer: A = 40
B = 50
C = 60
Substitute the values in the above formula, we get
Time taken together = (40 x 50 x 60)/(40 + 50 + 60)
= 16.21 days
Shortcut -Time and Work
Work done:
Work done per day =1/Time taken to do complete work.
Work done=Number of days worked $\times$ work done per day.
Question:
A can do a piece of work in 40 days. What is the fraction of work done by him in 25 days?
Answer: Per day work of A = 1/40
Number of days for which A worked = 25
Work done = (1/40) x 25=5/8
Question:
A can do a piece of work in 30 days. What is the fraction of work done by him in 45 days?
Answer: Per day work of A = 1/30
Number of days for which A worked = 45
Work done = (1/30) x 45=$1\dfrac{1}{2}$
Shortcut -Time and Work
Remaining work:
Remaining work=1-work done
Question:
A can do a work in 30 days, B in 40 days. A works for 9 days and the remaining work is done by B. What is the fraction of work done by B?
Answer: Remaining work for B = 1 – Work done by A
Work done by A = (1/30) x 9 = 3/10
Remaining work for B = 1 – (3/10) = 7/10
Question:
A can do a work in 40 days, B in 60 days. A works for 24 days and the remaining work is done by B. What is the fraction of work done by B?
Answer: Remaining work for B = 1 – Work done by A
Work done by A = (1/40) x 24 = 6/10
Remaining work for B = 1 – (6/10) = 4/10
= 2/5
Shortcut -Time and Work
Time taken to do the remaining work. Time taken to do the remaining work=Remaining work $\times $Time taken to do full work
Question:
A can do a work in 30 days, B in 40 days. A works for 9 days and the remaining work is done by B. What is the total number of days to complete the work?
Answer: Time taken to complete the work = Remaining work x time to finish full work
Remaining work for B = 1 – Work done by A
Work done by A = (1/30) x 9 = 3/10
Remaining work for B = 1 – 3/10 = 7/10
Time taken by B to complete the remaining work
= 7/10 x 40
= 28 days
Total time taken
= 28 + 9
= 37 days
Shortcut -Time and Work
Salary ratio of resources working together.
Salary of a resource is directly proportional to the per day work.
Question:
A can do a work in 20 days and B can do the same work in 40 days. They both work together and get a combined salary of Rs. 3000. What is the salary of A?
Answer: $S_{A}$ = Salary of A;
$ S_{B}$ = Salary of B
A = 20;
B = 40
$S_{A}: S_{B}$ = 40 : 20 = 2 : 1
$S_{A}$= [2/(2+1)] x 3000 = 2000
Question:
A can do a work in 30 days B in 40 days and C in 50 days. If they all work together, what will the ratio between their salaries?
Answer: A = 30;
B = 40;
C = 50
$S_{A}: S_{B} : S_{C}$ = (1/30) : (1/40) : (1/50) = 20 : 15 : 12