In this concept FastPath page you will learn how to solve Work and Time aptitude problems. This method when understood correctly will enable you to solve even the toughest variety of Work and Time problems in very less time. As a prerequisite you will need to know the concept of LCM. The rest of the problem solving technique is easy. Please do a thorough reading and get fair understanding of the below method before trying to directly solve any other Work and Time problems.
I decided to name this approach of solving as TRW (Time-Rate-Work) Method as it involves a table containing time(T), rate(R) and work(W) fields in which we tabulate the data received from the problem statement.
For any puzzle first form a TRW table as shown:
Make a mental note of the below 2 rules, you will know what this rule means in further steps.
Rules:
1) Work = Rate * Time
2) In the table when persons are combined only ‘rate’ can be summed.
Note that example 1 is dealt in very detailed manner, the method will seem long. Don’t give up just because the steps seem lengthy. See further examples, its solved in one shot!
Example 1
Steps to solve
Step 1
Draw the TRW table:
Step 2
Fill the table with only the time data obtained from the problem. Do not worry now about the “If they work together for 4 days then the fraction of work left is” part of the problem for now.
Step 3
In the work field of the table, fill in the LCM of the times (15 and 20). (You can put any common multiple, but putting the least common multiple (LCM) will scale down the mathematics to smaller numbers).
LCM of 15, 20 is 60. So fill 60 in the work field.
Step 4
We know that Rate = work / time. (From rule 1). So fill in the respective rates for A and B
This above table now means that:
A has to work for 15 days at the rate of 4 units of work per day to perform 60 units of work.
B has to work for 20 days at the rate of 3 units of work per day to perform 60 units of work.
Now let’s deal with “If they work together for 4 days then the fraction of work left is”
Since A and B work together, we extend the table as shown: (remember rule 2: In the table when persons are combined only ‘rate’ can be summed.) Therefore combined rate is 4 + 3 = 7
They work for 4 days,
Combined work they achieve is: Combined work =combined rate * time = 4 * 7 = 28
Therefore,
Work done: 28
Work left: 60 – 28 = 32
Fraction of work left = work left / total work = 32 / 60 = 8 / 15.
So option‘d’ is correct.
Example 2
Steps to solve
Step 1
Write the times (6, 8) the LCM (24) and calculate the rates as (4, 3)
So this means, 3200 is divided in ratio 4:3 and paid to A and B. We are not interested in that so don’t calculate yet.
Step 2
With help of C, i.e., A, B and C work together. We know time is 3, fill time as 3 for A + B + C. Work remaining is 24, fill work as 24 for A + B + C. Let rate of work of C be x which is to be found. So combined rate: 4 + 3 + x
=> Rate = work / time = 24 / 3 = 8
=> 4 + 3 + x = 7 + x = 8
=> x = 1
We can now infer that ₹ 3200 should be divided in ratio 4:3:1 among A, B and C respectively. So 4y+3y+1y = 3200
=> 8y = 3200
=> y = 400
So option ‘b’ is correct.
Example 3
Steps to solve
Step 1
Write the times (20, 12), the LCM (60) and calculate the rates as (3, 5).
A works alone for 4 days, so work done by A = 4 (days) * 3 (rate) = 12
Work remaining now is 60 – 12 = 48, this remaining work is done by A and B combined.
Step 2
So for A + B write down remaining work (48), rate (3 + 5 = 8)
Time = work / rate = 48 / 8 = 6
A works alone for first 4 days, the remaining work is completed by A and B combined for 6 days.
So the entire work lasted for 4 + 6 = 10 days.
So option ‘b’ is correct.
Example 4
Steps to solve
Step 1
Write down the time (8, 12, 6) and the LCM (24) and calculate rates as (3, 2, 4)
Now
A + B = 3
B + C = 2
A + B + C = 4
On solving A = 2, B = 1, C = 1
Step 2
For A and C time taken would be
I decided to name this approach of solving as TRW (Time-Rate-Work) Method as it involves a table containing time(T), rate(R) and work(W) fields in which we tabulate the data received from the problem statement.
For any puzzle first form a TRW table as shown:
Person
|
A
|
B
|
time ( T )
|
||
rate ( R )
|
||
work ( W )
|
Make a mental note of the below 2 rules, you will know what this rule means in further steps.
Rules:
1) Work = Rate * Time
2) In the table when persons are combined only ‘rate’ can be summed.
Note that example 1 is dealt in very detailed manner, the method will seem long. Don’t give up just because the steps seem lengthy. See further examples, its solved in one shot!
Example 1
A can do a work in 15 days and B in 20 days. If they work together for 4 days then the fraction of work left is:
Step 1
Draw the TRW table:
Person
|
A
|
B
|
time ( T )
|
||
rate ( R )
|
||
work ( W )
|
Step 2
Fill the table with only the time data obtained from the problem. Do not worry now about the “If they work together for 4 days then the fraction of work left is” part of the problem for now.
Person
|
A
|
B
|
time ( T )
|
15
|
20
|
rate ( R )
|
||
work ( W )
|
||
Step 3
In the work field of the table, fill in the LCM of the times (15 and 20). (You can put any common multiple, but putting the least common multiple (LCM) will scale down the mathematics to smaller numbers).
LCM of 15, 20 is 60. So fill 60 in the work field.
Person
|
A
|
B
|
time ( T )
|
15
|
20
|
rate ( R )
|
||
work ( W )
|
60
|
|
Step 4
We know that Rate = work / time. (From rule 1). So fill in the respective rates for A and B
Person
|
A
|
B
|
time ( T )
|
15
|
20
|
rate ( R )
|
4
|
3
|
work ( W )
|
60
|
|
This above table now means that:
A has to work for 15 days at the rate of 4 units of work per day to perform 60 units of work.
B has to work for 20 days at the rate of 3 units of work per day to perform 60 units of work.
Now let’s deal with “If they work together for 4 days then the fraction of work left is”
Since A and B work together, we extend the table as shown: (remember rule 2: In the table when persons are combined only ‘rate’ can be summed.) Therefore combined rate is 4 + 3 = 7
Person
|
A
|
B
|
A + B
|
time ( T )
|
15
|
20
|
|
rate ( R )
|
4
|
3
|
7
|
work ( W )
|
60
|
||
They work for 4 days,
Person
|
A
|
B
|
A + B
|
time ( T )
|
15
|
20
|
4
|
rate ( R )
|
4
|
3
|
7
|
work ( W )
|
60
|
||
Combined work they achieve is: Combined work =combined rate * time = 4 * 7 = 28
Person
|
A
|
B
|
A + B
|
time ( T )
|
15
|
20
|
4
|
rate ( R )
|
4
|
3
|
7
|
work ( W )
|
60
|
28
|
|
Therefore,
Work done: 28
Work left: 60 – 28 = 32
Fraction of work left = work left / total work = 32 / 60 = 8 / 15.
So option‘d’ is correct.
Example 2
A can do a piece of work in 6 days, B can do it in 8 days. A and B undertook to do it for ₹ 3200. With help of C, they complete the work in 3 days. How much should C be paid?
Step 1
Write the times (6, 8) the LCM (24) and calculate the rates as (4, 3)
Person
|
A
|
B
|
|
time ( T )
|
6
|
8
|
|
rate ( R )
|
4
|
3
|
|
work ( W )
|
24
|
||
So this means, 3200 is divided in ratio 4:3 and paid to A and B. We are not interested in that so don’t calculate yet.
Step 2
With help of C, i.e., A, B and C work together. We know time is 3, fill time as 3 for A + B + C. Work remaining is 24, fill work as 24 for A + B + C. Let rate of work of C be x which is to be found. So combined rate: 4 + 3 + x
Person
|
A
|
B
|
A+B+C
|
time ( T )
|
6
|
8
|
3
|
rate ( R )
|
4
|
3
|
4+3+x
|
work ( W )
|
24
|
24
|
|
=> Rate = work / time = 24 / 3 = 8
=> 4 + 3 + x = 7 + x = 8
=> x = 1
We can now infer that ₹ 3200 should be divided in ratio 4:3:1 among A, B and C respectively. So 4y+3y+1y = 3200
=> 8y = 3200
=> y = 400
So option ‘b’ is correct.
Example 3
A and B can do a piece of work in 20 days and 12 days respectively. A works alone and after 4 days B joins him until the completion of work. How long did the entire work last?
Step 1
Write the times (20, 12), the LCM (60) and calculate the rates as (3, 5).
Person
|
A
|
B
|
|
time ( T )
|
20
|
12
|
|
rate ( R )
|
3
|
5
|
|
work ( W )
|
60
|
||
A works alone for 4 days, so work done by A = 4 (days) * 3 (rate) = 12
Work remaining now is 60 – 12 = 48, this remaining work is done by A and B combined.
Step 2
So for A + B write down remaining work (48), rate (3 + 5 = 8)
Person
|
A
|
B
|
A + B
|
time ( T )
|
20
|
12
|
?
|
rate ( R )
|
3
|
5
|
8
|
work ( W )
|
60
|
48
|
|
Time = work / rate = 48 / 8 = 6
A works alone for first 4 days, the remaining work is completed by A and B combined for 6 days.
So the entire work lasted for 4 + 6 = 10 days.
So option ‘b’ is correct.
Example 4
A and B can do a work in 8 days, B and C do the same work in 12 days. A, B and C can finish it in 6 days. How many days will A and C take to do the work?
Step 1
Write down the time (8, 12, 6) and the LCM (24) and calculate rates as (3, 2, 4)
Person
|
A + B
|
B + C
|
A + B + C
|
time ( T )
|
8
|
12
|
6
|
rate ( R )
|
3
|
2
|
4
|
work ( W )
|
24
|
||
Now
A + B = 3
B + C = 2
A + B + C = 4
On solving A = 2, B = 1, C = 1
Step 2
For A and C time taken would be
Person
|
A + C
|
time ( T )
|
8
|
rate ( R )
|
2 + 1
|
work ( W )
|
24
|
From table above, even for A and C, time taken is 8 days.
So option ‘c’ is correct.
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