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Basic Concepts

(i) If A can do a piece of work in n days, then work done by him in 1 day = 1/n. (ii) If A's 1 day's work = 1/n, then A can finish the work whole work in n days.

Illustration 1:

Shikha can weave a sweater in 8 days.

So, her 1 day's work = 1/8 sweater.

(iii) If A is twice as good a workman as B, then: Ratio of work done by A and B = 2 : 1.Ratio of time taken by A and B = 1 : 2.

(iv) If the number of men to do a certain work be changed in the ratio m : n, then the ratio of time taken to finish the work, changes in the ratio n : m.

1/ time1 + 1 / time2 = 1 / total time

The basic formula for solving work problems is: 1/r + 1/s = 1/h where r and s are the number of hours it takes two people R and S respectively, to complete a job when working alone, and h is the number of hours it takes R and S to do the job when working together.

It may be noted that the relation between time and work is inverse. Students usually make the mistake of forgetting this. Hence, if we are given the following sum:

If 2 persons can do some work in 4 days, 4 persons will do the same work in:

1. 1 day

2. 2 days

3. 4 days

4. 8 days

The tendency of many students is to apply direct proportion and get the answer as 8 days.

This is wrong. More persons should do the work in less time, not more.

By inverse relationship, we should get half the time, or 2 days.

Concept 1: Two People Working Together

Explanation of concepts through illustration

 

Illustration 2:

A does a job in 8 days and B does it in 12 days. How much time will it take to complete the job if A and B work together?
1. 24/5
2. 5/24
3. 16/3
4. 3/16

Solution

One day’s work for A and B is 1/8 and 1/12 respectively.
If they work together it will take them: (1/8 +1/12) = (3 +2)/24 = 5/24.
Hence the work will b completed in 24/5 days (inverse).

 

Illustration 3:

A is twice as good as B. It takes 14 days if they work together. When can A alone finish the job?

Solution

Since A is twice as good as B, A will take half as many days as B.
Let A take x days and B take 2x days.

Then 1/x +1/2x = 1/14
On solving, we get: 3/2x = 1/14, or x = 21.
Hence A takes 21 days.

Concept 2: Using Percentage

Illustration 4:

X can do a piece of work in 48 days and with the help of Y, he finishes the job in 16 days. Y is what % faster than X?
1. 50%
2. 75%
3. 100%
4. 200%

Solution

If they both work together, then 1/48 + 1/Y = 1/16. Hence, 1/Y = 1/16 - 1/48 =2/48.
Thus, while X takes 48 days, Y takes 24 days, which is half the time taken by X, and hence 100% faster.

Concept 3: More then 2 People Working Together

Illustration 5:

A and B working together take 12 days to do a job. B and C take 15 days while C and A take 18 days to complete the same job. In how many days can C alone complete it?
1. 360/13
2. 360/5
3. 360/17
4. 360/7

Solution

Adding up the information together, we get: 2(A + B + C) = 1/12 + 1/15 + 1/18 = 37/180. Hence A + B + C = 37/360 C = (A + B + C) – (A + B) = (37/360 – 1/12) = 7/360Hence C takes = 360/7 days.

 

Illustration 6:

A can do a piece of work in 12 days and B in 10 days but with the help of C they finished the work in 4 days. C alone can do the work in how many days?

Solution

Here all three complete the work in 4 hours:
 Macintosh HD:Users:sanjeevkumar:Desktop:Screen Shot 2013-10-08 at 6.30.46 PM.png
Macintosh HD:Users:sanjeevkumar:Desktop:Screen Shot 2013-10-08 at 6.32.46 PM.png

 

Illustration 7:

A and B can do a piece of work in 18 days. B and C in 24 days. A and C can do this work in 36 days. In what time can they do it all working together?

Solution

A and B’s one day’s work = 1/18.
B and C’s one day’s work = 1/24.
C and A’s one day’s work = 1/36.
If we add all this it will give us the work of 2A, 2B and 2C in 1 day i.e.
(1/18)+(1/24)+(1/36)=1/8.
A, B and C’s one day’s work : 1/2 * 1/8 = 1/16.
They can complete the work in 16 days.

Concept 4: When People Join/Leave After Some Days

Illustration 8:

A does a job in 40 days. B can do the same job in 50 days. A started the job and after 5 days, B joined. After 10 days of the start, C joined and the job was over in 20 days. How much time will C alone take to complete the job.
1. 40 days
2. 50 days
3. 45 days
4. 55 days

Solution

A starts the job, B works for 5 days less and C works for 10 days less.
Hence A works for 20 days, B for 15 days and C for 10 days.
We can thus get the equation: 20/40 + 15/50 + 10/C = 1, where 1/C is C’s one day’s work.
Solving this, we get C = 50 days.

  

Illustration 9:

A can do a piece of work in 30 days, which B can do in 20 days. Both started the work but A left 5 days before the completion of the work. It took how many days to complete the work?

Solution

A left the job 5 days before the completion means for the last 5 days only B worked. First calculate B’ s five days work, which he did alone.
In 5 days B will do 5× 1/20 = 1⁄4th of the work. Remaining work 1-1⁄4 = 3⁄4, which A and B have done together.
A and B can do 1/20 + 1/30 work in 1 day.
Their one-day’s work is :
Macintosh HD:Users:sanjeevkumar:Desktop:Screen Shot 2013-10-08 at 6.38.11 PM.png

Concept 5: Men, Women and Children Working Together

Illustration 10:

If 4 men or 7 boys do a work in 29 days, 12 men and 8 boys will do the same work in:
1. 7 days
2. 8 days
3. 11 days
4. 12 days

Solution

We are given that 4m = 7b; hence we can say that 12m = 21b.Now the problem becomes: if 7b can do the work in 29 days, then how many days will (12m + 8b) take? Now (12m + 8b) = (21b + 8b) = 29b.If 7b do it in 29 days, one boy does it in 29 × 7 days.Hence 29 b will do it in (29 × 7)/29 = 7 days.

Concept 6: Machines Producing Goods

Illustration 11:

If machine X can produce 1,000 bolts in 4 hours and machine Y can produce 1,000 bolts in 5 hours, in how many hours can machines X and Y, working together at these constant rates, produce 1,000 bolts?

Solution

​Both machines will take: 1/4 + 1/5 = 1/h  9/20 = 1/hWorking together, machines X and Y can produce 1,000 bolts in 20/9 or 2 2/9 hours.





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