# 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

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?

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%

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

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) = 7/360 Hence 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?

Here all three complete the work in 4 hours:

** **

**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?

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

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?

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 = 1⁄4^{th} 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 :

# 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

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?

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