Can do a work in 12 day and A and B working together can do the same work in 8 days in what time can b alone do the work?

In time and work we will learn to calculate and find the time required to complete a piece of work and also find work done in a given period of time. We know the amount of work done by a person varies directly with the time taken by him to complete the work. (i) Suppose A can finish a piece of work in 8 days. Then, work done by A in 1 day = ¹/₈ [by unitary method]. (ii) Suppose that the work done by A in 1 day is ¹/₆ Then, time taken by A to finish the whole work = 6 days.

(i) Suppose if a person A can finish a work in n days. Then, work done by A in 1 day = 1/nᵗʰ part of the work.

(ii) Suppose that the work done by A in 1 day is \(\frac{1}{n}\)

Then, time taken by A to finish the whole work = n days. 

Problems on Time and Work :

1. Aaron alone can finish a piece of work in 12 days and Brandon alone can do it in 15 days. If both of them work at it together, how much time will they take to finish it?

Solution: Time taken by Aaron to finish the work = 12 days. Work done by Aaron in 1 day = ¹/₁₂

Time taken by Brandon to finish the work = 15 days. Work done by Brandon in 1 day = ¹/₁₅ Work done by (Aaron + Brandon) in 1 day = ¹/₁₂ + ¹/₁₅ = ⁹/₆₀ = ³/₂₀

Time taken by (Aaron + Brandon) to finish the work = \(\frac{20}{6}\) days, i.e., 6²/₃ days. 

Hence both can finish the work in 6²/₃ days.

2. A and B together can do a piece of work in 15 days, while B alone can finish it 20 days. In how many days can A alone finish the work?

Solution: Time taken by (A + B) to finish the work = 15 days. Time taken by B alone to finish the work 20 days. (A + B)’s 1 day’s work = ¹/₁₅ and B’s 1 day’s work = ¹/₂₀ A’s 1 day’s work = {(A + B)’s 1 day’s work} - {B’s 1 day’s work} = (¹/₁₅ - ¹/₂₀) = (4 - 3)/60 = ¹/₆₀

Therefore, A alone can finish the work in 60 days.

3. A can do a piece of work in 25 days and B can finish it in 20 days. They work together for 5 days and then A leaves. In how many days will B finish the remaining work?

Solution: Time taken by A to finish the work = 25 days. A’s 1 day’s work = ¹/₂₅ Time taken by B to finish the work = 20 days. B’s 1 day’s work = ¹/₂₀ (A + B)’s 1 day’s work = (¹/₂₅ + ¹/₂₀) = ⁹/₁₀₀ (A + B)’s 5 day’s work (5 × ⁹/₁₀₀) = 4̶5̶/1̶0̶0̶ = ⁹/₂₀ Remaining work (1 - ⁹/₂₀) = ¹¹/₂₀ Now, ¹¹/₂₀ work is done by B in 1 day Therefore, ¹¹/₂₀ work will be done by B in (11/2̶0̶ × 2̶0̶) days = 11 days.

Hence, the remaining work is done by B in 11 days.

4. A and B can do a piece of work in 18 days; B and C can do it in 24 days while C and A can finish it in 36 days. If A, B, C works together, in how many days will they finish the work?

Solution: Time taken by (A + B) to finish the work = 18 days. (A + B)’s 1 day’s work = ¹/₁₈ Time taken by (B + C) to finish the work = 24 days. (B + C)’s 1 day’s work = ¹/₂₄ Time taken by (C + A) to finish the work = 36 days. (C + A)’s 1 day’s work = ¹/₃₆

Therefore, 2(A + B + C)’s 1 day’s work = (¹/₁₈ + ¹/₂₄ + ¹/₃₆) = (4 + 3 + 2)/72 = \(\frac{9}{72}\) = ¹/₈

⇒ (A + B + C)’s 1 day’s work = (¹/₂ × ¹/₈) = ¹/₁₆

Therefore, A, B, C together can finish the work in 16 days.

5. A and B can do a piece of work in 12 days; B and C can do it in 15 days while C and A can finish it in 20 days. If A, B, C works together, in how many days will they finish the work? In how many days will each one of them finish it, working alone?

Solution: Time taken by (A + B) to finish the work = 12 days. (A + B)’s 1 day’s work = ¹/₁₂ Time taken by (B +C) to finish the work = 15 days. (B + C)’s 1 day’s work = ¹/₁₅ Time taken by (C + A) to finish the work = 20 days. (C + A)’s 1 day’s work = ¹/₂₀

Therefore, 2(A + B + C)’s 1 day’s work = (¹/₁₂ + ¹/₁₅ + ¹/₂₀) = \(\frac{12}{60}\) = ¹/₅

⇒ (A + B + C)’s 1 day’s work = (¹/₂ × ¹/₅) = ¹/₁₀

Therefore, A, B, C together can finish the work in 10 days.

Hence, A alone can finish the work in 30 days.

(¹/₁₀ – ¹/₂₀) = ¹/₂₀ Hence, B alone can finish the work in 20 days.

Hence, C alone can finish the work in 60 days.

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• The basic formula for solving is: 1/r + 1/s = 1/h
• Let us take a case, say a person Hrithik
• Let us say that in 1 day Hrithik will do 1/20th of the work and 1 day Dhoni will do 1/30th of the work. Now if they are working together they will be doing 1/20 + 1/30 = 5/60 = 1/12th of the work in 1 day. Now try to analyze, if two persons are doing 1/12th of the work on first day, they will do 1/12th of the work on second day, 1/12th of the work on third day and so on. Now adding all that when they would have worked for 12 days 12/12 = 1 i.e. the whole work would have been over. Thus the concept works in direct as well as in reverse condition.
• The conclusion of the concept is if a person does a work in ‘r’ days, then in 1 day- 1/rth of the work is done and if 1/sth of the work is done in 1 day, then the work will be finished in ‘s’ days. Thus working together both can finish 1/h (1/r + 1/s = 1/h) work in 1 day & this complete the task in ’h’ hours.
• The same can also be interpreted in another manner i.e. If one person does a piece of work in x days and another person does it in y days. Then together they can finish that work in xy/(x+y) days
• In case of three persons taking x, y and z days respectively, They can finish the work together in xyz/(xy + yz + xz) days

Example 1: If Aarti and Rita can do a job in 8 hours (working together at their respective constant rates) and Aarti can do the job alone in 12 hours. In how many hours can Rita do the job alone?

Sol: Let Rita does the work in R days. Using basic work formula the equation would be 1/12 + 1/R = 1/8 ⇒ 8R + 96 = 12R ⇒ 96 = 4R

⇒ 24 = R Working alone, Rita can do the job in 24 hours.

Or
Besides that one more approach can be applied in the work and time questions i.e. the unit approach. Time and work short tricks can be applied in this case, as the numbers used are 8 hours & 12 hours, let the work be equal to 24 units (which is the LCM of 8 & 12). Now as they finish the work in 8 hours working together, that implies together they do 24/8 = 3 units an hour. Working alone Aarti does this work in 12 hours, so Aarti alone does 24/12 = 2 units an hour. That means Rita will be doing 3 – 2 = 1 unit per hour. The total work is 24 units, which Rita can finish the work of her own in 24/1 = 24 hours.

Example 2: A can do a piece of work in 60 days, which B can do in 40 days. Both started the work but A left 10 days before the completion of the work. The work was finished in how many days?

Sol: A left the job 10 days before the completion. So, B worked alone for the last 10 days. First, we will calculate B’s 10 days work, which he did alone. In 10 days B will do 10 × 1/40 = 1/4th of the work. Remaining work 1 - ¼ = ¾ (Which A and B have done together). A and B can do 1/60 + 1/40 work in 1 day. Their one-day’s work is 1/60 + 1/40 = (2 + 3)/120 = 5/120 = 1/24. They can finish the work in 24 days. They would have done three-fourth of the work in 24 × 3/4 = 18 days.

⇒ Total days = 18 + 10 = 28.

Or
As discussed earlier in time work questions, time and work tricks like the unit approach can also be applied. In this case, as the numbers used are 60 & 40, let the work be equal to 120 units. That implies A does 120/60 = 2 units a day, whereas B alone does 120/40 = 3 units a day. That means working alone B would have done 3 × 10 = 30 units. The remaining 120 – 30 = 90 units of work has been done by them together. They do 2 + 3 = 5 units a day working together, thus they would have finished 90 units in 90/5 = 18 days. Hence the total work was finished in 18 + 10 = 28 days.

Example 3: A can do a piece of work in 24 days and B in 20 days but with the help of C they finished the work in 8 days. C alone can do the work in how many days?

Sol: Using work formula here (1/A) + (1/B) + (1/C) = (1/8) (1/C) = (1/8) - (1/A) - (1/B) ⇒ (1/C) = (1/8) - (1/24) - (1/20) ⇒ (1/C) = (1/30)

C can do this work in 30 days.

Or
You can take the total work to be equal to 120 units (the LCM of 24, 20 & 8). That implies A does 120/24 = 5 units a day, B does 120/20 = 6 units a day. Together they finished the work in 8 days means they are doing 120/8 = 15 units a day. Let the units done by C per day be = c. Now as per the statement 5 + 6 + c = 15 ⇒ c = 4 units. Now if C does 4 units a day, he can finish the work in 120/4 = 30 days.

Example 4: If machine X can produce 1,000 bolts in 8 hours and machine Y can produce 1,000 bolts in 24 hours. In how many hours can machines X and Y, working together at these constant rates, produce 1,000 bolts?

Sol: Using formula for work: 1/8 + 1/24 = 1/h ⇒ 4/24 = 1/6. Working together, machines X and Y can produce 1,000 bolts in 6 hours.

Example 5: A and B can do a piece of work in 36 days, B and C in 48 days, A and C can do this work in 72 days. In what time can they do it all working together?

Sol: A and B’s one day’s work = 1/36. B and C’s one day’s work = 1/48. C and A’s one day’s work = 1/72. If we add all this it will give us the work of 2A, 2B and 2C in 1 day i.e. (1/36) + (1/48) + (1/72) + (1/16) That also implies that A, B and C’s one day’s work will be half of this i.e. (1/2) x (1/16) = (1/32)

From here it can found that they will complete the work in 32 days.

Example 6:  A can do as much work in 6 days as C in 10 days. B can do as much work in 6 days as C can do in 4 days. What time would B require to do a work if A takes 48 days to finish it?

Sol. A : C :: 6 : 10 or (A/C) = (3/5) and B : C :: 6 : 4 or (B/C) = (3/2), (B/A) =(B/C) x (C/A) = (3/2) x (5/3) = (5/2)
Therefore, B = (5/2) x A ⇒ (5/2) x 48 = 120 days.

Example 7:  A can do a piece of work in 48 days and B in 72 days but with the help of C they finished the work in 24 days. Out of the total payment of Rs. 3000, how much should be given to C?

Sol: The payment made to anybody is in the proportion of the work done and not in the ratio of days spent. Using work and time formula in 24 days working alone A & B would have done 24/48 = 1/2 and 24/72 = 1/3 of the work. That means they together did 1/2 + 1/3 = 5/6th of the work. Remaining 1/6th of the work must be done by C, the only person present. Now as he did 1/6th of the work, he should be paid 1/6th of the money i.e. 3000 × 1/6 = Rs. 500.

In this article we learned, how to solve the time and work questions by applying the basic time and work formula and by using the unit’s approach. Here using the unit’s approach, you make your calculations simple and you can solve the question without writing much. You can make it a point to use this approach in time work problems.