Time and 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.

General Rules

(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.

Now, A’s 1 day’s work

= {(A + B + C)’s 1 day’s work} - {(B + C)’s 1 day’s work}

= (¹/₁₀ - ¹/₁₅) = ¹/₃₀

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

B’s 1 day’s work

{(A + B + C)’s 1 day’s work} - {(C + A)’s 1 day’s work}

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

C’s 1 days work

= {(A + B + C)’s 1 day’s work} - {(A + B)’s 1 day’s work}

= (¹/₁₀ – ¹/₁₂) = ¹/₆₀

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