Solved Examples: Time & Work

Table of Contents
1. Time
2. Work
3. What to focus on (examination-oriented)
4. Rules and standard formulae
5. Key concepts (concise)
6. Examples
7. Additional notes and common tips
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Time

Time is the measure of duration or interval during which work or tasks are performed. In the context of time and work problems, time quantifies how long a person or group requires to complete a defined job.

Work

Work denotes the total job or task to be completed. In problems of this topic, work is usually considered as a single whole (1 unit). Fractional parts of this whole represent partial completion of the job.

What to focus on (examination-oriented)

• Understand the fundamental relation: Work = Rate × Time.
• Learn how to express an individual's rate of work as a fraction of the job done per unit time (for example, "Raj can do a job in 10 days" means Raj's rate = 1/10 job per day).
• Solve problems with multiple workers working together or leaving/joining the work at different times.
• Deal with fractional completion of work and remaining work after certain days.
• Understand efficiency (efficiency ∝ rate) and how efficiency ratios convert to time ratios.
• Practice common problem types: combined work, relative efficiency, work done in parts, and problems where workers have different efficiencies.

Rules and standard formulae

• If M1 persons complete W1 units of work in D1 days and M2 persons complete W2 units of work in D2 days, then the standard relation between people, work and days is based on proportionality: more people or more days increase total work done. The typical formula used is derived from the idea that Work = Number of persons × Rate per person × Time.
• If individuals have productivities (rates) E1 and E2 (jobs per day), then the combined rate is the sum of individual rates. Thus combined rate = E1 + E2.
• If person A can do a whole job in n days, A's one-day work (rate) = 1/n (fraction of job done in one day).
• If person X can finish a job in D1 days and person Y can finish the same job in D2 days, then together their one-day work = 1/D1 + 1/D2. The time taken by both together = reciprocal of that sum.
• If person X is twice as efficient as person Y, then X's time = half of Y's time to finish the same job.
• If X and Y together finish a job in D1 days and X alone takes D2 days, then Y's time alone can be found by subtracting rates: Y's one-day rate = 1/D1 - 1/D2, so Y's time = reciprocal of that.

Key concepts (concise)

• Rate (one-day work): If a person does 1 job in d days, rate = 1/d.
• Combined rate: Sum of individual rates when working together.
• Time for combined work: Time = 1 / (sum of one-day works).
• Efficiency: If efficiencies are in ratio a:b, then inverse gives times ratio b:a.
• Fractional completion: After t days, work done = rate × t; remaining = 1 - work done.

Examples

Example 1: Sam can finish a task is 12 days, and Adam can finish it in 15 days. Calculate how much time together they will take to complete the same job?
(a) 20 / 3
(b) 16 / 3
(c) 9 / 2
(d) 18 / 7
Ans: (a) Sam's 1-day effort. \( \displaystyle \frac{1}{12} \)
Adam's 1-day effort. \( \displaystyle \frac{1}{15} \)
Combined 1-day effort.
\( \displaystyle \frac{1}{12} + \frac{1}{15} = \frac{3}{20} \)
Time to finish together (reciprocal of combined rate).
\( \displaystyle \frac{1}{3/20} = \frac{20}{3} \) days

Example 2: George and Victor can finish making a painting in 12 hours, Victor and Sam in 15 hours, Sam and George in 20 hours. Find in how many hours will they together finish the painting?
(a) 21
(b) 10
(c) 21/2
(d) 11
Ans: (b)
Let G, V, S denote one-hour rates of George, Victor and Sam respectively.
G + V = \( \displaystyle \frac{1}{12} \)
V + S = \( \displaystyle \frac{1}{15} \)
S + G = \( \displaystyle \frac{1}{20} \)
Add the three equations.
\( \displaystyle 2(G + V + S) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \)
Simplify the right-hand side.
\( \displaystyle 2(G+V+S) = \frac{5 + 4 + 3}{60} = \frac{12}{60} = \frac{1}{5} \)
Therefore, combined one-hour work.
\( \displaystyle G+V+S = \frac{1}{10} \)
Time required together.
\( \displaystyle \frac{1}{1/10} = 10 \) hours

Example 3: M can do a task in 25 days, and S can finish a similar job in 20 days. They work as one for five days and M goes away. Calculate the days required by S to do the work then?
(a) 12
(b) 11
(c) 17
(d) 21.5
Ans: (b)
M and S one-day work together.
\( \displaystyle \frac{1}{25} + \frac{1}{20} = \frac{4 + 5}{100} = \frac{9}{100} \)
Work done in 5 days.
\( \displaystyle 5 \times \frac{9}{100} = \frac{45}{100} = \frac{9}{20} \)
Remaining work. \( \displaystyle 1 - \frac{9}{20} = \frac{11}{20} \)
S's one-day work. \( \displaystyle \frac{1}{20} \)
Days S needs to finish remaining work.
\( \displaystyle \frac{11/20}{1/20} = 11 \) days

Example 4: Jack is three times as good as Rose. Jack can finish a task in 60 days less than Rose. Calculate the time in which they can complete the job together.
(a) 43/3
(b) 45/2
(c) 47/2
(d) 49/2
Ans: (b)
Let Rose take \(x\) days to finish the job alone.
Then Jack, being three times as good, will take \( \displaystyle \frac{x}{3} \) days.
Given Jack takes 60 days less than Rose.
\( \displaystyle \frac{x}{3} = x - 60 \)
Solve for \(x\).
\( \displaystyle x = 90 \)
Times: Rose = 90 days, Jack = 30 days.
Rose + Jack one-day work.
\( \displaystyle \frac{1}{90} + \frac{1}{30} = \frac{1 + 3}{90} = \frac{4}{90} = \frac{2}{45} \)
Time together.
\( \displaystyle \frac{1}{2/45} = \frac{45}{2} \) days

Example 5: Jack and Vinny can finish their target in 12 days. Vinny and Sid together can complete the goal in 15 days. If Jack is twice as good as Sid, calculate the number of days required by Vinny alone to finish the target?
(a) 21
(b) 22
(c) 25/2
(d) 20
Ans: (d)
Let Sid's one-day work be \(s\) (job/day).
Then Jack is twice as good, so Jack's one-day work = \(2s\).
Let Vinny's one-day work be \(v\).
From Jack + Vinny = 1/12 one-day work.
\( \displaystyle 2s + v = \frac{1}{12} \)
From Vinny + Sid = 1/15 one-day work.
\( \displaystyle v + s = \frac{1}{15} \)
Subtract the second equation from the first.
\( \displaystyle (2s+v) - (v+s) = \frac{1}{12} - \frac{1}{15} \)
Simplify left and right.
\( \displaystyle s = \frac{1}{12} - \frac{1}{15} = \frac{5 - 4}{60} = \frac{1}{60} \)
Now find \(v\).
\( \displaystyle v = \frac{1}{15} - s = \frac{1}{15} - \frac{1}{60} = \frac{4 - 1}{60} = \frac{3}{60} = \frac{1}{20} \)
Days Vinny alone needs = reciprocal of \(v\).
\( \displaystyle 20 \) days

Example 6: Max and Samantha can together finish the project in 12 days. If Samantha alone takes 30 days to complete, then calculate the days in which Max will finish the project alone?
(a) 25
(b) 20
(c) 16
(d) 19
Ans: (b)
Combined one-day work = \( \displaystyle \frac{1}{12} \)
Samantha's one-day work = \( \displaystyle \frac{1}{30} \)
Max's one-day work = combined - Samantha's.
\( \displaystyle \frac{1}{12} - \frac{1}{30} = \frac{5 - 2}{60} = \frac{3}{60} = \frac{1}{20} \)
Max alone will take 20 days.

Example 7: Mary is three times as good as Jonny. If they finish a task together in 15 days, calculate the days taken by Mary alone to complete the work.
(a) 20
(b) 25
(c) 30
(d) 17
Ans: (a)
Ratio of one-day work Mary : Jonny = 3 : 1.
Together one-day work = \( \displaystyle \frac{1}{15} \)
Mary's share of the combined one-day work = \( \displaystyle \frac{3}{3+1} \times \frac{1}{15} = \frac{3}{4} \times \frac{1}{15} = \frac{1}{20} \)
Mary alone will take 20 days.

Example 8: Max can finish writing one chapter in 2 days. While Alex can do the same chapter in 5 days, calculate the number of days taken them together to complete the chapter?
(a) 5/9
(b) 1/2
(c) 10/7
(d) 8/11
Ans: (c)
Max one-day work = \( \displaystyle \frac{1}{2} \)
Alex one-day work = \( \displaystyle \frac{1}{5} \)
Combined one-day work = \( \displaystyle \frac{1}{2} + \frac{1}{5} = \frac{5 + 2}{10} = \frac{7}{10} \)
Time together = reciprocal of combined rate.
\( \displaystyle \frac{10}{7} \) days

Example 9: Mike and Nik can finish a project in 45 days and 40 days individually. Both of them started the project together, but Mike quits subsequently by some days, and Nik finishes the outstanding in 23 days. Find in how many days Mike quits?
(a) 8/3
(b) 7/2
(c) 9
(d) 9/5
Ans: (c)
Nik's one-day work = \( \displaystyle \frac{1}{40} \)
Nik's work done alone after Mike quits = \( \displaystyle 23 \times \frac{1}{40} = \frac{23}{40} \)
Remaining work when Nik started alone = \( \displaystyle 1 - \frac{23}{40} = \frac{17}{40} \)
Combined one-day work of Mike and Nik = \( \displaystyle \frac{1}{45} + \frac{1}{40} = \frac{8 + 9}{360} = \frac{17}{360} \)
Time (days) they spent working together to do \( \displaystyle \frac{17}{40} \) of work = \( \displaystyle \frac{17/40}{17/360} = \frac{17}{40} \times \frac{360}{17} = 9 \) days

Example 10: Ross and Sam can finish an assignment in 15 days together. Ross can complete this Assignment alone in 20 days. Calculate the days taken by Sam to do the same assignment alone?
(a) 25
(b) 60
(c) 30
(d) 33
Ans: (b)
Ross + Sam one-day work = \( \displaystyle \frac{1}{15} \)
Ross's one-day work = \( \displaystyle \frac{1}{20} \)
Sam's one-day work = combined - Ross's = \( \displaystyle \frac{1}{15} - \frac{1}{20} = \frac{4 - 3}{60} = \frac{1}{60} \)
Sam alone will take 60 days.

Additional notes and common tips

• Always represent the whole job as 1. Express each person's one-day work as a fraction of 1.
• When workers join or leave, compute work done in segments: sum of rates × time for each segment, then subtract from 1 to get remaining work.
• When given efficiency ratios, convert efficiencies to rates and then to times by reciprocals.
• Check units (days, hours) and convert them consistently before adding rates.
• For three or more workers where pairwise times are given, use system of linear equations or the trick of summing pairwise equations (as in Example 2) to obtain the combined rate.

Summary (optional)

Time and work problems reduce to finding rates (one-day work), adding or subtracting rates when workers work together or separately, and taking reciprocals to convert rates into times. Practise setting up equations and performing operations on simple fractions carefully; this solves the majority of problems in this topic.