Important Formulas Time and Work - Quantitative Reasoning for GMAT PDF

Table of Contents
1. Time and Work Important Concepts
2. Time and Work Important Formulas
3. Time and Work Relationships to Remember
4. Time and Work Tricks
5. Worked Examples
6. How to Set Up Equations - Practical Advice
7. Summary
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Time and Work Important Concepts

Understanding time and work begins with two simple ideas: each worker performs a certain fraction of the whole job per unit time, and the total job is completed when the sum of these fractions reaches 1. Commonly used terms and definitions are listed below.

• Work (or total job) is the complete task to be done; it is customary to take the total work = 1 unit unless otherwise specified.
• Rate or efficiency of a person is the fraction of the work that the person completes in one unit of time (usually one day or one hour).
• 1 day's work of a person who completes the job in n days = 1/n.
• Man-hours (or person-hours) is a measure of total effort: if M men work for T hours (or days), total effort = M × T man-hours (or man-days).
• If a worker's 1 day's work = 1/D, then that worker will finish the job in D days.
• The fraction of work done by a worker in t days = t/D when the worker's full job time is D days.
• Basic proportional insights:
  • More men ⇒ more work can be done in the same time.
  • More men ⇒ same work can be finished in less time.
  • More work ⇒ more time required (for same number of workers).

Time and Work Important Formulas

These formulas are the standard tools for solving time-and-work problems.

• One worker's daily rate: If A can finish the work in n days, then A's 1 day's work = 1/n.
• Time from rate: If A's 1 day's work = 1/n, then A's total time to finish = n days.
• Combined work (two workers): If A does job in x days and B in y days, then A's daily work = 1/x, B's daily work = 1/y, and together their daily work = 1/x + 1/y. Time taken together = xy / (x + y).
• Combined work (three workers): If A, B, C take x, y, z days respectively, then time together = xyz / (xy + yz + zx).
• One worker from combined time: If A alone takes x days and A and B together take y days, then B alone takes xy / (x - y) days (provided x > y).
• General rate relation (different jobs or different working hours): If M1 men finish W1 work in D1 days and M2 men finish W2 work in D2 days, then M1 × D1 / W1 = M2 × D2 / W2.
• Including different working hours per day: If M1 men work T1 hours per day and M2 men work T2 hours per day, then M1 × D1 × T1 / W1 = M2 × D2 × T2 / W2.
• Work remaining / completed: Work done in t days by A = (1/x) × t when A's full time is x days. Remaining work = 1 - work done.

Time and Work Relationships to Remember

These relationships are useful as quick checks or mental shortcuts when setting up equations.

• Men and Work: More men ⇒ can do more work in the same time. Fewer men ⇒ can do less work in the same time.
• Work and Time: More work ⇒ takes more time (for a fixed number of workers). Less work ⇒ takes less time.
• Men and Time: More men ⇒ can complete a given work in less time. Fewer men ⇒ take more time for the same work.
• Man-days perspective: Total work = (number of workers) × (days worked) × (per-worker daily rate). Often useful to convert all scenarios to man-days to compare.

Time and Work Tricks

These tricks are standard shortcuts and formulae often used in competitive problems. Variable letters are as follows unless noted: capital letters X, Y, Z or lowercase x, y, z stand for the number of days a single worker or machine takes to complete the job; M, M1, M2 are numbers of men; D, D1, D2 are days; T1, T2 are hours per day; W, W1, W2 are quantities of work.

• Trick 1: If M1 men finish W1 work in D1 days and M2 men finish W2 work in D2 days, then M1 × D1 / W1 = M2 × D2 / W2.
• Trick 2: If M1 men finish W1 work in D1 days working T1 hours each day and M2 men finish W2 work in D2 days working T2 hours each day, then M1 × D1 × T1 / W1 = M2 × D2 × T2 / W2.
• Trick 3 (two workers): If A completes a job in x days and B in y days, then their combined time = xy / (x + y).
• Trick 4 (three workers): If A, B, C take x, y, z days respectively, their combined time = xyz / (xy + yz + zx).
• Trick 5 (find B from A & together): If A alone takes x days and A+B together take y days, then B alone takes xy / (x - y) days (x > y).
• Trick 6 (basic conversions): One day's work = Total work / Total number of working days. Total work = (one day's work) × (total days). Remaining work = 1 - work done so far.
• Trick 7 (one worker leaves early - B leaves m days before completion): If A takes X days and B takes Y days and B leaves m days before completion, total time taken = X(Y + m) / (X + Y).
• Trick 8 (A leaves m days before completion): If A leaves m days before completion, total time taken = Y(X + m) / (X + Y).
• Trick 9 (food/consumption analogy): If food is available for a men for a days and after b days B men join, then remaining food will last total men for A(a - b) / (A + B) days.
• Trick 10 (finish remaining work after working together): If A and B together finish a job in a days, they worked together for b days and then one of them finished the rest alone in d days, then the time taken by that worker alone to finish the whole job = ad / (a - b). (Derivation given below in worked examples.)

Worked Examples

Example 1: A can do a piece of work in 12 days and B can do the same work in 18 days. How long will they take working together?

Sol.

Daily rate of A = 1/12.

Daily rate of B = 1/18.

Combined daily rate = 1/12 + 1/18.

Combine the fractions: 1/12 + 1/18 = (3 + 2)/36 = 5/36.

Time to complete together = 1 / (combined rate) = 36/5 days.

Ans. 7.2 days (i.e., 36/5 days).

Example 2: A completes a job in 10 days and B completes it in 15 days. They start working together but B leaves m = 2 days before completion. Find the total time taken to finish the job.

Sol.

Let total time be t days.

Daily rate of A = 1/10.

Daily rate of B = 1/15.

B works for (t - 2) days and A works for t days.

Total work done = t × (1/10) + (t - 2) × (1/15) = 1.

Multiply through by 30: 3t + 2(t - 2) = 30.

3t + 2t - 4 = 30.

5t = 34.

t = 34/5 = 6.8 days.

Ans. 6.8 days (i.e., 34/5 days).

How to Set Up Equations - Practical Advice

• Always take the total work as 1 unless the problem gives a concrete quantity of work.
• Write each worker's daily rate clearly (1/x if worker finishes alone in x days).
• When workers work for different numbers of days, multiply rate by the exact number of days they worked.
• When workers work different hours per day, incorporate hours per day into the effective daily rate (rate × hours).
• When people leave or join, account separately for the days each person worked and sum their contributions to equal 1.
• For problems about machines, pipes, or food consumption, map the quantities to the same rate/time framework: machines → rate, pipes → filling/draining rates, food → consumption per person per day.

Summary

Time-and-work problems reduce to calculating and summing rates. Key formulas to remember are 1/x for a single worker's daily rate, combined time for two workers xy/(x + y), and the general approach of converting scenarios into contributions that add to 1 (the whole job). Use man-days or man-hours when comparing different workforce sizes or working hours, and apply the trick formulae above to handle joining/leaving and differing daily hours. Practise the standard patterns to gain speed and accuracy.