CAT Quant – Time and Work – Important Formulas and Concepts

Basic Assumptions

  • Work to be done is usually considered as a whole 1 unit.
  • If a person does some work in a certain number of days, we assume (unless explicitly stated) he does the work uniformly i.e same amount of work every day.
    • Example – A does some work in —> 15 days; He does —> (1/15)th of the work in 1 day.
  • If there is more than 1 person (workforce) carrying out the work, w assume that each person does the same amount of work everyday (unless otherwise specified). They share the work equally.
    • Example – 2 People can do a work in —> 8 days; 1 man do it in —> 16 days. Thus each person can do —> (1/16)th of work per day.

Unitary Method

  • The method used in “Time and Work” is Unitary Method i.e “time taken per unit method”.
    • Number of persons required to complete Unit Work.
    • Work completed by unit person in Unit Time
  • Note – A useful approach to look at time and work is in terms of Percentage(%) as it gives a direct comparison and a clear picture.

LCM Method

  • Another approach to solving Time and Work problem is through LCM. Different cases using LCM to solve time and work problems are explained below with the help of examples.
    • Case 1 – If A completes a work in 10 days and B completes a work in 15 days. How much time will they together take to complete the work. $$ A = 10 \ B =15 \longrightarrow LCM = 30 \\ A’s Work {\LARGE[} \frac{30}{15} {\LARGE]} = 3 \ ; \ B’s Work {\LARGE[} \frac{30}{15} {\LARGE]} = 2 \\ (A + B) \ Together = (3+5) = 5 \\ Time \ taken \ to \ complete \ the \ work \ together = {\LARGE[} \frac{30}{5} {\LARGE]} = 6 \ days$$
    • Case 2 – If A and B completes a work in 15 days and B alone completes the work in 20 days. How much time will A take to complete the work alone. $$(A+B) = 15 \ B =20 \longrightarrow LCM = 60 \\ (A+B)’s Work {\LARGE[} \frac{60}{15} {\LARGE]} = 4 \ ; \ B’s Work {\LARGE[} \frac{60}{20} {\LARGE]} = 3 \\ [(A+B) – B] = (4-3) = 1 \\ Time \ taken \ by \ A \ to \ complete \ the \ work \ alone = {\LARGE[} \frac{30}{5} {\LARGE]} = 6 \ days$$

Formulas

  • The basic formula that applies to Time and Work problems are as – $$Work \ done = Men \ (Work \ Rate) * Time \\ W = M * T$$
  • The following Conclusions can be drawn out –
    • When “Time” is constant —> W \( \infty \) M
    • When “Men” is constant —> W \( \infty \) T
    • When “Work” is constant —> M \( \infty \) 1/T
  • Man-days – The number of men multiplied by the number of days they take to complete the work is the Man-Days required to do the work.
  • Efficiency = 1/ Time Taken
  • Case 1 – If M1 men can do W1 work in D1 days working H1 hours per day and M2 men can do W2 work in D2 days working H2 hours per day (where all men work at the same rate) then, $$\frac{M_1D_1H_1}{W_1} = \frac{M_2D_2H_2}{W_2}$$
  • Case 2 – If A can do a piece of work in p days and B can do it in q days then A and B together can complete the same work in – $$\frac{1}{p} + \frac{1}{q} = \frac{pq}{p+q} \ days $$

Read concepts and formulas for: Averages and Alligations

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