Understanding Time and Work: Formulas, Shortcuts, and Tricks for Success

Understanding Time and Work: Formulas, Shortcuts, and Tricks for Success

Understanding Time and Work: Formulas, Shortcuts, and Tricks for Success

Time and Work problems are a classic challenge in competitive exams and real-world tasks. These problems involve determining how long a task will take when multiple people or machines work together or separately. If you’re gearing up for placement exams or any competitive test, mastering this topic is essential.

In this guide, we’ll break down the key Time and Work formulas, provide shortcuts, and walk you through tricks that will help you tackle problems with ease.


What is Time and Work?

Time and work problems deal with the amount of work done by individuals or groups over a specific period. The rate of work is crucial to understanding these problems, which can be described with the formula:

Work (W)=Rate (R)×Time (T)\text{Work (W)} = \text{Rate (R)} \times \text{Time (T)}

When simplifying, the rate of work is inversely proportional to time. For instance, if R (rate) is the efficiency of work, the total time taken can be represented as T=1RT = \frac{1}{R}.


Time and Work Formulas You Should Know

Let’s dive into some essential Time and Work formulas that will help you simplify problems.

1. One Worker Completing a Task

  • If A can complete a job in n days, then A’s one day work = 1n\frac{1}{n}
  • Total work completed = No. of days × Efficiency

2. Comparing Efficiencies

  • If A is 3 times as efficient as B, the ratio of work done by A and B = 3:1
  • The time taken by A and B to complete the same task will be in the ratio 1:3.

3. Work Done by Multiple Workers

  • When multiple people work together, you can calculate their total work per day by summing their individual work rates. The formula for total time when two workers are involved is:

Total time=xyx+y(where x and y are the individual times taken by the workers)\text{Total time} = \frac{xy}{x + y} \quad \text{(where x and y are the individual times taken by the workers)}


Solving Time and Work Problems: Step-by-Step Approach

Time and Work problems typically fall into one of these categories:

  • Working simultaneously or alternately
  • Teams of different efficiency levels

The key is to approach each problem methodically. Let’s walk through an example using both the Per Day Work Method and the LCM Method.


Example: Solving Time and Work Problem with Two Workers

Problem: If A completes a task in 10 days, and B completes it in 12 days, how long will it take if they work together?

Approach 1: Per Day’s Work Method
  • A’s work in 1 day = 110\frac{1}{10}
  • B’s work in 1 day = 112\frac{1}{12}

Total work done by both in 1 day = 110+112=1160\frac{1}{10} + \frac{1}{12} = \frac{11}{60}

Total time to complete the task = 6011\frac{60}{11} days ≈ 5.45 days

Approach 2: LCM Method
  • Let’s assume the total work is 60 units (LCM of 10 and 12).
  • A’s efficiency = 6 units/day, B’s efficiency = 5 units/day.
  • Total efficiency = 6 + 5 = 11 units/day.

Time to complete 60 units of work = 6011\frac{60}{11} days ≈ 5.45 days

Tip: Both methods give the same result, but the choice of method depends on the complexity of the problem.


Understanding the Chocolate Method for Intuitive Problem Solving

The Chocolate Method is a fun and intuitive approach where work is treated as chocolates. Let’s explore this method with another example:

Example: A completes a task in 9 days and B in 18 days. How many days will they take if they work together?

Solution:
  1. Assume the total work is the LCM of 9 and 18 = 18 chocolates.
  2. A’s efficiency = 2 chocolates/day, B’s efficiency = 1 chocolate/day.
  3. Together, they eat 3 chocolates/day.

Time to finish the work = 183=6\frac{18}{3} = 6 days.


Man-Days Concept in Time and Work

In some problems, you’ll encounter the man-days concept, where you’re asked to calculate the work based on the number of workers and the time they work. The formula is:

Work (W)=Number of people (P)×Number of days (D)×Number of hours per day (H)\text{Work (W)} = \text{Number of people (P)} \times \text{Number of days (D)} \times \text{Number of hours per day (H)}

Example: Solving Man-Days Problems

Problem: 12 men can complete a task in 14 days. How many days will it take for 28 men to complete 10 tasks?

Solution:

  • Work for 12 men = 12 × 14 = 168 man-days for 1 task.
  • Work for 28 men = 16828=6\frac{168}{28} = 6 days for 1 task.
  • For 10 tasks, it will take 6 × 10 = 60 days.

Time and Work Problems Involving Wages

When wages are divided among workers based on the work done, the formula is:

Wages of A:Wages of B=Work done by A:Work done by B\text{Wages of A} : \text{Wages of B} = \text{Work done by A} : \text{Work done by B}

Let’s see how this works with an example:

Problem: A can complete a task in 10 days, and B can complete it in 15 days. If they work together and A works for 1 day, how much should B earn if the total wage is 1000?

Solution:

  • LCM of 10 and 15 = 30 (total work).
  • A’s efficiency = 3 chocolates/day, B’s efficiency = 2 chocolates/day.
  • A’s work in 1 day = 3 chocolates, remaining work = 27 chocolates.
  • Ratio of work = 3:27=1:93 : 27 = 1 : 9.

Thus, B’s share = 910×1000=900\frac{9}{10} \times 1000 = 900.


Pipes and Cisterns: Time and Work in Action

In Pipes and Cisterns problems, the focus is on filling or emptying a tank. If a pipe fills a tank, it’s called an inlet pipe; if it empties the tank, it’s called an outlet pipe.

The key formula for filling/emptying a tank is:

Net work per hour=1x−1y(where x is the time for filling and y is the time for emptying)\text{Net work per hour} = \frac{1}{x} – \frac{1}{y} \quad \text{(where x is the time for filling and y is the time for emptying)}

Example: Pipes and Cisterns Problem

Problem: A cistern has three taps: P, Q, and R. P fills it in 10 minutes, Q in 15 minutes, and R empties it in 12 minutes. If all three taps are opened, how long will it take to fill the cistern?

Solution:

  • Rate of P = 6 liters/min, Rate of Q = 4 liters/min, Rate of R = 5 liters/min.
  • Net filling rate = 6+4−5=56 + 4 – 5 = 5 liters/min.
  • Time to fill the cistern = 605=12\frac{60}{5} = 12 minutes.

Conclusion

Time and Work problems are all about understanding the relationship between work, time, and efficiency. With the right formulas, shortcuts, and a structured approach, you can solve these problems efficiently.

Accenture Numerical Ability Questions and Answers
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