Mastering Work and Wages for Competitive Exams

Welcome to the module on Work and Wages! This is a crucial topic in quantitative aptitude sections of competitive exams like the CAT. Understanding the relationship between the amount of work done, the time taken, and the wages earned is key to solving a variety of problems efficiently.

Core Concepts: Work, Time, and Wages

The fundamental principle is that wages are directly proportional to the amount of work done. If individuals work at the same rate, their wages will be in the same ratio as the work they complete. However, when individuals work at different rates, the distribution of wages becomes more nuanced.

Wages are distributed based on the work done by each individual.

If multiple people work together, their total wages are distributed among them according to the proportion of the total work each person contributes. This is often simplified by assuming wages are directly proportional to the work done.

Consider a scenario where A, B, and C work together to complete a task. If A completes 1/3 of the work, B completes 1/2, and C completes 1/6, and the total wage for the task is $W$ , then A's wage will be (1/3) * W, B's wage will be (1/2) * W, and C's wage will be (1/6) * W. This assumes they all worked for the same duration or that their individual contributions are measured over the entire project duration.

Work Rate and Wages

When individuals work at different rates, their wages are typically distributed in proportion to their individual work rates. The work rate is the amount of work an individual can complete in a unit of time (e.g., per day).

If A can do a piece of work in 10 days and B can do it in 15 days, and they work together for a total wage of ₹1500, in what ratio should the wages be divided?

A's rate is 1/10 work per day, and B's rate is 1/15 work per day. Their work rates are in the ratio 1/10 : 1/15, which simplifies to 3:2. Therefore, the wages should be divided in the ratio 3:2. A gets (3/5) * 1500 = ₹900, and B gets (2/5) * 1500 = ₹600.

A key insight is that if A, B, and C can complete a work in $d_A$ , $d_B$ , and $d_C$ days respectively, their daily work rates are $1/d_A$ , $1/d_B$ , and $1/d_C$ . The ratio of work done by them in any given time period will be the same as the ratio of their daily work rates. Consequently, their wages will also be distributed in this ratio.

Scenario	Basis for Wage Distribution	Formula/Ratio
Individuals work for the same duration	Proportion of total work completed	Wage_A : Wage_B = Work_A : Work_B
Individuals work at different rates (but complete the same task)	Ratio of their individual work rates	Wage_A : Wage_B = Rate_A : Rate_B = (1/Days_A) : (1/Days_B)

Work and Wages with Different Durations

When individuals work for different durations, the amount of work done is the product of their rate and the time they worked. Wages are then distributed based on this product.

Work done = Rate × Time.

If A works for $t_A$ days and B works for $t_B$ days, and their daily rates are $R_A$ and $R_B$ , the work done by A is $R_A imes t_A$ and by B is $R_B imes t_B$ . Wages are distributed in the ratio $(R_A imes t_A) : (R_B imes t_B)$ .

Let's say A can do a work in 10 days and B in 15 days. They are paid a total of ₹1200. If A works for 5 days and B works for 10 days, we first find their daily rates: $R_A = 1/10$ and $R_B = 1/15$ . The work done by A is $(1/10) imes 5 = 1/2$ . The work done by B is $(1/15) imes 10 = 2/3$ . The ratio of work done is $(1/2) : (2/3)$ , which simplifies to $3:4$ . Therefore, A gets $(3/7) imes 1200$ and B gets $(4/7) imes 1200$ .

Remember: The core principle is always about the amount of work done. Whether it's due to different rates or different durations, the wages follow the work.

Advanced Scenarios and Tips

Some problems involve men, women, and children working together, or situations where efficiency changes. Always break down the problem into individual work rates and the total work done by each person or group.

What is the most critical factor determining wage distribution in any work scenario?

The amount of work done by each individual.

Practice a variety of problems, starting from simple direct proportionality to more complex scenarios involving combined efforts and varying durations. This will build your confidence and speed.

Learning Resources

Time, Speed, Distance and Work - Concepts and Formulas(documentation)

Provides a comprehensive overview of fundamental concepts, formulas, and solved examples for Time and Work problems, including wage distribution.

Work and Wages - Quantitative Aptitude(documentation)

A dedicated section on Work and Wages with explanations and practice questions, focusing on the relationship between work, time, and earnings.

Time and Work - CAT Quantitative Aptitude(blog)

This blog post covers Time and Work, including how wages are distributed based on individual contributions and efficiency, relevant for CAT aspirants.

Work and Wages - Explanation and Examples(documentation)

Offers clear explanations and examples for Work and Wages problems, highlighting the logic behind wage distribution based on work done.

Time and Work - Practice Questions with Solutions(documentation)

A collection of practice questions on Time and Work, many of which involve wage distribution scenarios, with detailed solutions.

Understanding Work and Wages in Aptitude Tests(blog)

A forum discussion that breaks down Time and Work concepts, including Work and Wages, with insights and problem-solving strategies from experienced users.

Work and Wages - Basic Concepts(documentation)

Explains the fundamental principles of Work and Wages, focusing on how to approach problems involving multiple workers and their earnings.

CAT Quantitative Aptitude: Time and Work(video)

A video tutorial explaining Time and Work concepts, including how to solve problems related to wages and work distribution.

Work and Wages - Solved Examples(documentation)

Provides a set of solved examples for Work and Wages problems, demonstrating step-by-step solutions for various scenarios.

Time and Work - Key Concepts for CAT(blog)

This blog post covers essential concepts for Time and Work in CAT, including a section on how wages are determined based on work done.