Solving Work and Wages Problems: Finding Individual Share

This problem involves the concept of work and wages, where the wages earned by individuals working together are distributed based on the amount of work they contribute. In such cases, the wages are usually divided in the ratio of the work done by each person.

Since they work together for the same amount of time to finish the work, the work done by each person is proportional to their individual daily work rate (or efficiency). The faster a person works (i.e., the less time they take to finish the work alone), the more work they do in a given amount of time, and thus they earn a larger share of the wages.

Calculating Individual Daily Work Rates

First, let's determine the portion of work each person can complete in one day:

• A can do the work in 10 days. So, A's daily work rate is $\frac{1}{10}$ of the total work.
• B can do the work in 15 days. So, B's daily work rate is $\frac{1}{15}$ of the total work.
• C can do the work in 20 days. So, C's daily work rate is $\frac{1}{20}$ of the total work.

Determining the Ratio of Work Done

Since A, B, and C work together until the work is finished, the total work done by each person is proportional to their daily work rate. The ratio of the work done by A, B, and C is the ratio of their daily work rates:

Ratio of work done by A : B : C = (A's daily work) : (B's daily work) : (C's daily work)

Ratio = $\frac{1}{10} : \frac{1}{15} : \frac{1}{20}$

To simplify this ratio, we find the Least Common Multiple (LCM) of the denominators (10, 15, and 20). The LCM of 10, 15, and 20 is 60.

Multiply each fraction in the ratio by the LCM (60):

• For A: $\frac{1}{10} \times 60 = 6$
• For B: $\frac{1}{15} \times 60 = 4$
• For C: $\frac{1}{20} \times 60 = 3$

So, the simplified ratio of the work done by A : B : C is 6 : 4 : 3.

Calculating C's Share of Wages

The total wages received for the work is Rs. 2,600. This total amount is to be distributed among A, B, and C in the ratio of the work they have done, which is 6 : 4 : 3.

The sum of the ratio parts is $6 + 4 + 3 = 13$.

This means that the total wage of Rs. 2,600 is divided into 13 equal parts. C's share corresponds to 3 parts of this total.

C's wage = (C's ratio part / Total ratio parts) × Total wages

C's wage = $\frac{3}{13} \times 2600$

Now, we calculate the value:

C's wage = $3 \times \frac{2600}{13}$

C's wage = $3 \times 200$

C's wage = Rs. 600

Therefore, C's wage is Rs. 600.

Revision Table: Work and Wages Summary

Additional Information: Key Concepts in Work and Wages

Understanding work and wages problems involves a few key concepts:

• Work Rate: This is the amount of work a person can do in a unit of time (like a day or an hour). If a person can complete a work in 'n' days, their daily work rate is $\frac{1}{n}$.
• Combined Work Rate: If multiple people work together, their individual work rates are added up to find the combined work rate. If A's rate is $R_A$ and B's rate is $R_B$, their combined rate is $R_A + R_B$.
• Time Taken Together: If the combined work rate of a group is $R_{combined}$, the time taken by the group to complete the entire work is $\frac{1}{R_{combined}}$.
• Wage Distribution: When people work together for the same duration, wages are distributed in the ratio of their individual work rates. If they work for different durations, the wages are distributed in the ratio of (individual work rate × time worked).

In this problem, since A, B, and C worked together for the same amount of time to finish the task, the wage distribution was simply based on their daily work rate ratio.