A works twice as fast as B, whereas A and B together can work three times as fast as C. They all work together on a job and get a sum of Rs 17,136 for the completion of the job. The share of A is:
Rs. 8,568
Understanding the Work and Wages Problem
This question involves calculating the share of an individual (A) from a total sum received for completing a job, based on the work efficiency of different individuals (A, B, and C).
In work and wages problems, the amount of wages received by each person is directly proportional to the amount of work done by them. When they work for the same duration, the work done is proportional to their efficiency or work rate.
The key information given is:
- A works twice as fast as B.
- A and B together work three times as fast as C.
- Total earnings for the job are Rs 17,136.
Our goal is to find the ratio of their work rates (efficiency) because the wages will be distributed in that same ratio.
Calculating the Efficiency Ratios
Let's denote the work rate (efficiency) of A, B, and C as $r_A$, $r_B$, and $r_C$ respectively.
From the first statement:
A works twice as fast as B. This means $r_A = 2 \times r_B$.
From the second statement:
A and B together work three times as fast as C. This means the combined rate of A and B is three times the rate of C.
So, $r_A + r_B = 3 \times r_C$.
Now, we can substitute the first equation into the second equation to relate the rates of B and C:
$(2 \times r_B) + r_B = 3 \times r_C$
$3 \times r_B = 3 \times r_C$
Dividing both sides by 3, we get:
$r_B = r_C$
So, the efficiency of B is equal to the efficiency of C.
Now we can find the ratio of their efficiencies $r_A : r_B : r_C$.
- We know $r_A = 2 \times r_B$.
- We know $r_B = r_C$.
If we consider $r_B$ as 1 unit, then:
- $r_B = 1$
- $r_C = r_B = 1$
- $r_A = 2 \times r_B = 2 \times 1 = 2$
Therefore, the ratio of their work rates is $r_A : r_B : r_C = 2 : 1 : 1$.
Distributing the Wages Based on Efficiency Ratio
Since the wages are distributed in proportion to the work done, and they work together for the same time, the wages will be distributed in the ratio of their efficiencies.
Ratio of shares of A, B, and C = $r_A : r_B : r_C = 2 : 1 : 1$.
The total ratio parts are $2 + 1 + 1 = 4$.
The total amount of wages received is Rs 17,136.
A's share is $\frac{\text{A's ratio part}}{\text{Total ratio parts}} \times \text{Total Wages}$.
A's share = $\frac{2}{4} \times 17136$
A's share = $\frac{1}{2} \times 17136$
A's share = $8568$
So, the share of A is Rs 8,568.
Summary of Share Calculation
Here is a breakdown of how the total wages are distributed:
| Person | Efficiency Ratio | Fraction of Total Wage | Calculated Share |
|---|---|---|---|
| A | 2 | $\frac{2}{4}$ | $\frac{2}{4} \times 17136 = 8568$ |
| B | 1 | $\frac{1}{4}$ | $\frac{1}{4} \times 17136 = 4284$ |
| C | 1 | $\frac{1}{4}$ | $\frac{1}{4} \times 17136 = 4284$ |
Total Share = Rs 8568 (A) + Rs 4284 (B) + Rs 4284 (C) = Rs 17136.
This confirms that the shares are calculated correctly based on the efficiency ratio.
Revision Table: Work and Wages Concepts
| Concept | Explanation | Relation to Problem |
|---|---|---|
| Work Rate (Efficiency) | The amount of work done by a person or team in a unit of time. | Used to establish the ratio A:B:C. |
| Total Work | The entire job or task to be completed. | Implicitly represented by the total wage sum. |
| Wage Distribution | Wages are divided among workers in proportion to the work they complete. | The basis for sharing the Rs 17,136. |
| Ratio and Proportion | Mathematical tools used to compare quantities and distribute values proportionally. | Applied to determine individual shares from the total wage based on efficiency ratios. |
Additional Information: Work and Time Relation
While this problem focuses on wage distribution based on efficiency when working together for the same time, it's useful to remember the relationship between work, rate, and time:
Work = Rate $\times$ Time
If the work is constant (completing the same job), then Rate is inversely proportional to Time. A faster worker (higher rate) takes less time to complete the same work alone.
In this problem, because A, B, and C work together on the same job for the same duration, their individual contributions to the total work are directly proportional to their rates (efficiencies).
- If A has rate $r_A$, B has $r_B$, C has $r_C$, and they work for time $T$.
- Work done by A = $r_A \times T$
- Work done by B = $r_B \times T$
- Work done by C = $r_C \times T$
- Ratio of work done = $(r_A \times T) : (r_B \times T) : (r_C \times T) = r_A : r_B : r_C$.
This confirms why the wage distribution is based directly on the ratio of their rates (efficiencies) when they work together on the same task for the same amount of time.
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