Understanding Work and Wages Share Calculation

In problems involving work and wages, the wages are usually distributed among individuals in proportion to the amount of work done by each person. If all individuals work for the same duration, the work done is proportional to their efficiency or daily work rate.

Calculating Individual Work Rates

First, let's find the amount of work each person can do in one day. This is their daily work rate or efficiency.

• A can do the work in 12 days.
• A's daily work rate = \( \frac{1}{\text{Number of days A takes}} = \frac{1}{12} \) of the work per day.
• B can do the work in 16 days.
• B's daily work rate = \( \frac{1}{\text{Number of days B takes}} = \frac{1}{16} \) of the work per day.

Calculating Combined Work Rate

With the help of C, A and B completed the work in 6 days. This means A, B, and C together completed the work in 6 days.

• A, B, and C together can do the work in 6 days.
• Combined daily work rate of A, B, and C = \( \frac{1}{\text{Number of days A, B, C take}} = \frac{1}{6} \) of the work per day.

Finding C's Work Rate

The combined work rate of A, B, and C is the sum of their individual daily work rates. We can find C's daily work rate by subtracting the combined work rate of A and B from the combined work rate of A, B, and C.

• Combined daily work rate of A and B = A's rate + B's rate = \( \frac{1}{12} + \frac{1}{16} \)
• To add these fractions, find a common denominator, which is 48.
• \( \frac{1}{12} = \frac{1 \times 4}{12 \times 4} = \frac{4}{48} \)
• \( \frac{1}{16} = \frac{1 \times 3}{16 \times 3} = \frac{3}{48} \)
• Combined daily work rate of A and B = \( \frac{4}{48} + \frac{3}{48} = \frac{7}{48} \) of the work per day.
• C's daily work rate = (A+B+C)'s rate - (A+B)'s rate = \( \frac{1}{6} - \frac{7}{48} \)
• To subtract, find a common denominator, which is 48.
• \( \frac{1}{6} = \frac{1 \times 8}{6 \times 8} = \frac{8}{48} \)
• C's daily work rate = \( \frac{8}{48} - \frac{7}{48} = \frac{1}{48} \) of the work per day.

Determining the Ratio of Shares

Since A, B, and C worked together for the same number of days (6 days), their shares of the wages will be proportional to their daily work rates.

• Ratio of daily work rates A : B : C = \( \frac{1}{12} : \frac{1}{16} : \frac{1}{48} \)
• To convert this ratio into whole numbers, multiply each fraction by the least common multiple (LCM) of 12, 16, and 48, which is 48.
• Ratio A : B : C = \( \left(\frac{1}{12} \times 48\right) : \left(\frac{1}{16} \times 48\right) : \left(\frac{1}{48} \times 48\right) \)
• Ratio A : B : C = \( 4 : 3 : 1 \)

The total ratio parts are \( 4 + 3 + 1 = 8 \).

Calculating A's Share

The total amount for the work is Rs. 6000. A's share is proportional to their ratio part (4) out of the total ratio parts (8).

• A's share = \( \left(\frac{\text{A's ratio part}}{\text{Total ratio parts}}\right) \times \text{Total Amount} \)
• A's share = \( \left(\frac{4}{8}\right) \times 6000 \)
• A's share = \( \frac{1}{2} \times 6000 \)
• A's share = Rs. 3000

The share of A in the total amount is Rs. 3000.