Daily Wages Analysis: Work Cost Calculation
This problem involves calculating the total cost of a specific work when two individuals, A and B, undertake the task together. We are given their individual daily wages and the total amount each received for completing the work separately. The goal is to find the cost if they collaborate on the same work.
Individual Wages: Calculate Time Taken
First, we need to determine how many days each person (A and B) takes to complete the work individually. This can be found by dividing the total wage received by their respective daily wages.
| Person | Total Wage (₹) | Daily Wage (₹) | Days Taken |
|---|---|---|---|
| A | 63.00 | 3.50 | $ \frac{63}{3.50} = 18 $ |
| B | 75.00 | 2.50 | $ \frac{75}{2.50} = 30 $ |
So, A completes the work in 18 days, and B completes the same work in 30 days.
Combined Rate: Calculate Work Done Together Per Day
Now, we find out how much work A and B can do together in one day. This is their combined work rate.
- Work done by A in 1 day = $\frac{1}{18}$ of the work.
- Work done by B in 1 day = $\frac{1}{30}$ of the work.
- Combined work rate of A and B per day = (Work done by A in 1 day) + (Work done by B in 1 day)
- Combined work rate = $\frac{1}{18} + \frac{1}{30}$
Calculating Combined Rate
To add these fractions, we find a common denominator. The least common multiple (LCM) of 18 and 30 is 90.
Combined work rate = $\frac{5}{90} + \frac{3}{90} = \frac{5+3}{90} = \frac{8}{90}$
Simplifying the fraction, the combined work rate is $\frac{4}{45}$ of the work per day.
Work Together Time: Calculate Duration for Combined Effort
The time taken to complete the work together is the reciprocal of their combined work rate.
Time taken together = $\frac{1}{\text{Combined work rate}}$
Time taken together = $\frac{1}{4/45} = \frac{45}{4}$ days
Converting this to a decimal: $\frac{45}{4} = 11.25$ days.
Combined Wage: Summing Individual Wages
When working together, their combined daily earning is the sum of their individual daily wages.
Combined daily wage = Daily Wage of A + Daily Wage of B
Combined daily wage = ₹$3.50 + ₹$2.50 = ₹$6.00$
Total Work Cost: Final Calculation
The total cost of the work when A and B do it together is their combined daily wage multiplied by the number of days they take to complete the work together.
Total Cost = (Combined daily wage) $\times$ (Time taken together)
Total Cost = ₹$6.00 \times \frac{45}{4}$
Total Cost = $6 \times 11.25$
Total Cost = ₹$67.50$
Conclusion: Total Cost Calculated
Therefore, if both A and B work together to complete the task, the total cost of the work will be ₹$67.50$.
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