Math Problem Statement

A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for Rs. 3200. With the help of C, they completed the work in 3 days. How much is to be paid to C?

Solution

Let’s break down the problem step by step.

Step 1: Calculate A and B's individual work rates

  • A can complete the work in 6 days, so A’s work rate is: Rate of A=16 of the work per day.\text{Rate of A} = \frac{1}{6} \text{ of the work per day}.

  • B can complete the work in 8 days, so B’s work rate is: Rate of B=18 of the work per day.\text{Rate of B} = \frac{1}{8} \text{ of the work per day}.

Step 2: Calculate the combined work rate of A, B, and C

  • A, B, and C together completed the work in 3 days. Therefore, the combined work rate of A, B, and C is: Rate of A, B, and C together=13 of the work per day.\text{Rate of A, B, and C together} = \frac{1}{3} \text{ of the work per day}.

Since A and B together can complete the work in: Rate of A and B together=16+18=424+324=724 of the work per day.\text{Rate of A and B together} = \frac{1}{6} + \frac{1}{8} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \text{ of the work per day}.

Thus, the rate of C’s work is: Rate of C=(13724).\text{Rate of C} = \left(\frac{1}{3} - \frac{7}{24}\right). Let’s calculate that: 13=824,so824724=124.\frac{1}{3} = \frac{8}{24}, \quad \text{so} \quad \frac{8}{24} - \frac{7}{24} = \frac{1}{24}. Therefore, C's work rate is 124\frac{1}{24} of the work per day.

Step 3: Determine the payment for C

  • The total payment for completing the work is Rs. 3200.
  • Since C’s work rate is 124\frac{1}{24}, C completed 124\frac{1}{24} of the work.
  • The amount C should be paid is: Payment for C=124×3200=133.33(Rs).\text{Payment for C} = \frac{1}{24} \times 3200 = 133.33 \, \text{(Rs)}.

Thus, C should be paid Rs. 133.33.


Would you like a more detailed explanation of any step? Or do you have any other questions?

Here are some related questions you might find helpful:

  1. How to calculate combined work rates of two or more people?
  2. What is the method to determine individual payments in joint work problems?
  3. How do you deal with fractions of work and rates in real-world problems?
  4. How can you solve similar work problems involving different time frames?
  5. How do you approach rate-based work problems when multiple workers are involved?

Tip: When solving work-rate problems, always ensure to express the rate of each person in the same unit (work/day or work/hour) to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Work and Time
Rates of Work
Fractional Work

Formulas

Rate of Work = 1 / Time
Combined Rate = Rate of A + Rate of B + Rate of C
Payment = Work fraction completed by C * Total Payment

Theorems

Work-rate Theorem
Work Completion Theorem

Suitable Grade Level

Grades 7-9