Two men undertook to do a job for ₹ 1400. One of them can do it alone in 7 days and the other in 8 days. With the assistance of a boy they together completed the work in 3 days. How much money did the boy get ?

Answer

To solve the problem step by step, we will first determine the work done by each person and then calculate the share of the boy.

### Step 1: Calculate the work done by A and B in one day.

- Let the total work be represented as 1 unit.
- A can complete the work in 7 days, so A's work in one day = \( \frac{1}{7} \) units.
- B can complete the work in 8 days, so B's work in one day = \( \frac{1}{8} \) units.

### Step 2: Calculate the combined work done by A and B in one day.

- Combined work of A and B in one day = \( \frac{1}{7} + \frac{1}{8} \).
- To add these fractions, we need a common denominator. The least common multiple of 7 and 8 is 56.

\[ \frac{1}{7} = \frac{8}{56} \quad \text{and} \quad \frac{1}{8} = \frac{7}{56} \]

- Therefore,

\[ \text{Combined work} = \frac{8}{56} + \frac{7}{56} = \frac{15}{56} \]

### Step 3: Calculate the total work done by A, B, and the boy in 3 days.

- In 3 days, the work done by A and B together is:

\[ 3 \times \frac{15}{56} = \frac{45}{56} \]

- Since they completed the work together with the boy, the total work done by A, B, and the boy in 3 days is 1 unit (the whole job).

### Step 4: Calculate the work done by the boy in 3 days.

- Let the work done by the boy in one day be \( x \).
- Therefore, in 3 days, the boy does \( 3x \).

- The equation for total work becomes:

\[ \frac{45}{56} + 3x = 1 \]

### Step 5: Solve for \( x \).

- Rearranging the equation gives:

\[ 3x = 1 - \frac{45}{56} = \frac{56}{56} - \frac{45}{56} = \frac{11}{56} \]

- Now, divide by 3:

\[ x = \frac{11}{56} \div 3 = \frac{11}{168} \]

### Step 6: Calculate the total work done by the boy in 3 days.

- The total work done by the boy in 3 days is:

\[ 3x = 3 \times \frac{11}{168} = \frac{33}{168} = \frac{11}{56} \]

### Step 7: Calculate the share of money for A, B, and the boy.

- Total payment for the job is ₹1400.
- The share of A and B can be calculated based on the work done:

- Work done by A in 3 days:

\[ 3 \times \frac{1}{7} = \frac{3}{7} \]

- Work done by B in 3 days:

\[ 3 \times \frac{1}{8} = \frac{3}{8} \]

- Total work done by A, B, and the boy:

\[ \frac{3}{7} + \frac{3}{8} + \frac{11}{56} \]

- To add these fractions, we need a common denominator, which is 56:

\[ \frac{3}{7} = \frac{24}{56} \quad \text{and} \quad \frac{3}{8} = \frac{21}{56} \]

- Thus, the total work done becomes:

\[ \frac{24}{56} + \frac{21}{56} + \frac{11}{56} = \frac{56}{56} = 1 \]

### Step 8: Calculate the share of the boy.

- The share of the boy based on his work:

\[ \text{Boy's share} = \frac{\text{Work done by boy}}{\text{Total work}} \times \text{Total payment} \]

\[ = \frac{\frac{11}{56}}{1} \times 1400 = \frac{11 \times 1400}{56} = 275 \]

### Final Answer:

The boy gets ₹275.