Total wages for a work is Rs.1280.A alone can do a piece of work in 8 days, while B alone can do it in 12 days. If A and B work on alternate days, then find the share of A.

Options

A. Rs.800
B. Rs. 500
C. Rs. 600
D. Rs. 700

Answer

To solve the problem step by step, we need to determine how much work A and B can do together when they work on alternate days, and then calculate A's share of the total wages based on the amount of work done.

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 8 days, so A's work in one day = 1/8 of the work.
- B can complete the work in 12 days, so B's work in one day = 1/12 of the work.

**Hint:** To find the work done in one day, use the formula: Work done = Total work / Number of days.

### Step 2: Calculate the total work done by A and B in two days.
- In the first day, A works: 1/8 of the work.
- In the second day, B works: 1/12 of the work.
- Total work done in two days = (1/8) + (1/12).

To add these fractions, find a common denominator:
- The LCM of 8 and 12 is 24.
- Convert the fractions:
- 1/8 = 3/24
- 1/12 = 2/24
- Total work in two days = 3/24 + 2/24 = 5/24.

**Hint:** When adding fractions, always find a common denominator.

### Step 3: Determine how many complete cycles of two days are needed to finish the work.
- The total work is 1 (whole work).
- Work done in 2 days = 5/24.
- To find how many such cycles are needed to complete the work:
- Let x be the number of cycles:
- x * (5/24) = 1
- x = 24/5 = 4.8 cycles.

Since A and B work on alternate days, we can complete 4 full cycles (8 days) and then check how much work is left.

**Hint:** To find how many cycles are needed, set up the equation: Total work = Number of cycles * Work done in each cycle.

### Step 4: Calculate the work done in 8 days.
- In 8 days (4 cycles), work done = 4 * (5/24) = 20/24 = 5/6 of the work.
- Remaining work = 1 - 5/6 = 1/6.

**Hint:** To find the remaining work, subtract the work completed from the total work.

### Step 5: Determine who works on the 9th day.
- On the 9th day, it is A's turn to work.
- A can do 1/8 of the work in one day.

**Hint:** Alternate the workers based on the number of days worked.

### Step 6: Calculate how much work A can do on the 9th day.
- Work done by A on the 9th day = 1/8.
- Since the remaining work is 1/6, A can only do 1/6 of the work on the 9th day.

**Hint:** Compare the work A can do in one day with the remaining work to see how much is actually done.

### Step 7: Calculate the total work done by A.
- A worked for 5 days in the first 8 days (4 cycles) and did 1/6 of the work on the 9th day.
- Total work done by A = (5/8) + (1/6).

To add these fractions, find a common denominator:
- The LCM of 8 and 6 is 24.
- Convert the fractions:
- 5/8 = 15/24
- 1/6 = 4/24
- Total work done by A = 15/24 + 4/24 = 19/24.

**Hint:** Always find a common denominator when adding fractions.

### Step 8: Calculate A's share of the total wages.
- Total wages = Rs. 1280.
- A's share = (Work done by A / Total work) * Total wages.
- A's share = (19/24) * 1280.

Calculating A's share:
- A's share = (19 * 1280) / 24 = 19 * 53.33 = Rs. 1013.33 (approximately).

**Hint:** To find a person's share, multiply their work fraction by the total wages.

### Final Answer:
A's share of the wages is approximately Rs. 1013.33.