A can complete a piece of work in ‘a’ days by working ‘m’ hours per day, and B can complete the same work in ‘b’ days by working ‘n’ hours per day. If each of them reduces his working hours per day by 4 hours, each of them would take 5 days more to finish the work (m, n < 16). Which of the following is the time taken if A and B started working together such that A worked for ‘m’ hours per day and B worked for ‘n’ hours per day? (m,
n, a, b are all integers)
1. 5 days
2. 4 days
3. 2.5 days
4. 10/3 days
5. 3 1/2 days

Let's break down the problem step by step.

### Step 1: Determine Work Done by A and B
- A completes the work in \(a\) days working \(m\) hours per day.
- The total work done by A is:  
  \[ \text{Work by A} = a \times m \text{ (total hours of work)} \]

- Similarly, B completes the work in \(b\) days working \(n\) hours per day.
- The total work done by B is:  
  \[ \text{Work by B} = b \times n \text{ (total hours of work)} \]

Since both A and B are completing the same amount of work, the total work is equal:
\[
a \times m = b \times n
\]
Let’s call this total work \(W\), so:
\[
W = a \times m = b \times n
\]

### Step 2: Incorporate the Information About Reduced Hours
If A and B reduce their working hours by 4 per day, they each take 5 days more to complete the work.
- A’s new working hours per day: \(m - 4\)
- B’s new working hours per day: \(n - 4\)

So, the new time taken by A with reduced hours is \(a + 5\) days, and the new time taken by B is \(b + 5\) days. Using this:
- The work done by A with reduced hours:
  \[
  W = (a + 5) \times (m - 4)
  \]
- The work done by B with reduced hours:
  \[
  W = (b + 5) \times (n - 4)
  \]

### Step 3: Use the Information to Find \(m\) and \(n\)
We now have the equations:
\[
a \times m = (a + 5) \times (m - 4)
\]
\[
b \times n = (b + 5) \times (n - 4)
\]

### Step 4: Solve for the Time Taken When They Work Together
When A and B work together, A works for \(m\) hours per day and B works for \(n\) hours per day. Together, their combined rate of work is the sum of their individual rates:
- A's rate of work: \( \frac{1}{a \times m} \)
- B's rate of work: \( \frac{1}{b \times n} \)

Their combined rate of work:
\[
\text{Combined rate} = \frac{1}{a \times m} + \frac{1}{b \times n} = \frac{W}{a \times m} + \frac{W}{b \times n}
\]

Since \(a \times m = b \times n\), their combined work rate simplifies to:
\[
\text{Combined rate} = \frac{2}{W}
\]

Thus, the total time taken when they work together is:
\[
\text{Time taken} = \frac{W}{2}
\]

Since \(W = a \times m = b \times n\), the time taken is half the time taken by either A or B working alone.

### Step 5: Answer
Therefore, the time taken is half the time taken by either A or B. Given the choices:
- 5 days
- 4 days
- 2.5 days
- 10/3 days
- 3.5 days

The correct answer is **2.5 days**.

### Relative Questions:
1. How would the result change if A and B increased their working hours by 2 instead of reducing them by 4?
2. What if A worked faster and could complete the work in half the time compared to B?
3. How does reducing the number of hours per day affect the total work done in other real-life scenarios?
4. Could you calculate the work rate for three people instead of two, and how does this affect the total time?
5. What mathematical assumptions are made in combining the work rates of two people?

**Tip**: When solving work-rate problems, always check if you can simplify the rates before solving for the total time.

A can complete a piece of work in ‘a’ days by working ‘m’ hours per day, and B can complete the same work in ‘b’ days by working ‘n’ hours per day. If each of them reduces his working hours per day by 4 hours, each of them would take 5 days more to finish the work (m, n < 16). Which of the following is the time taken if A and B started working together such that A worked for ‘m’ hours per day and B worked for ‘n’ hours per day? (m, n, a, b are all integers).

Learn how to solve a work-rate problem where two individuals, A and B, complete a task together at different rates. This step-by-step guide uses algebraic equations to calculate the time taken when both work simultaneously. Suitable for students in Grades 10-12.

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