Context: Two workers Bumrah and Siraj can complete a piece of work in 40 days if they work alternately starting with Bumrah and followed by Siraj. Bumrah takes 'p' days to complete the same work working alone while Siraj takes 'q' days to complete the same work working alone. If p and q are integers, then how many ordered pairs of (p, q) are possible?

Answer question

To solve the problem, let's break it down step by step.

### Step 1: Determine the work done by each worker
Let the total work be \(W = 1\).

- **Bumrah's work rate:** \( \text{Bumrah's rate} = \frac{1}{p} \) (Bumrah completes the work in \(p\) days)
- **Siraj's work rate:** \( \text{Siraj's rate} = \frac{1}{q} \) (Siraj completes the work in \(q\) days)

### Step 2: Work done in one cycle (two days)
Since they work alternately, one cycle consists of Bumrah working on the first day and Siraj on the second day.

- Work done by Bumrah in one day = \( \frac{1}{p} \)
- Work done by Siraj in one day = \( \frac{1}{q} \)

Total work done in one cycle (two days) is:

\[
\text{Work in one cycle} = \frac{1}{p} + \frac{1}{q}
\]

### Step 3: Determine the number of cycles needed to complete the work
They complete the entire work in 40 days, which means they work for 20 cycles since each cycle is 2 days long.

Thus, the total work done in 20 cycles is:

\[
20 \times \left(\frac{1}{p} + \frac{1}{q}\right) = 1
\]

### Step 4: Set up the equation
From the above, we get:

\[
\frac{20}{p} + \frac{20}{q} = 1
\]

Multiplying through by \(pq\):

\[
20q + 20p = pq
\]

Rearrange the equation:

\[
pq - 20p - 20q = 0
\]

Add 400 to both sides:

\[
pq - 20p - 20q + 400 = 400
\]

Factor the left-hand side:

\[
(p - 20)(q - 20) = 400
\]

### Step 5: Find integer solutions for \( (p-20)(q-20) = 400 \)
We need to find the integer factor pairs of 400. The factor pairs are:

\[
(1, 400), (2, 200), (4, 100), (5, 80), (8, 50), (10, 40), (16, 25), (20, 20)
\]

These pairs correspond to:

\[
(p-20, q-20) = (1, 400), (2, 200), (4, 100), (5, 80), (8, 50), (10, 40), (16, 25), (20, 20)
\]

### Step 6: Calculate the possible ordered pairs \( (p, q) \)
For each factor pair \((a, b)\), the corresponding \((p, q)\) is:

\[
(p, q) = (a + 20, b + 20)
\]

This gives us the ordered pairs:

\[
(21, 420), (22, 220), (24, 120), (25, 100), (28, 70), (30, 60), (36, 45), (40, 40)
\]

Each of these pairs is unique and valid since \(p\) and \(q\) are both integers.

### Step 7: Count the number of solutions
There are 8 factor pairs, so there are 8 possible ordered pairs \((p, q)\).

### Final Answer:
There are **8 ordered pairs** of \((p, q)\) that satisfy the condition.

Would you like more details or have any questions?

**Relative Questions:**

1. How would the solution change if Bumrah and Siraj worked simultaneously instead of alternately?
2. Can the number of cycles change if they work for a different number of days?
3. What happens if \(p\) and \(q\) are not integers?
4. What if the order of workers starting is reversed?
5. How does adding or subtracting from \(p\) and \(q\) affect the number of pairs?

**Tip:** When working with factor pairs, always consider all possible combinations, including reversing the factors, to ensure all potential solutions are covered.

Learn how to solve a work allocation problem involving Bumrah and Siraj, where they alternate to complete a piece of work in 40 days. Explore the mathematical breakdown, including determining work rates and calculating ordered pairs (p, q) for integer values of days. Suitable for high school students studying algebra and equations.

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