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.