Step-by-Step Algebraic Solution

To solve this problem, we can set up equations based on the information provided and then use the given price range to find a unique solution.

1. Define Variables and Formulate Equations

• Let N be the original number of friends.
• Let P be the price of the CD player in Rs.

From the initial plan, the contribution per person was P/N. After two friends withdrew, the number of friends became (N-2) and the contribution became (P/N + 1).

This gives us the equation for the total price:

P = (N - 2) × (P/N + 1)

2. Simplify the Equation

Let's solve the equation to find a relationship between P and N.

P = (N - 2) × ( (P + N) / N )

P × N = (N - 2) × (P + N)

PN = PN + N² - 2P - 2N

0 = N² - 2P - 2N

2P = N² - 2N

2P = N(N - 2)

P = N(N - 2) / 2

3. Use the Price Range Constraint

We are told that the price P is an integer between 1000 and 1100.

1000 < P < 1100

Substituting our expression for P:

1000 < N(N - 2) / 2 < 1100

Multiplying by 2:

2000 < N(N - 2) < 2200

4. Find the Value of N

We are looking for an integer N where the product of N and (N-2) is between 2000 and 2200. We can estimate N by noting that N(N-2) is approximately N². Since $\sqrt{2000}$ is approximately 45, we can test values of N around 45.

• If N = 46, N(N-2) = 46 × 44 = 2024. This is in the range.
• If N = 47, N(N-2) = 47 × 45 = 2115. This is also in the range.
• If N = 48, N(N-2) = 48 × 46 = 2208. This is outside the range.

So, the original number of friends N could be 46 or 47. However, the price P must be an integer.

• If N = 46, P = 46 × 44 / 2 = 1012. This is an integer.
• If N = 47, P = 47 × 45 / 2 = 1057.5. This is not an integer.

Therefore, the original number of friends (N) must have been 46.

Conclusion

The question asks for the number of friends who actually contributed, which is N - 2.

Number who contributed = 46 - 2 = 44.