Question

A group of well wishers intended to contribute kshs. 4500 for purchase of school uniform for a needy student. Before the actual payment was made, two more well wishers joined the group and as a result each person contributed ksh. 75 less

Answer 1

Let's solve the problem step by step.

Let the original number of well-wishers be \( n \).

1. **Original Contribution per Person:**
If the total amount contributed is Kshs. 4500, then the contribution per person before the two new well-wishers joined is:
\[
\text{Contribution per person} = \frac{4500}{n}
\]

2. **New Number of Well-Wishers:**
When two additional well-wishers joined, the total number of well-wishers becomes \( n + 2 \).

3. **Contribution per Person After New Members:**
Now, the contribution per person becomes:
\[
\text{New contribution per person} = \frac{4500}{n + 2}
\]

4. **Relationship Between Contributions:**
According to the problem, the new contribution per person is Kshs. 75 less than the original one. Therefore, we can write the equation:
\[
\frac{4500}{n + 2} = \frac{4500}{n} - 75
\]

5. **Cross-Multiplying to Solve the Equation:**
Let's cross-multiply to eliminate the fractions:
\[
4500n = 4500(n + 2) - 75n(n + 2)
\]

Simplifying further:
\[
4500n = 4500n + 9000 - 75n^2 - 150n
\]
Subtract \( 4500n \) from both sides:
\[
0 = 9000 - 75n^2 - 150n
\]

Rearranging gives:
\[
75n^2 + 150n - 9000 = 0
\]

6. **Dividing the Entire Equation by 75:**
\[
n^2 + 2n - 120 = 0
\]

7. **Factoring the Quadratic Equation:**
The quadratic can be factored as:
\[
(n + 12)(n - 10) = 0
\]
So, the solutions are:
\[
n + 12 = 0 \quad \text{or} \quad n - 10 = 0
\]
This gives:
\[
n = -12 \quad \text{(not valid)} \quad \text{or} \quad n = 10
\]

8. **Conclusion:**
The original number of well-wishers is \( n = 10 \).

9. **Verify Contributions:**
- Original contribution per person:
\[
\frac{4500}{10} = 450 \, \text{Kshs}
\]
- After two more join (total of 12):
\[
\frac{4500}{12} = 375 \, \text{Kshs}
\]
- Difference:
\[
450 - 375 = 75 \, \text{Kshs} \, \text{(which checks out)}
\]

Thus, the solution is consistent, and the original number of well-wishers was 10.