Let's solve the problem step by step.
Let the original number of well-wishers be \( n \).
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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: \[ \text{Contribution per person} = \frac{4500}{n} \]
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New Number of Well-Wishers: When two additional well-wishers joined, the total number of well-wishers becomes \( n + 2 \).
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Contribution per Person After New Members: Now, the contribution per person becomes: \[ \text{New contribution per person} = \frac{4500}{n + 2} \]
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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: \[ \frac{4500}{n + 2} = \frac{4500}{n} - 75 \]
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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: \[ 75n^2 + 150n - 9000 = 0 \]
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Dividing the Entire Equation by 75: \[ n^2 + 2n - 120 = 0 \]
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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: \[ n = -12 \quad \text{(not valid)} \quad \text{or} \quad n = 10 \]
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Conclusion: The original number of well-wishers is \( n = 10 \).
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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.