To find the number of people at the beginning of the year, we use the formula for percentage increase:

$\text{Final Number} = \text{Initial Number} \times (1 + \text{Percentage Increase})$

Given:

• Final Number = $230$
• Percentage Increase = $15\% = 0.15$

We rearrange the formula to solve for the Initial Number:

$\text{Initial Number} = \frac{\text{Final Number}}{1 + \text{Percentage Increase}}$

$\text{Initial Number} = \frac{230}{1 + 0.15} = \frac{230}{1.15} \approx 200$

Answer:

The number of people at the beginning of the year was 200.

Would you like further details or have any questions?

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Here are 5 related questions for further exploration:

1. What would the number of people be if the increase was 20% instead of 15%?
2. How do you calculate percentage decrease if the population dropped instead?
3. Can you determine the percentage increase if the population at the beginning was 180 and ended at 230?
4. What is the formula to calculate the initial value given any final value and percentage change?
5. How would the result differ if the percentage increase was compounded monthly?

Tip: Always convert percentages to decimals before using them in equations to simplify calculations.