Calculating Population 2 Years Ago: A Step-by-Step Guide

This problem asks us to find the population of a city two years ago, given its present population and the percentage changes that occurred over the last two years.

Let's break down the information given:

• Increase in population in the first year = 30%
• Decrease in population in the second year = 15%
• Present population = 11,050

We want to find the population 2 years ago. Let's call this original population \(P\).

When a quantity increases by a certain percentage, say \(x%\), the new quantity is the original quantity multiplied by \( (1 + \frac{x}{100}) \). When a quantity decreases by a certain percentage, say \(y%\), the new quantity is the original quantity multiplied by \( (1 - \frac{y}{100}) \).

In this problem, the changes happen consecutively:

1. In the first year, the population \(P\) increased by 30%. The population at the end of the first year became: \(P \times (1 + \frac{30}{100}) = P \times (1 + 0.30) = P \times 1.30\)
2. In the second year, this new population (from the end of year 1) decreased by 15%. The population at the end of the second year (which is the present population) became: \((P \times 1.30) \times (1 - \frac{15}{100}) = (P \times 1.30) \times (1 - 0.15) = (P \times 1.30) \times 0.85\)

We are given that the present population is 11,050. So, we can set up the equation:

Now, let's solve for \(P\).

First, calculate the product \(1.30 \times 0.85\):

Substitute this back into the equation:

To find \(P\), divide the present population by the combined percentage factor:

Performing the division:

So, the population 2 years ago was 10,000.

Let's verify this:

• Starting population: 10,000
• After 1st year (30% increase): \(10000 + 30\% \text{ of } 10000 = 10000 + 3000 = 13000\)
• After 2nd year (15% decrease on 13000): \(13000 - 15\% \text{ of } 13000 = 13000 - (0.15 \times 13000) = 13000 - 1950 = 11050\)

The calculated present population matches the given present population (11,050), confirming our result.

The population 2 years ago was 10,000.

Revision Table: Population Change Concepts

Additional Information: Understanding Successive Percentage Changes

In problems involving successive percentage changes, it is important to remember that each subsequent percentage change is applied to the *new* value obtained after the previous change, not the original value. This is why we multiplied by \((1 + 0.30)\) first, and then multiplied the result by \((1 - 0.15)\).

A common mistake is to simply combine the percentage changes (e.g., 30% increase and 15% decrease might feel like a net 15% increase, but this is incorrect). As shown in our calculation, a 30% increase followed by a 15% decrease results in a net change factor of \(1.30 \times 0.85 = 1.105\), which is a net increase of 10.5% over the original value.

This concept is widely applicable in areas like finance (compound interest, depreciation), population dynamics, and calculating price changes.