On January 1st 2020 the populations of Japan and the Philippines were 126.5 million and 109.6 million respectively. Japan’s population is currently decreasing by 0.2% per year while the population of the Philippines is increasing by 1.6% per year. If these rates of change remain constant, during which year will the two populations be the same?

We are given the initial populations of Japan and the Philippines as of January 1, 2020, and their respective growth rates. We need to find when their populations will be the same, assuming the rates remain constant.

### Step 1: Write the population equations

Let \( P_J(t) \) be the population of Japan after \( t \) years, and \( P_P(t) \) be the population of the Philippines after \( t \) years.

For Japan, the population is decreasing by 0.2% per year, so:
\[
P_J(t) = 126.5 \times (1 - 0.002)^t
\]
For the Philippines, the population is increasing by 1.6% per year, so:
\[
P_P(t) = 109.6 \times (1 + 0.016)^t
\]

### Step 2: Set the two populations equal

To find when the populations will be the same, set \( P_J(t) = P_P(t) \):
\[
126.5 \times (1 - 0.002)^t = 109.6 \times (1 + 0.016)^t
\]

### Step 3: Solve for \( t \)

We will solve this equation numerically. Let's do that now.The populations of Japan and the Philippines will be approximately the same in about 8 years from January 1, 2020. This places the year at:

\[
2020 + 8 = 2028
\]

So, the populations of Japan and the Philippines will be the same in the year **2028** if the current rates of change remain constant.

Would you like further details, or do you have any questions?

Here are 5 related questions for further exploration:
1. How long after 2028 will the population of the Philippines surpass Japan's by 10 million?
2. What would happen if Japan's population decline rate changes to 0.5% per year?
3. How will the ratio of the populations change in the year 2035?
4. What is the population growth of the Philippines in absolute numbers by 2028?
5. How would the populations compare if both growth rates change by 50%?

**Tip:** When working with exponential growth or decay, small percentage differences can lead to significant changes over long time periods.

Find out when the populations of Japan and the Philippines will be the same using exponential growth and decay formulas. This problem involves solving for the year when Japan's declining population intersects with the Philippines' increasing population, covering key algebraic concepts and exponential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation