If a town of 25,000 people is growing at a rate of approximately 1% per year, the population of the...

1. Translate the problem requirements: We need to find the population after 5 years when starting with 25,000 people and growing at 1% per year. This means each year the population becomes 101% of the previous year's population.
2. Set up the compound growth pattern: Recognize that 1% annual growth means multiplying by 1.01 each year, so after 5 years we multiply the initial population by \((1.01)^5\).
3. Calculate the growth multiplier using approximation: Since answer choices are 1,000 apart, we can approximate \((1.01)^5\) without precise decimal calculations.
4. Apply the multiplier to find final population: Multiply 25,000 by our calculated growth factor and match to the closest answer choice.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking in everyday terms. We have a town with 25,000 people today. Each year, the population grows by 1 percent. This means that if we have 100 people this year, next year we'll have 101 people \((100 + 1\% \text{ of } 100 = 100 + 1 = 101)\).

So growing at 1% per year means the population becomes 101% of what it was the previous year. In decimal form, \(101\% = 1.01\).

We need to find what the population will be after 5 years of this growth pattern.

Process Skill: TRANSLATE - Converting the percentage growth language into a clear mathematical relationship

2. Set up the compound growth pattern

Here's how compound growth works with a simple example: Imagine we start with 1,000 people.

• After Year 1: \(1,000 \times 1.01 = 1,010\) people
• After Year 2: \(1,010 \times 1.01 = 1,020.1\) people
• After Year 3: \(1,020.1 \times 1.01 = 1,030.3\) people

Notice that each year we multiply by 1.01 again. So after 5 years, we multiply by 1.01 a total of 5 times.

This means: \(\text{Final Population} = \text{Starting Population} \times (1.01)^5\)

For our problem: \(\text{Final Population} = 25,000 \times (1.01)^5\)

3. Calculate the growth multiplier using approximation

Now we need to figure out what \((1.01)^5\) equals. Since our answer choices are 1,000 people apart, we don't need to be super precise - we can use smart approximation.

Let's calculate \((1.01)^5\) step by step:

• \((1.01)^1 = 1.01\)
• \((1.01)^2 = 1.01 \times 1.01 = 1.0201\)
• \((1.01)^3 = 1.0201 \times 1.01 \approx 1.0303\)
• \((1.01)^4 = 1.0303 \times 1.01 \approx 1.0406\)
• \((1.01)^5 = 1.0406 \times 1.01 \approx 1.051\)

So \((1.01)^5 \approx 1.051\), meaning the population will be about 105.1% of the original after 5 years.

Process Skill: SIMPLIFY - Using approximation when answer choices allow for it

4. Apply the multiplier to find final population

Now we multiply our starting population by our growth factor:

\(\text{Final Population} = 25,000 \times 1.051 = 26,275\) people

Looking at our answer choices:

1. 26,000
2. 27,000
3. 28,000
4. 29,000
5. 30,000

26,275 is closest to 26,000.

Final Answer

The population of the town in 5 years will be closest to 26,000 people.

Answer: A

Verification: Our calculation of 26,275 is only 275 people away from choice A (26,000), but 725 people away from choice B (27,000), confirming that A is indeed the closest answer.

Common Faltering Points

Errors while devising the approach

1. Confusing compound growth with simple growth: Students often think that 1% growth per year for 5 years means adding 5% to the original population \((25,000 + 5\% = 26,250)\). They fail to recognize that each year's growth is calculated on the new population size, not the original size. This leads them to use simple interest formula instead of compound growth.

2. Misunderstanding the growth multiplier: Students may incorrectly think that 1% growth means multiplying by 0.01 instead of 1.01. They don't realize that 1% growth means the new population is 101% of the previous year's population \((100\% + 1\% = 101\% = 1.01)\).

Errors while executing the approach

1. Calculation errors in computing \((1.01)^5\): Students often make arithmetic mistakes when calculating powers, especially when doing step-by-step multiplication. For example, they might incorrectly calculate \((1.01)^2\) as 1.02 instead of 1.0201, leading to a significant error in the final result.

2. Rounding errors accumulating through calculations: Students may round too early in their calculations (e.g., rounding 1.0201 to 1.02 in step 2), causing the final answer to drift significantly from the correct value. This premature rounding can lead them to select answer choice B (27,000) instead of A (26,000).

Errors while selecting the answer

1. Not recognizing 'closest to' requirement: Students might calculate the exact value (26,275) but then look for an answer choice that matches this precise number. When they don't find 26,275 as an option, they may incorrectly assume they made an error rather than selecting the closest available choice (26,000).