Constant Percent Rate and Growth Factors

Lesson 1 of 2: Definition and Conversion

In this lesson:

• Define constant percent rate and connect it to exponential functions
• Explain why percent rate produces accelerating growth
• Convert percent rates to growth factors

What You Will Learn Today

1. Define constant percent rate as a fixed percentage of current value
2. Explain why constant percent rate produces exponential behavior
3. Connect the percent rate to the base of an exponential function
4. Convert percent rates to growth/decay factors
5. Identify doubles, halves, and half-life as disguised percent rates

Interest Comparison: When Do Models Diverge?

Both start at $1,000. Both earn 6% in Year 1. But from Year 2 onward, they diverge.

Constant Percent Rate: Percentage Fixed, Amount Changes

Same amount → linear. Same percent → exponential.

Why the Amounts Diverge After Year One

Simple interest: adds 6% of the original $1,000 = $60 every year.

Compound interest: adds 6% of the current balance — a growing number.

Year 2: 6% of $1,060 = $63.60 (not $60)

Year 3: 6% of $1,123.60 = $67.42 (not $60)

Why Percent Rate Produces Exponential Behavior

Each period: new value = old value factor

The new value becomes the base for the next calculation.

• Larger balance → larger 6% interest → even larger balance
• This self-reinforcing behavior = exponential

Growth accelerates over time; decay approaches zero but never arrives.

Constant Rate vs. Percent Rate: Year 5

Constant rate:

Constant percent rate:

After Year 1, percent rate always produces more.

Which gives more in Year 10? Year 20?

From Percent Rate to Growth Factor: The Algebra

"Grows by 6% per year": new = old + 6% of old

Growth factor = — the "1" keeps the original; "0.06" adds the growth.

Growth Factors for Common Percent Rates

Factor → growth. The closer to 1, the slower the growth.

Decay Factors for Common Percent Rates

Decays 15%/yr:

Factor Reference: Growth and Decay Values

: decay. : no change. : growth.

Verbal Phrases as Disguised Percent Rates

Quick Check: Compute the Growth Factor

Given each description, find the growth or decay factor:

1. A population grows 12% per year.
2. A substance decays 25% per hour.
3. A quantity doubles every month.

Write the factor for each before the next slide.

Practice: Convert Eight Percent Rates Now

1. Grows 5% per year
2. Decays 10% per hour
3. Halves every period
4. Depreciates 18% annually
5. Triples each generation
6. Loses 3% per month
7. Increases by a factor of 1.25
8. Grows 0.5% per day

Find each growth or decay factor.

Answers to Eight Factor Conversions

3. (halving)
5. (tripling)
7. (already a factor)

Key Takeaways from Lesson One

✓ Constant percent rate: same PERCENTAGE applied to changing base
✓ Growth factor: (growth) or (decay)
✓ Factor : growth; $0 < $ factor : decay

MULTIPLY by factor — don't add the initial percent amount every year
Factor = , not — the "1" keeps the original amount
Decay never reaches zero — approaches but never touches

Coming Up in Lesson Two

Deck 2 covers:

• Recognizing constant percent rate language in verbal descriptions
• Distinguishing constant rate (linear) from constant percent rate (exponential)
• Writing exponential functions from descriptions
• Interpreting and in context and making predictions