Simple And Compound Interest Growth And Decay

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Notes

Simple Interest

**Simple interest** is calculated only on the original amount (principal), so each interest payment is the same.
Total interest = principal $\times (rate/100) \times$ time (years).
Final amount = principal + total interest.
Example: $250 at 4% for 6 years → interest per year = $10, total interest = $60, final amount = $310.
To find the rate: divide total interest by $(principal \times time)$ .

Compound Interest

**Compound interest** is calculated on the running total, so interest is earned on interest.
Final amount = principal $\times (1 + \frac{r}{100})^{n}$ , where r is the annual percentage rate and n is the number of years.
The multiplier is (1 + r/100); e.g., 5% → multiplier 1.05.
For different rates over time, apply each multiplier sequentially.
Reverse problems: original amount = final amount $\div (1 + \frac{r}{100})^{n}$ .

Depreciation

**Depreciation** is a percentage decrease in value over time (e.g., cars, phones).
Final value = principal $\times (1 - \frac{r}{100})^{n}$ , where r is the annual depreciation rate.
The multiplier is (1 - r/100); e.g., 15% depreciation → multiplier 0.85.
Amount lost = original value - final value.

Exponential Growth & Decay

**Exponential growth**: quantity increases by a fixed percentage each period; multiplier $k > 1$ .
**Exponential decay**: quantity decreases by a fixed percentage each period; multiplier $0 < k < 1$ .
General model: $B = A \times k^{n}$ , where A is initial amount, k is multiplier, n is number of periods.
Examples: population growth, bacterial growth (growth); cooling, radioactive decay (decay).
To find $k: k = (\frac{B}{A})^(\frac{1}{n})$ . To find n: use trial and improvement.

Key Differences: Simple vs Compound

Simple interest: interest only on principal; linear growth.
Compound interest: interest on principal + accumulated interest; exponential growth.
For the same rate and time, compound interest yields a higher final amount than simple interest.
Always check whether the question asks for interest earned or final amount.

Common Exam Tips

Identify whether the problem involves simple interest, compound interest, depreciation, or exponential growth/decay.
Use the correct multiplier: growth $= 1 + (rate/100)$ , decay $= 1 - (rate/100)$ .
Round answers as instructed (e.g., nearest hundred, nearest dollar).
For reverse problems, rearrange the formula to solve for the unknown.

Simple Interest vs Compound Interest Growth

Exponential Growth and Decay Curves

Practice questions

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1.What is simple interest?
Easy
- AInterest calculated only on the original amount
- BInterest calculated on the running total each year
- CInterest that increases each year
- DInterest that decreases each year
2.Which multiplier represents a 4% increase?
Easy
- A1.4
- B0.96
- C1.04
- D0.04
3.A car depreciates by 15% each year. What is the multiplier for one year?
Easy
- A1.15
- B0.15
- C0.85
- D1.85
4.Paula invests $600 at a rate of r% per year simple interest. At the end of 10 years, the total interest earned is $90. Find r.
Medium
- A1.5
- B15
- C0.15
- D1.5%
5.Jan invests $800 at a rate of 3% per year simple interest. Calculate the value of her investment at the end of 4 years.
Medium
- A$896
- B$824
- C$96
- D$800
6.Toby invested £7500 for 2 years in a savings account. He was paid 4% per annum compound interest. How much money did Toby have at the end of 2 years?
Medium
- A£8112
- B£7800
- C£8100
- D£8112.00
7.Eric invests an amount in a bank that pays compound interest at a rate of 2.16% per year. At the end of 5 years, the value of his investment is $6999.31. Calculate the amount Eric invests.
Hard
- A$6300
- B$6500
- C$6200
- D$6000
8.The population of a town decreases exponentially at a rate of 1.7% per year. The population now is 250 000. Calculate the population at the end of 5 years, correct to the nearest hundred.
Hard
- A229 500
- B229 400
- C230 000
- D228 800

Summary

## Simple & Compound Interest, Growth & Decay ### Simple Interest - Interest calculated only on the **original amount** (principal). - Each interest payment is the same. - **Formula:** Total interest = Principal × (rate/100) × time. - Final amount = Principal + Total interest. - Example: $250 at 4% for 6 years → interest = 250 × 0.04 × 6 = $60; total = $310. ### Compound Interest - Interest calculated on the **running total** (principal + accumulated interest). - Use a **multiplier**: for r% increase, multiplier = 1 + r/100. - **Formula:** Final amount = P × (1 + r/100)^n, where P = principal, r = rate (%), n = number of years. - To find original amount (reverse): P = Final amount / (1 + r/100)^n. - Example: $1200 at 4% for 7 years → 1200 × 1.04^7 ≈ $1579. ### Depreciation - Value **decreases** by a fixed percentage each year. - Multiplier = 1 - r/100 (e.g., 15% decrease → 0.85). - **Formula:** Final value = P × (1 - r/100)^n. - Example: £20,000 car depreciates 15% p.a. for 3 years → 20000 × 0.85^3 = £12,282.50. ### Exponential Growth & Decay - General model: B = A × k^n, where A = initial amount, B = new amount, k = multiplier (k>1 for growth, 0<k<1 for decay), n = number of periods. - **Growth:** e.g., population increases 1.6% p.a. → k = 1.016. - **Decay:** e.g., population decreases 1.7% p.a. → k = 0.983. - To find k: k = (B/A)^(1/n). - To find n: use trial and improvement. - Round answers as required (e.g., nearest hundred, nearest thousand). ### Key Points - **Simple vs Compound:** Simple interest uses only original amount; compound uses running total. - **Multiplier method:** Works for both growth and decay. - **Reverse problems:** Divide final amount by multiplier^n to find original. - **Check question:** Are they asking for interest earned or total amount?

Key terms

Simple Interest: Interest calculated only on the original principal amount; each interest payment is the same.
Compound Interest: Interest calculated on the running total (principal plus accumulated interest); interest earns interest.
Multiplier: A decimal factor used to increase or decrease an amount by a percentage; for growth: 1 + r/100, for decay: 1 - r/100.
Depreciation: The decrease in value of an asset over time, usually calculated as a percentage decrease each year.
Exponential Growth: A quantity increases by a fixed percentage each time period; modelled by B = A × k^n with k > 1.
Exponential Decay: A quantity decreases by a fixed percentage each time period; modelled by B = A × k^n with 0 < k < 1.
Principal: The original amount of money invested or borrowed.
Interest Rate: The percentage at which interest is calculated, usually per year.

Slides

Simple Interest

Simple interest is calculated only on the original amount (principal).
Each interest payment is the same amount each year.
Total interest $= (principal \times$ rate $\times time)$ where rate is as a decimal.
Final amount = principal + total interest.

If you invest $500 at 4% simple interest per year for 3 years, what is the total interest earned?

$20
$60
$80
$120

Compound Interest

Compound interest is calculated on the running total (principal + previous interest).
Use a multiplier: for r% increase, multiplier $= 1 + \frac{r}{100}$ .
Final amount = principal $\times (multiplier)^{n}$ , where n is number of years.
This formula is not given in the exam.

To increase $300 by 5% each year for 2 years, the calculation is $300 $\times ____^{2}$ .

Depreciation

Depreciation is when an item loses value over time (e.g. cars).
It works like compound interest but with a percentage decrease.
Multiplier for a decrease of r% is $1 - \frac{r}{100}$ .
Final value = original $\times (multiplier)^{n}$ .

A car worth $20,000 depreciates by 15% each year. What is its value after 3 years?

$12,282.50
$13,500
$11,000
$15,000

Exponential Growth & Decay

Exponential growth: quantity increases by a fixed percentage each period.
Exponential decay: quantity decreases by a fixed percentage each period.
Model: $B = A \times k^{n}$ , where A is initial amount, k is multiplier, n is number of periods.
$k > 1$ for growth, $0 < k < 1$ for decay.

Which of these are true about exponential growth and decay?

Exponential growth has a multiplier greater than 1.
Exponential decay has a multiplier greater than 1.
Population increase is an example of exponential growth.
Radioactive decay is an example of exponential growth.

Reverse Compound Interest

Sometimes you know the final amount and need to find the original principal.
Rearrange the formula: original = final amount $\div (multiplier)^{n}$ .
Example: If $432 after 3 years at 20% compound interest, original = $432 $\div 1.20^{3} =$ $250.

If an investment grows to $6999.31 after 5 years at 2.16% compound interest, the original amount is $6999.31 $\div ____^{5}$ .

Finding the Rate or Time

To find the rate, rearrange $B = A \times k^{n}$ to find k, then convert to percentage.
To find time n, use trial and improvement with whole numbers.
Example: If a population of 25000 grows to 41626.83 in 13 years at $4\%, k = 1.04$ .

A population of 45,000 increases at 1.6% per year. What is the population after 5 years (to nearest hundred)?

48,700
48,800
48,900
49,000

Simple Interest Formula

Simple interest: $I = P \times r \times t$ , where r is rate as decimal, t is time in years.
Total amount: $A = P + I = P(1 + rt)$ .
Example: $600 at r% for 10 years gives $90 interest. So $r = 90/(600\times 10) = 0.015 = 1.5\%$ .

Paula invests $600 at r% simple interest. After 10 years, total interest is $90. So $r = ____\%$ .

Comparing Investments

To compare simple vs compound interest, calculate final amounts separately.
Compound interest usually gives more over longer periods.
Always check if the question asks for total interest or final amount.

Which investment yields more after 8 years: $2000 at 2.5% simple interest or $2000 at 2% compound interest?

Simple interest
Compound interest
They are equal
Cannot determine

Exponential Decay Example

A town population of 250,000 decreases at 1.7% per year.
Multiplier $= 1 - 0.017 = 0.983$ .
Population after 5 years $= 250,000 \times 0.983^{5}$ ≈ 229,500 (to nearest hundred).

A population of 250,000 decreases at 1.7% per year. After 5 years, population ≈ $____ (nearest hundred)$ .

Key Exam Tips

Check if question asks for interest earned or final amount.
Round answers as instructed (e.g. nearest dollar, hundred, thousand).
For compound interest, remember the multiplier method.
The compound interest formula is not given in the exam.

Which formula correctly gives the final amount after n years with compound interest at r%?

$A = P(1 + rn)$
$A = P(1 + \frac{r}{100})^{n}$
$A = P(1 + r)^{n}$
$A = P(1 + \frac{r}{100})n$

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