Simple And Compound Interest

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Notes

Simple Interest

Interest is money added regularly to an original amount; with **simple interest**, each payment is the same because it is based only on the starting amount.
To calculate total simple interest: find a percentage (rate) of the starting amount using a multiplier, then multiply by the number of time periods.
Total balance = starting amount + total interest earned.
Example: $250 at 4% per year for 6 years → interest per year $= 0.04 \times 250 =$ $10, total interest $= 10 \times 6 =$ $60, final balance $= 250 + 60 =$ $310.
To find the interest rate when given total interest: divide total interest by number of years, then find what percentage that is of the original amount.
Always check if the question asks for the **interest earned** or the **total amount** at the end.

Compound Interest

**Compound interest** is calculated on the running total, not just the starting amount, so interest is earned on previously earned interest.
To calculate compound interest, use a multiplier for the percentage increase and apply it repeatedly for each time period.
For an increase of r% per year, the multiplier is $(1 + \frac{r}{100})$ . Final amount $= P \times (multiplier)^{n}$ , where P is the original amount and n is the number of years.
Example: $1200 at 4% per year for 7 years → multiplier $= 1.04$ , final amount $= 1200 \times 1.04^{7}$ ≈ $1579.
The compound interest formula is: Final balance $= P(1 + \frac{r}{100})^{n} (not$ given in exam).
**Depreciation** uses a multiplier less than 1; e.g., a 15% decrease has multiplier 0.85.

Comparing Simple and Compound Interest

Simple interest grows linearly; compound interest grows exponentially.
For the same rate and time, compound interest yields a higher final amount than simple interest (after the first year).
Over many years, compound interest eventually always gives a greater amount because interest is earned on interest.
Example: $1000 at 10% compound vs 12% simple: after 4 years, compound $= 1000\times 1.1^{4} =$ $1464.10, simple $= 1000 + 1000\times 0.12\times 4 =$ $1480; after 6 years, compound = $1771.56, simple = $1720.

Solving Problems with Simple Interest

To find the rate n% when total amount is known: subtract original to get total interest, divide by years to get annual interest, then find percentage of original.
Example: £9000 invested for 5 years becomes £11700 → total interest = £2700, annual interest = £540, rate $= \frac{540}{9000} = 0.06 = 6\%$ .
To find time: use total interest $= P \times (\frac{r}{100}) \times t$ , solve for t.
Always round answers as required (e.g., nearest dollar, nearest hundred).

Solving Problems with Compound Interest

To find the final amount: use multiplier method or formula $P(1 + \frac{r}{100})^{n}$ .
To find the number of years to reach a target: use trial and improvement or logarithms (not required at Core level).
Example: $1750 at 6.5% compound → target $2000: try $n=2$ → $1750\times 1.065^{2}$ ≈ $1985, $n=3$ → $1750\times 1.065^{3}$ ≈ $2114, so at least 3 years.
To find the interest earned: subtract the original amount from the final amount.
Example: $8400 at 3.5% for 2 years → final $= 8400\times 1.035^{2}$ ≈ $8998.29, interest = $8998.29 - $8400 = $598.29.

Common Mistakes and Tips

**Double-check** whether the question uses simple or compound interest.
Ensure you answer the exact question: interest earned or total amount?
Round answers appropriately (e.g., to the nearest dollar, nearest cent).
For compound interest, the multiplier is (1 + r/100); for depreciation, it is $(1 - \frac{r}{100})$ .

Simple Interest Growth Over 5 Years

Compound Interest Growth Over 5 Years

Simple vs Compound Interest Comparison

Depreciation Over 5 Years

Practice questions

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1.Simple interest is calculated on which amount?
Easy
- AThe original amount only
- BThe current balance each year
- CThe total amount at the end
- DThe interest earned each year
2.Ethan invests $6400 at a rate of 2.6% per year simple interest. Calculate the total value of his investment at the end of 3 years.
Easy
- A$6899.20
- B$6656
- C$6720
- D$6998.40
3.Jamie invests $12000 at a rate of 5% per year compound interest. Calculate the value of his investment at the end of 3 years.
Easy
- A$13891.50
- B$13800
- C$12600
- D$13900
4.Jan invests $800 at a rate of 3% per year simple interest. Calculate the value of her investment at the end of 4 years.
Easy
- A$896
- B$824
- C$872
- D$900
5.Sam invests $4500 into a savings account which pays 3.5% compound interest each year. Find the total amount in the account after 6 years.
Easy
- A$5532.67
- B$5445
- C$5670
- D$5590
6.Cody invests $1750 in a savings account which pays 6.5% compound interest each year. How many years will it take for the balance to reach at least $2000?
Easy
- A3 years
- B2 years
- C4 years
- D5 years
7.Trina invests $16000 at a rate of 5% per year compound interest. Work out the value of her investment at the end of 4 years.
Medium
- A$19448.10
- B$19200
- C$16800
- D$20000
8.Gino invests $6000 for 5 years at a rate of 1.2% per year compound interest. Calculate the value of his investment at the end of the 5 years. Give your answer correct to the nearest dollar.
Medium
- A$6369
- B$6360
- C$6372
- D$6000

Summary

## Simple & Compound Interest (Core IGCSE Maths) ### Simple Interest - Interest is calculated **only on the original amount** (principal). - Each interest payment is the same. - **Formula:** Total interest = Principal × (rate/100) × time (years). - **Final amount** = Principal + Total interest. - Example: $6400 at 2.6% for 3 years → Interest = 6400 × 0.026 × 3 = $499.20; Total = $6899.20. ### Compound Interest - Interest is calculated on the **running total** (principal + accumulated interest). - Use a **multiplier**: For r% increase, multiplier = 1 + r/100. - **Final amount** = Principal × (multiplier)^(number of years). - Formula (not given in exam): Final = P × (1 + r/100)^n. - Example: $12000 at 5% for 3 years → multiplier = 1.05; Final = 12000 × 1.05^3 = $13891.50. ### Key Differences - Simple interest: linear growth (same interest each year). - Compound interest: exponential growth (interest on interest). - Compound interest eventually yields more than simple interest for the same rate and time. ### Exam Tips - Check if question asks for **interest earned** or **total amount**. - Round answers as instructed (e.g., nearest dollar). - For compound interest, ensure you apply the multiplier the correct number of times. - To find rate or time, rearrange formulas. ### Depreciation (similar to compound interest but decrease) - Multiplier = 1 - r/100. - Final value = Principal × (multiplier)^n.

Key terms

Simple Interest: Interest calculated only on the original principal amount, resulting in equal interest payments each period.
Compound Interest: Interest calculated on the principal plus any previously earned interest, causing the amount to grow exponentially.
Principal: The original amount of money invested or borrowed.
Multiplier: A decimal factor used to increase (1 + r/100) or decrease (1 - r/100) an amount by a given percentage.
Depreciation: A decrease in value over time, calculated similarly to compound interest but with a reduction multiplier.

Slides

Simple Interest

Interest is money added regularly to an original amount.
Simple interest is based only on the starting amount.
Each interest payment is the same amount.

Which of the following best describes simple interest?

Interest calculated on the starting amount only
Interest calculated on the running total
Interest that changes each year
Interest that is added once

Calculating Simple Interest

Find a percentage of the starting amount using a multiplier (e.g., 0.05 for 5%).
Multiply this by the number of time periods (e.g., years).
To find the total balance, add the interest to the starting amount.

To find simple interest, multiply the starting amount by $the ____ and$ the number of years.

Compound Interest

Compound interest is calculated on the running total, not just the starting amount.
Each year, interest is added to the new balance.
This causes the amount to grow faster than simple interest.

Which of these are true about compound interest?

Interest is calculated on the original amount only.
Interest is calculated on the current balance.
The amount grows faster than with simple interest.

Calculating Compound Interest

Use a multiplier: for a 5% increase, multiply by 1.05.
Apply the multiplier for each year: final $= P \times (multiplier)^{n}$ .
The formula is: final $= P \times (1 + \frac{r}{100})^{n}$ , where r is the percentage rate and n is the number of years.

What is the multiplier for a compound interest rate of 4% per year?

0.04
1.04
4.0
1.4

Example: Simple Interest

Nigel invests $250 at 4% simple interest for 6 years.
Interest per year $= 0.04 \times 250 =$ $10.
Total interest $= 10 \times 6 =$ $60.
Final amount $= 250 + 60 =$ $310.

If $500 is invested at 3% simple interest for 4 years, the total interest earned is $____.

Example: Compound Interest

Jasmina invests $1200 at 4% compound interest for 7 years.
Multiplier $= 1.04$ , applied 7 times: $1200 \times 1.04^{7}$ .
Final amount ≈ $1579 (to nearest dollar).

What is the value after 3 years of $800 invested at 5% compound interest?

$920
$926.10
$960
$840

Finding the Rate or Time

For simple interest, you can find the rate by dividing annual interest by the principal.
For compound interest, you can rearrange the formula to find n or r.
Example: Noah invested £9000 and got £11700 after 5 years simple interest. Interest per year $= (11700-9000)/5 =$ £540. Rate $= \frac{540}{9000} = 0.06 = 6\%$ .

If $2000 grows to $2600 in 5 years with simple interest, the annual interest rate $is ____\%$ .

Depreciation

Depreciation is when value decreases by a percentage each year.
Use a multiplier less than 1: e.g., 15% decrease means multiply by 0.85.
Formula: final $= P \times (1 - \frac{r}{100})^{n}$ .

A car worth $16000 depreciates by 15% each year. What is its value after 6 years?

$6034.39
$6400
$13600
$16000

Key Differences

Simple interest: same amount each year, linear growth.
Compound interest: amount increases each year, exponential growth.
Over time, compound interest always yields more than simple interest (for positive rates).

Which statements are true?

Simple interest grows linearly.
Compound interest grows exponentially.
Simple interest always gives more than compound interest.

Flashcards

What is simple interest?

Interest calculated only on the original amount; each interest payment is the same.

How do you calculate total simple interest earned?

Find a percentage of the starting amount using a multiplier, then multiply by the number of time periods.

How do you find the total balance after simple interest?

Add the total interest earned to the starting amount.

Ethan invests $6400 at 2.6% per year simple interest. Calculate total value after 3 years.

Interest per year $= 6400 \times 0.026 =$ $166.40; total interest $= 166.40 \times 3 =$ $499.20; total value $= 6400 + 499.20 =$ $6899.20.

Jan invests $800 at 3% per year simple interest. Calculate value after 4 years.

Interest per year $= 800 \times 0.03 =$ $24; total interest $= 24 \times 4 =$ $96; value $= 800 + 96 =$ $896.

What is compound interest?

Interest calculated on the running total (original plus accumulated interest), not just the starting amount.

How do you calculate compound interest using a multiplier?

Multiply the original amount by $(1 + \frac{r}{100})^{n}$ , where r is the percentage rate and n is the number of periods.

Jamie invests $12000 at 5% per year compound interest. Calculate value after 3 years.

Multiplier = 1.05; value $= 12000 \times 1.05^{3} =$ $13891.50.

Sam invests $4500 at 3.5% compound interest. Find total amount after 6 years.

Multiplier = 1.035; amount $= 4500 \times 1.035^{6}$ ≈ $5533.51.

Cody invests $1750 at 6.5% compound interest. How many years to reach at least $2000?

Solve $1750 \times 1.065^{n} \ge 2000$ . $n = 3$ years gives $2113.12; $n = 2$ gives $1985.04. So 3 years.

Trina invests $16000 at 5% per year compound interest. Work out value after 4 years.

Value $= 16000 \times 1.05^{4} =$ $19448.10.

Gino invests $6000 for 5 years at 1.2% compound interest. Value to nearest dollar?

Value $= 6000 \times 1.012^{5}$ ≈ $6369.43 → $6369.

Lucia invests $5000 at 4.5% compound interest for 7 years. Value?

Value $= 5000 \times 1.045^{7}$ ≈ $6803.67.

Ella invests $4000 for 3 years at 1.2% compound interest. Value?

Value $= 4000 \times 1.012^{3}$ ≈ $4145.73.

Javier invests $750 for 3 years at 1.8% compound interest. Value?

Value $= 750 \times 1.018^{3}$ ≈ $791.24.

Hawa invests $750 at 3.5% compound interest for 3 years. Value?

Value $= 750 \times 1.035^{3}$ ≈ $831.55.

$12000 invested at n% simple interest for 3 years becomes $12900. Find n.

Interest $= 12900 - 12000 =$ $900; per year $= \frac{900}{3} =$ $300; $n = (\frac{300}{12000})\times 100 = 2.5\%$ .

Sonia invests $5000 for 6 years. Bank A: 3.5% compound. Bank B: 22% increase at end. Which is better and by how much?

Bank $A: 5000\times 1.035^{6}$ ≈ $6146.34; Bank $B: 5000\times 1.22 =$ $6100; Bank A better by $46.34.

Riya invests $30000 at 2.5% compound interest for 7 years. Value to nearest dollar?

Value $= 30000 \times 1.025^{7}$ ≈ $35647.44 → $35647.

Six years ago farmer invested $3750 at 4% compound interest. Value after 6 years to nearest dollar?

Value $= 3750 \times 1.04^{6}$ ≈ $4745.70 → $4746.

Paula invests $600 at r% simple interest for 10 years, earns $90 interest. Find r.

Interest per year $= \frac{90}{10} =$ $9; $r = (\frac{9}{600})\times 100 = 1.5\%$ .

$8400 invested for 2 years at 3.5% compound interest. Total interest earned?

Final amount $= 8400 \times 1.035^{2} =$ $8998.29; interest $= 8998.29 - 8400 =$ $598.29.

Harry invests $800 for 2 years at 3% compound interest. Interest received?

Final amount $= 800 \times 1.03^{2} =$ $848.72; interest $= 848.72 - 800 =$ $48.72.

Tomas borrows $5000 for 3 years at 2.5% compound interest. Total payback?

Amount $= 5000 \times 1.025^{3} =$ $5384.45.

Kim invests £13000 at 10% simple interest for 5 years. Total value?

Interest per year $= 13000\times 0.10 =$ £1300; total interest $= 1300\times 5 =$ £6500; total $= 13000+6500 =$ £19500.

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