Compound Interest Revision Notes for Leaving Cert Mathematics

Compound Interest

What is compound interest?

Compound interest is the interest earned on both the original amount invested (the principal) and any interest that has already been added to that amount. Unlike simple interest, compound interest grows faster because each year's interest is calculated on a larger base amount.

When you invest money, three key terms are important to understand:

Principal: The original amount of money invested
Rate per annum: The percentage interest rate charged or earned each year
Amount or Final amount: The total value after interest has been added

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The key difference between simple and compound interest is that compound interest is calculated on the growing total each year, while simple interest is always calculated on the original principal only. This makes compound interest much more powerful for long-term investments.

For example, if you invest €100 at 5% per year, after one year you earn €5 interest, giving you €105 total. In the second year, you earn 5% on the full €105, not just the original €100.

The compound interest formula

The pattern of compound interest calculations leads us to a useful formula:

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The Compound Interest Formula:

$F = P(1 + i)^n$

Where:

F = final amount
P = principal (starting amount)
i = interest rate expressed as a decimal
n = time in years

The term (1 + i) is called the multiplier.

This formula can also be stated in words as:

Amount after t years = original amount × (multiplier)^t

Converting percentages to decimals

To use the formula, you must convert percentage rates to decimals:

4% becomes 0.04, so multiply by 1.04
4.5% becomes 0.045, so multiply by 1.045
12% becomes 0.12, so multiply by 1.12

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Worked Example: Finding Compound Interest

Question: Find the compound interest on €2800 for 3 years at 7.5% per annum.

Solution: Using the formula $F = P(1 + i)^n$

P = €2800
i = 7.5% = 0.075
n = 3 years

$F = €2800(1.075)^3$ $F = €2800 × 1.242 = €3478.43$

Interest earned = €3478.43 - €2800 = €678.43

Finding the rate and the principal

The compound interest formula can be rearranged to find missing values when other information is given.

Finding the interest rate

When you know the principal, final amount, and time period, you can calculate the interest rate. The key relationship is that if €300 grows to €318 in one year, the interest earned is €18.

Using the formula: Rate = (Interest ÷ Principal) × 100%

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Worked Example: Finding the Rate

Question: If €650 amounts to €702 in one year, find the rate.

Solution: Interest = €702 - €650 = €52

Rate = (Interest ÷ Principal) × 100% Rate = (€52 ÷ €650) × 100% = 8%

Therefore the rate is 8%.

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Worked Example: Different Rates for Different Years

Question: A woman invested €6000 in a Building Society for two years. The rate of interest for the first year was 3% per annum. She did not withdraw any money at the end of the first year. At the end of the second year her total investment was worth €6427.20. What was the rate of interest for the second year?

Solution: Amount at end of first year = €6000 × 1.03 = €6180

Amount at end of second year = €6427.20 Interest for second year = €6427.20 - €6180 = €247.20

Rate for second year = (Interest ÷ Principal) × 100% Rate = (€247.20 ÷ €6180) × 100% = 4%

The interest rate for the second year was 4%.

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Worked Example: Finding the Principal

Question: What sum of money, invested at 4% per annum compound interest, will amount to €3149.62 after 3 years?

Solution: Using $F = P(1 + i)^n$ €3149.62 = P(1.04)³ €3149.62 = P × 1.1249

$P = €3149.62 ÷ 1.1249 = :success[€2800]$

The amount invested was €2800.

Annual equivalent rate (AER)

Some institutions, such as credit card companies, charge interest on a monthly basis rather than annually. The Annual Equivalent Rate (AER) shows what the true annual interest rate is.

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Why AER Matters

Financial institutions may advertise low monthly rates, but the AER reveals the true cost when these rates are compounded over a full year. This helps consumers make better comparisons between different financial products.

For example, if 2% compound interest is charged each month for 12 months: Amount owed = €100 × $(1.02)^{12}$ = €126.82

This means €26.82 interest is charged on €100, giving an annual rate of 26.8%.

Therefore, 2% compound interest per month is equivalent to an annual interest rate of 26.8%. This annual rate is called the Annual Equivalent Rate or AER.

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Worked Example: Calculating AER

Question: An investment bond gives a 20% return when invested for 8 years. Calculate the AER for this bond, correct to one decimal place.

Solution: If the return is 20%, the final amount is 1.20 times the sum invested.

$(multiplier)^8 = 1.2$ $multiplier = (1.2)^{1/8} = 1.02305$

Therefore the AER is 2.3%.

Depreciation

Depreciation is when something loses value over time. This follows the same mathematical pattern as compound interest, but in reverse.

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Understanding Depreciation

Depreciation uses the same mathematical principles as compound interest growth, but with a multiplier less than 1. Instead of adding interest, we're removing value at a consistent percentage each year.

If a car depreciates by 20% each year, its value at the end of the first year will be 80% of its original value. To find 80% of an amount, multiply by 0.8.

Depreciation calculations

If a car costs €25,000 and depreciates by 15% each year:

Value at end of year 1 = €25,000 × 0.85
Value at end of year 2 = €25,000 × $(0.85)^2$
Value at end of year 3 = €25,000 × $(0.85)^3$

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Depreciation Formula:

$Final\ value = Original\ value × (1 - depreciation\ rate)^n$

Note that the multiplier $(1 - depreciation\ rate)$ is always less than 1 for depreciation.

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Worked Example: Depreciation Problem

Question: A machine depreciates in value by 10% per annum. If the machine is worth €58,320 at the end of 3 years, find its value when new.

Solution: Let P be the value when new. After 3 years: Value = $P(0.9)^3$

$P(0.9)^3 = €58,320$ $P(0.729) = €58,320$ $P = €58,320 ÷ 0.729 = :success[€80,000]$

The machine was worth €80,000 when new.

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Key Points to Remember:

Compound interest formula: $F = P(1 + i)^n$ where interest is added to the growing total each year
The multiplier $(1 + i)$ is key to all calculations - greater than 1 for growth, less than 1 for depreciation
To find missing values, rearrange the main formula or use Rate = (Interest ÷ Principal) × 100%
AER shows the true annual rate when interest is charged more frequently than yearly
Depreciation uses the same principles but with a multiplier less than 1

Compound Interest (Leaving Cert Mathematics): Revision Notes