Compound Interest and Depreciation

What you'll learn

Turn percentage increases and decreases into decimal multipliers.
Use compound interest to find a final amount or total interest.
Use depreciation for values that go down over time.
Compare different saving options fairly.

1. Percentages as multipliers

Before compound interest, you need to be confident with percentage multipliers.

The original amount is 100% of itself. So:

for an increase, the multiplier is bigger than 1
for a decrease, the multiplier is smaller than 1

Definition

Multiplier

A multiplier is the decimal you multiply by for a percentage change. An increase of 6% uses 1.06; a decrease of 6% uses 0.94.

Example

One percentage increase

A tablet costs £450. Its price increases by 8%. Find the new price.

A bar model shows that an 8% increase means adding 8% to the original 100%, giving a multiplier of 1.08.

An increase of 8% means the multiplier is 1.08.
Multiply the starting price by the multiplier: $450 \times 1.08 = 486$ .
The new price is £486.

2. Compound interest

Interest is extra money earned on savings or an investment.

Definition

Compound interest

Compound interest means each year's interest is added on, then the next year's interest is calculated on the new larger amount. “Per annum” means each year.

If the starting amount is $P$ , the annual percentage rate is $r$ , and the number of years is $n$ , then the final amount $A$ is:

P\left(1 + \frac{r}{100}\right)^n

The power tells you how many times the multiplier is used.

Key Idea

Repeated increases

For compound interest, do not add the interest rate each year. Multiply by the yearly multiplier once for each year.

Example

Finding the final amount

Ravi invests £3600 for 4 years at 2.5% compound interest per annum. Find how much is in the account at the end.

A timeline shows the same multiplier, 1.025, applied once for each of the 4 years.

A 2.5% increase means the multiplier is 1.025.
There are 4 years, so calculate $3600 \times 1.025^4$ .
This gives 3973.726..., so round to the nearest penny.
Ravi has £3973.73 at the end of 4 years.

3. Total interest

Sometimes the question asks for the total interest, not the final amount.

Definition

Total interest

Total interest is the extra money earned. It is the final amount minus the starting amount.

Example

Using a given formula

The value $V$ of an investment after $n$ years is given by $\times 1.018^n$ .

Find the annual interest rate and the total interest after 3 years.

The formula tree highlights that the multiplier 1.018 means an increase of 0.018, or 1.8%, and that total interest is final amount minus original amount.

The multiplier is 1.018, so the increase above 1 is 0.018.
This means the annual interest rate is 1.8%.
Substitute $n = 3$ : $\times 1.018^3$ .
This gives £6329.87 to the nearest penny.
Subtract the original investment: £6329.87 minus £6000 is £329.87.

Common Mistake

Interest vs final amount

If a question asks for total interest, do not stop when you have found the final amount. You must subtract the original investment.

4. Depreciation

Definition

Depreciation

Depreciation means a value goes down over time, often because an item gets older or is worth less.

For a depreciation of 20%, the item keeps 80% of its value, so the multiplier is 0.80.

If the same depreciation rate happens every year, use:

P\left(1 - \frac{r}{100}\right)^n

Example

Depreciation over several years

A new scooter costs £1800. It loses 25% of its value in the first year, then 10% in each of the next two years. Find its value after 3 years.

A depreciation timeline shows a 25% loss first, then two separate 10% losses on the reduced value.

Losing 25% means keeping 75%, so the first multiplier is 0.75.
Losing 10% means keeping 90%, so the next two multipliers are both 0.90.
Calculate $1800 \times 0.75 \times 0.90^2 = 1093.50$ .
The scooter is worth £1093.50 after 3 years.

5. Different changes in different years

Not every year has to have the same percentage change. Use a separate multiplier for each year.

A rise uses a multiplier above 1. A fall uses a multiplier below 1.

Example

Increase, increase, then decrease

A house is bought for £300,000. Its value rises by 2% in the first year, rises by 5% in the second year, then falls by 4% in the third year. Find the value after 3 years.

The value changes are shown as three separate multipliers, with increases above 1 and the decrease below 1.

The three multipliers are 1.02, 1.05 and 0.96.
Multiply by each one once: $300000 \times 1.02 \times 1.05 \times 0.96 = 308448$ .
The house is worth £308,448 after 3 years.

Common Mistake

Adding the percentages

Do not just combine the percentage changes in your head. Each percentage is taken from the new value, not always from the original value.

6. Finding the number of years

Sometimes you are given the final amount and asked to find $n$ , the number of years.

At Grade 4, a reliable method is to try whole-number powers on your calculator until the value matches or passes the target.

Example

Finding n from a final amount

A savings account starts with £3000 and grows by 2% compound interest per year. After $n$ years, the account has £3378.49. Find $n$ .

A trial timeline shows powers of 1.02 being tested until the target amount is reached.

The multiplier is 1.02.
Try 5 years: $3000 \times 1.02^5 = 3312.24$ .
Try 6 years: $3000 \times 1.02^6 = 3378.49$ .
So $n = 6$ . The money has been invested for 6 years.

Example

When something halves in value

A machine depreciates by 15% each year. Find how many years it takes to be worth less than half its original value.

A threshold diagram compares repeated depreciation by 0.85 with the half-value line at 0.5.

Use 1 as the original value. Half of this is 0.5.
A 15% decrease means the multiplier is 0.85.
Try powers: $0.85^4 = 0.522...$ , which is still above half.
Try the next year: $0.85^5 = 0.443...$ , which is below half.
It first becomes worth less than half after 5 years.

Tip

Calculator accuracy

Use the full calculator value until the final answer, then round money to 2 decimal places. Rounding too early can change the final penny.

7. Comparing investments

Definition

Simple interest

Simple interest is calculated only on the original amount each year. Compound interest is calculated on the growing amount.

When comparing options, work out the final amount for each one using the correct method, then choose the larger amount.

Example

Compound interest or simple interest

Leah invests £7000 for 5 years. Option A gives 2.4% compound interest per annum. Option B gives 2.5% simple interest per annum. Which option should she choose?

A side-by-side comparison shows compound interest multiplying each year, while simple interest adds the same amount each year.

For Option A, use the compound multiplier 1.024: $7000 \times 1.024^5 = 7881.30$ .
So Option A gives a final amount of £7881.30.
For Option B, 2.5% of £7000 is £175 each year.
Over 5 years, the simple interest is £875, so the final amount is £7875.
Option A is better because £7881.30 is more than £7875.

Example

Comparing banks with changing rates

Noah invests £5000 for 3 years. Bank A pays 1.4% compound interest each year. Bank B pays 2.5% in the first year, then 1% for each extra year. Which bank gives more interest?

Bank A: $5000 \times 1.014^3 = 5212.95$ , so the interest is £212.95.
Bank B: $5000 \times 1.025 \times 1.01^2 = 5228.01$ , so the interest is £228.01.
Bank B gives more interest by £15.06.

Exam technique

In the exam

Underline whether the question wants the final amount, total interest, difference, or number of years.
Write the multiplier before you calculate: increases are above 1, decreases are below 1.
For comparisons, calculate each option separately and make a clear final statement.
Round money to 2 decimal places, but only at the end.

Self review

Check yourself

What multiplier would you use for a 7% decrease?
If an investment grows by 3% for 4 years, why is using 12% as one increase not the compound method?
After finding a final amount, how do you find the total interest?

Compound Interest and Depreciation

What you'll learn

Turn percentage increases and decreases into decimal multipliers.
Use compound interest to find a final amount or total interest.
Use depreciation for values that go down over time.
Compare different saving options fairly.

1. Percentages as multipliers

Before compound interest, you need to be confident with percentage multipliers.

The original amount is 100% of itself. So:

for an increase, the multiplier is bigger than 1
for a decrease, the multiplier is smaller than 1

Definition

Multiplier

A multiplier is the decimal you multiply by for a percentage change. An increase of 6% uses 1.06; a decrease of 6% uses 0.94.

Example

One percentage increase

A tablet costs £450. Its price increases by 8%. Find the new price.

A bar model shows that an 8% increase means adding 8% to the original 100%, giving a multiplier of 1.08.

An increase of 8% means the multiplier is 1.08.
Multiply the starting price by the multiplier: $450 \times 1.08 = 486$ .
The new price is £486.

2. Compound interest

Interest is extra money earned on savings or an investment.

Definition

Compound interest

Compound interest means each year's interest is added on, then the next year's interest is calculated on the new larger amount. “Per annum” means each year.

If the starting amount is $P$ , the annual percentage rate is $r$ , and the number of years is $n$ , then the final amount $A$ is:

P\left(1 + \frac{r}{100}\right)^n

The power tells you how many times the multiplier is used.

Key Idea

Repeated increases

For compound interest, do not add the interest rate each year. Multiply by the yearly multiplier once for each year.

Example

Finding the final amount

Ravi invests £3600 for 4 years at 2.5% compound interest per annum. Find how much is in the account at the end.

A timeline shows the same multiplier, 1.025, applied once for each of the 4 years.

A 2.5% increase means the multiplier is 1.025.
There are 4 years, so calculate $3600 \times 1.025^4$ .
This gives 3973.726..., so round to the nearest penny.
Ravi has £3973.73 at the end of 4 years.

3. Total interest

Sometimes the question asks for the total interest, not the final amount.

Definition

Total interest

Total interest is the extra money earned. It is the final amount minus the starting amount.

Example

Using a given formula

The value $V$ of an investment after $n$ years is given by $\times 1.018^n$ .

Find the annual interest rate and the total interest after 3 years.

The formula tree highlights that the multiplier 1.018 means an increase of 0.018, or 1.8%, and that total interest is final amount minus original amount.

The multiplier is 1.018, so the increase above 1 is 0.018.
This means the annual interest rate is 1.8%.
Substitute $n = 3$ : $\times 1.018^3$ .
This gives £6329.87 to the nearest penny.
Subtract the original investment: £6329.87 minus £6000 is £329.87.

Common Mistake

Interest vs final amount

If a question asks for total interest, do not stop when you have found the final amount. You must subtract the original investment.

4. Depreciation

Definition

Depreciation

Depreciation means a value goes down over time, often because an item gets older or is worth less.

For a depreciation of 20%, the item keeps 80% of its value, so the multiplier is 0.80.

If the same depreciation rate happens every year, use:

P\left(1 - \frac{r}{100}\right)^n

Example

Depreciation over several years

A new scooter costs £1800. It loses 25% of its value in the first year, then 10% in each of the next two years. Find its value after 3 years.

A depreciation timeline shows a 25% loss first, then two separate 10% losses on the reduced value.

Losing 25% means keeping 75%, so the first multiplier is 0.75.
Losing 10% means keeping 90%, so the next two multipliers are both 0.90.
Calculate $1800 \times 0.75 \times 0.90^2 = 1093.50$ .
The scooter is worth £1093.50 after 3 years.

5. Different changes in different years

Not every year has to have the same percentage change. Use a separate multiplier for each year.

A rise uses a multiplier above 1. A fall uses a multiplier below 1.

Example

Increase, increase, then decrease

A house is bought for £300,000. Its value rises by 2% in the first year, rises by 5% in the second year, then falls by 4% in the third year. Find the value after 3 years.

The value changes are shown as three separate multipliers, with increases above 1 and the decrease below 1.

The three multipliers are 1.02, 1.05 and 0.96.
Multiply by each one once: $300000 \times 1.02 \times 1.05 \times 0.96 = 308448$ .
The house is worth £308,448 after 3 years.

Common Mistake

Adding the percentages

Do not just combine the percentage changes in your head. Each percentage is taken from the new value, not always from the original value.

6. Finding the number of years

Sometimes you are given the final amount and asked to find $n$ , the number of years.

At Grade 4, a reliable method is to try whole-number powers on your calculator until the value matches or passes the target.

Example

Finding n from a final amount

A savings account starts with £3000 and grows by 2% compound interest per year. After $n$ years, the account has £3378.49. Find $n$ .

A trial timeline shows powers of 1.02 being tested until the target amount is reached.

The multiplier is 1.02.
Try 5 years: $3000 \times 1.02^5 = 3312.24$ .
Try 6 years: $3000 \times 1.02^6 = 3378.49$ .
So $n = 6$ . The money has been invested for 6 years.

Example

When something halves in value

A machine depreciates by 15% each year. Find how many years it takes to be worth less than half its original value.

A threshold diagram compares repeated depreciation by 0.85 with the half-value line at 0.5.

Use 1 as the original value. Half of this is 0.5.
A 15% decrease means the multiplier is 0.85.
Try powers: $0.85^4 = 0.522...$ , which is still above half.
Try the next year: $0.85^5 = 0.443...$ , which is below half.
It first becomes worth less than half after 5 years.

Tip

Calculator accuracy

Use the full calculator value until the final answer, then round money to 2 decimal places. Rounding too early can change the final penny.

7. Comparing investments

Definition

Simple interest

Simple interest is calculated only on the original amount each year. Compound interest is calculated on the growing amount.

When comparing options, work out the final amount for each one using the correct method, then choose the larger amount.

Example

Compound interest or simple interest

Leah invests £7000 for 5 years. Option A gives 2.4% compound interest per annum. Option B gives 2.5% simple interest per annum. Which option should she choose?

A side-by-side comparison shows compound interest multiplying each year, while simple interest adds the same amount each year.

For Option A, use the compound multiplier 1.024: $7000 \times 1.024^5 = 7881.30$ .
So Option A gives a final amount of £7881.30.
For Option B, 2.5% of £7000 is £175 each year.
Over 5 years, the simple interest is £875, so the final amount is £7875.
Option A is better because £7881.30 is more than £7875.

Example

Comparing banks with changing rates

Noah invests £5000 for 3 years. Bank A pays 1.4% compound interest each year. Bank B pays 2.5% in the first year, then 1% for each extra year. Which bank gives more interest?

Bank A: $5000 \times 1.014^3 = 5212.95$ , so the interest is £212.95.
Bank B: $5000 \times 1.025 \times 1.01^2 = 5228.01$ , so the interest is £228.01.
Bank B gives more interest by £15.06.

Exam technique

In the exam

Underline whether the question wants the final amount, total interest, difference, or number of years.
Write the multiplier before you calculate: increases are above 1, decreases are below 1.
For comparisons, calculate each option separately and make a clear final statement.
Round money to 2 decimal places, but only at the end.

Self review

Check yourself

What multiplier would you use for a 7% decrease?
If an investment grows by 3% for 4 years, why is using 12% as one increase not the compound method?
After finding a final amount, how do you find the total interest?

Compound Interest and Depreciation

What you'll learn

1. Percentages as multipliers

2. Compound interest

3. Total interest

4. Depreciation

5. Different changes in different years

6. Finding the number of years

7. Comparing investments

Recap questions

You've reached the end

Compound Interest and Depreciation

What you'll learn

1. Percentages as multipliers

2. Compound interest

3. Total interest

4. Depreciation

5. Different changes in different years

6. Finding the number of years

7. Comparing investments

Recap questions

You've reached the end

Compound Interest and Depreciation

What you'll learn

1. Percentages as multipliers

2. Compound interest

3. Total interest

4. Depreciation

5. Different changes in different years

6. Finding the number of years

7. Comparing investments

Recap questions

You've reached the end

Compound Interest and Depreciation Revision Guide

Compound Interest and Depreciation

What you'll learn

1. Percentages as multipliers

2. Compound interest

3. Total interest

4. Depreciation

5. Different changes in different years

6. Finding the number of years

7. Comparing investments

Recap questions

You've reached the end

Compound Interest and Depreciation Revision Guide