How to Solve: Simple and Compound Interest

How to Solve: Simple and Compound Interest

GCSE & A-Level Maths

Introduction

"Mastering simple and compound interest doesn’t just help you pass your exam—it helps you avoid losing thousands of pounds in real life. Banks, loans, and investments all use these formulas, and examiners love testing them. In GCSE Maths, this topic appears in at least 3-5 marks per paper, and in A-Level, it’s a gateway to financial maths questions worth 8-12 marks. Get this right, and you’re not just scoring points—you’re gaining financial superpowers."

What You Need To Know First

Before diving in, make sure you understand: 1. Percentage increase/decrease – How to calculate a % of a number and adjust values. 2. Rearranging formulas – Solving for any variable (e.g., finding time when given interest). 3. Indices (powers) – Especially for compound interest (e.g., (1.05^3)).

If any of these are shaky, pause and review them first.

Key Vocabulary

Formulas To Know

1. Simple Interest

Formula: [ I = P \times r \times t ] Variables: - ( I ) = Interest earned/paid (£) - ( P ) = Principal (£) - ( r ) = Annual interest rate (as a decimal, e.g., 5% = 0.05) - ( t ) = Time in years

MEMORISE THIS – Not always given on exam sheets.

Total Amount (A): [ A = P + I ] or [ A = P(1 + r \times t) ]

2. Compound Interest

Formula: [ A = P \times (1 + r)^t ] Variables: - ( A ) = Total amount after time ( t ) (£) - ( P ) = Principal (£) - ( r ) = Annual interest rate (as a decimal) - ( t ) = Time in years

MEMORISE THIS – Given on most exam sheets, but you must know how to use it.

Interest Earned (I): [ I = A - P ]

If compounded more than once per year (e.g., monthly, quarterly): [ A = P \times \left(1 + \frac{r}{n}\right)^{n \times t} ] - ( n ) = Number of times interest is compounded per year (e.g., 12 for monthly).

Given on exam sheet – But understand when to use it.

Step-by-Step Method

Simple Interest – Step-by-Step

1. Identify the variables – Write down ( P ), ( r ), and ( t ). Convert % to decimal (e.g., 5% → 0.05).
2. Plug into the formula – ( I = P \times r \times t ).
3. Calculate the interest – Multiply the numbers.
4. Find the total amount (if needed) – ( A = P + I ).

Compound Interest – Step-by-Step

1. Identify the variables – Write down ( P ), ( r ), and ( t ). Convert % to decimal.
2. Check compounding frequency – If not yearly, adjust ( r ) and ( t ) (e.g., monthly = ( r/12 ), ( t \times 12 )).
3. Plug into the formula – ( A = P \times (1 + r)^t ).
4. Calculate the total amount – Use a calculator for the power.
5. Find the interest (if needed) – ( I = A - P ).

Worked Examples

Example 1 – Basic Simple Interest

Question: £800 is invested at 3% simple interest per year for 4 years. Calculate the total interest earned.

Step-by-Step Solution: 1. Identify variables:
- ( P = £800 )
- ( r = 3\% = 0.03 )
- ( t = 4 ) years 2. Plug into formula:
( I = P \times r \times t )
( I = 800 \times 0.03 \times 4 ) 3. Calculate:
( I = 800 \times 0.12 = £96 ) 4. Answer: The total interest earned is £96.

What we did and why: - We used the simple interest formula because the question specified simple interest. - We converted the percentage to a decimal (3% → 0.03) before multiplying. - The time was already in years, so no conversion was needed.

Example 2 – Medium Compound Interest

Question: £1,200 is invested at 4% compound interest per year for 5 years. Calculate the total amount after 5 years.

Step-by-Step Solution: 1. Identify variables:
- ( P = £1,200 )
- ( r = 4\% = 0.04 )
- ( t = 5 ) years 2. Plug into formula:
( A = P \times (1 + r)^t )
( A = 1,200 \times (1 + 0.04)^5 ) 3. Calculate inside brackets first:
( 1 + 0.04 = 1.04 ) 4. Apply the power:
( 1.04^5 = 1.2166529 ) (use calculator) 5. Multiply by principal:
( A = 1,200 \times 1.2166529 = £1,459.98 ) (rounded to 2 d.p.) 6. Answer: The total amount after 5 years is £1,459.98.

What we did and why: - We used the compound interest formula because the question specified compound interest. - We calculated ( (1 + r)^t ) first, then multiplied by ( P ). - Rounding to 2 decimal places is standard for money.

Example 3 – Exam-Style (Disguised Question)

Question: Jamie invests £2,500 in a savings account. The account pays 2.5% compound interest per year. After 3 years, Jamie withdraws the money. How much more interest would Jamie have earned if the interest was compounded monthly instead of yearly?

Step-by-Step Solution: Part 1: Yearly Compounding 1. Identify variables:
- ( P = £2,500 )
- ( r = 2.5\% = 0.025 )
- ( t = 3 ) years 2. Plug into formula:
( A = 2,500 \times (1 + 0.025)^3 ) 3. Calculate:
( 1.025^3 = 1.076890625 )
( A = 2,500 \times 1.076890625 = £2,692.23 ) 4. Interest earned:
( I = 2,692.23 - 2,500 = £192.23 )

Part 2: Monthly Compounding 1. Adjust variables:
- ( r = 0.025 / 12 ) (monthly rate)
- ( t = 3 \times 12 = 36 ) months 2. Plug into formula:
( A = 2,500 \times \left(1 + \frac{0.025}{12}\right)^{36} ) 3. Calculate:
( 1 + \frac{0.025}{12} = 1.0020833 )
( 1.0020833^{36} = 1.077605 )
( A = 2,500 \times 1.077605 = £2,694.01 ) 4. Interest earned:
( I = 2,694.01 - 2,500 = £194.01 )

Part 3: Difference in Interest 1. Subtract:
( 194.01 - 192.23 = £1.78 ) 2. Answer: Jamie would earn £1.78 more with monthly compounding.

What we did and why: - The question was disguised—it didn’t explicitly say "compound interest," but the mention of "compounded monthly" gave it away. - We had to adjust the rate and time for monthly compounding. - We calculated both scenarios and found the difference, which is a common exam trap.

Common Mistakes

Exam Traps

1-Minute Recap

"Right, listen up—this is your last-minute cram. Simple interest is just ( I = P \times r \times t ). Convert the percentage to a decimal, multiply, and you’re done. Compound interest is ( A = P \times (1 + r)^t )—same deal, but you’re raising to a power. If it’s compounded monthly or quarterly, divide the rate and multiply the time. Always check if the question wants the total amount or just the interest—subtract ( P ) if it’s interest. Watch out for time units—years, months, days? Convert them. And don’t round until the very end. That’s it. Go smash those questions."