Percentage Basics and Interest

Ever wondered why shops can advertise "20% OFF" and you immediately know it's a good deal? Percentages help us understand proportional changes quickly.

To calculate percentages with a calculator, use decimal multipliers. For example, to find 27% of 90, multiply 90 × 0.27 = 24.3. For percentage increases, multiply by a value greater than 1 (e.g., increasing by 20% means multiplying by 1.2). For decreases, multiply by a value less than 1 (e.g., decreasing by 20% means multiplying by 0.8).

When comparing interest types, simple interest adds the same amount each year (like £400 on an £8000 investment at 5%), while compound interest calculates interest on both the original amount and previously earned interest $£8000 × 1.05³ = £9261 after 3 years$. Compound interest always outperforms simple interest over longer periods!

Quick Tip: For mental calculations, use percentage rules like 50% = ½, 25% = ¼, 10% = 1/10, and 1% = 1/100 to break down complex problems into manageable steps.

Understanding Interest Types

Simple and compound interest might seem similar at first, but they behave quite differently over time.

Simple interest adds the same fixed amount each year, calculated only on the original investment. It grows in a straight line on a graph and is easier to calculate mentally.

Compound interest calculates interest on both your original investment and any previously earned interest. This creates an exponential growth curve that starts slowly but accelerates dramatically over time.

The difference becomes striking over longer periods. For example, after 20 years, an £8000 investment at 5% would be worth £16,000 with simple interest, but a whopping £21,226.38 with compound interest!

Remember this: Albert Einstein reportedly called compound interest "the eighth wonder of the world." It's why starting to save early, even with small amounts, can be so powerful!

Repeated Percentage Changes

Have you ever wondered how your money grows when you leave it invested for several years? This is where repeated percentage changes come in.

When calculating multiple percentage changes in sequence, simply chain the decimal multipliers together. For example, if £2000 increases by 13%, then decreases by 10%, then increases by 7%, calculate: 2000 × 1.13 × 0.9 × 1.07 = £2176.4.

For compound interest over multiple years, we use the formula: Total = P$1 + r/100$^n, where P is the principal amount (initial investment), r is the interest rate percentage, and n is the number of years. For example, £100 invested at 3% compound interest for 7 years equals £100 × 1.03^7 = £122.99.

Pro tip: When doing calculations with percentages on your calculator, always convert to decimal multipliers first - it's much faster and less prone to errors than working with percentages directly!

Reverse Percentages

You've seen a sale sign advertising "20% OFF" and the new price is £64. How do you work out the original price? This is where reverse percentages come in handy.

One approach is to reason that if 80% equals £64 (after the 20% reduction), then 20% equals £16, making the original price £80. This method works by identifying what percentage the final amount represents.

There's also a quick "cheat" method using decimal multipliers. To undo a percentage increase, divide by the decimal multiplier (e.g., to undo a 20% increase, divide by 1.2). To undo a percentage decrease, divide by the decimal multiplier (e.g., to undo a 15% decrease, divide by 0.85).

Test yourself: A t-shirt costs £35 after a 30% increase. What was the original price? $Answer: £35 ÷ 1.3 = £26.92$

Percentage Growth Problems

Population growth, investment returns, and inflation all involve repeated percentage changes over time. These problems test your ability to apply percentage calculations to real situations.

When solving problems about repeated percentage growth, use the formula with decimal multipliers: Initial Value × $1 + rate$^number of periods. For example, if 1300 rabbits increase by 5% every week, after 14 weeks the population would be 1300 × 1.05^14 ≈ 2574 rabbits.

A clever shortcut for determining if a quantity will double is to check if $1 + rate$^periods ≥ 2. In our rabbit example, 1.05^14 = 1.97, which is less than 2, so the population won't quite double in 14 weeks.

Interesting fact: The "Rule of 72" gives a quick estimate of how many periods it takes for something to double at a given percentage growth rate: divide 72 by the percentage rate. For 5% growth, it takes approximately 72 ÷ 5 = 14.4 periods to double!

Iteration

Iterations are like mathematical recipes that tell you how to get from one term to the next in a sequence. They're incredibly useful in computer programming and modelling real-world scenarios.

In iteration, we use the notation U₁ for the first term, and U_{n+1} for the next term in the sequence. The rule U_{n+1} = 5U_n + 1 means "to get the next term, multiply the current term by 5 and add 1." Starting with U₁ = 4, we calculate U₂ = 5 × 4 + 1 = 21, then U₃ = 5 × 21 + 1 = 106, and so on.

Different iteration rules create different patterns. For example, U_{n+1} = U_n/4 with U₁ = 32 creates a decreasing sequence: 32, 8, 2, 0.5, etc. This particular sequence divides each term by 4 to get the next.

Applied thinking: Iterations are used to model population growth, compound interest, and even weather forecasts! They allow mathematicians and scientists to see how systems evolve over time when following specific rules.

Percentage Problem Solving

Real-life percentage problems often involve multiple steps and require careful organisation of information. These skills are directly applicable to budget planning, business decisions, and data analysis.

When solving complex percentage problems, start by identifying what you know and what you need to find. Create a table to organise the information clearly. For example, in a school trip problem with 200 people (90% pupils, 10% teachers), and 3/10 of all people from secondary school, a table showing the breakdown helps visualise the solution.

Breaking down the information step by step helps tackle what seems complicated at first. If teachers from primary and secondary schools are in ratio 4:1, we can work out that there must be 16 primary teachers and 4 secondary teachers, then fill in the rest of our table.

Exam tip: In percentage problem-solving questions, double-check your final answer by verifying that all your values add up correctly both vertically and horizontally in your table. This quick check can help catch calculation errors!