Welcome to Percentage Change and Simple Interest!

Have you ever seen a sign in a shop window screaming "20% off all trainers!", or heard a news reporter talk about a "5% pay rise"? Percentages are everywhere in the world around us, especially when we talk about shopping, wages, and bank accounts.

In this chapter, you will learn how to calculate increases and decreases, discover how to work backwards to find an original price, and understand how simple interest works in everyday financial mathematics. Don't worry if maths sometimes feels overwhelming — we will break every single idea down into small, friendly steps!

Note: This chapter is part of Ratio, Proportion and Rates of Change. For other topics such as Unit Conversions or Direct and Inverse Proportion, take a look at their dedicated chapters.

1. Quick Refresh: What is a Percentage?

The word percent comes from the Latin per centum, meaning "out of one hundred". Think of a percentage as a way of describing a number of parts per hundred.

Because percentages mean "out of \(100\)", we can easily write any percentage as a fraction or a decimal:

• \(50\%\) means \(\frac{50}{100}\), which simplifies to \(\frac{1}{2}\), or as a decimal: \(0.5\)

• \(25\%\) means \(\frac{25}{100}\), which simplifies to \(\frac{1}{4}\), or as a decimal: \(0.25\)

• \(10\%\) means \(\frac{10}{100}\), which simplifies to \(\frac{1}{10}\), or as a decimal: \(0.1\)

• \(100\%\) means the entire whole: \(\frac{100}{100} = 1\)

Percentages as Multipliers (Operators):
A percentage can act as an operator on a quantity. To find a percentage of an amount, you can simply multiply that amount by the decimal form. For example, to find \(35\%\) of \(£80\), you can calculate \(£80 \times 0.35 = £28\).

Percentages Greater Than \(100\%\):
Can a percentage be bigger than \(100\%\)? Yes! If a quantity is more than the original whole, it represents more than \(100\%\). For instance, \(150\%\) of a quantity means the original whole plus another half of it (\(1.5\) times the original).

Key Takeaway: Always remember that the original whole amount is your starting baseline of \(100\%\).

2. Percentage Increase and Percentage Decrease

When an amount gets bigger, we call it a percentage increase. Examples include price rises, population growth, or salary increases.
When an amount gets smaller, we call it a percentage decrease. Examples include sale discounts, price cuts, or weight loss.

Method 1: The Step-by-Step Addition/Subtraction Method

This method is fantastic for mental maths or non-calculator questions.

Step 1: Calculate the percentage of the amount.
Step 2: For an increase, add that value to the original amount. For a decrease, subtract that value from the original amount.

Worked Example 1 (Increase): A board game normally costs \(£40\). Its price increases by \(15\%\). Find the new price.

Step 1: Find \(15\%\) of \(£40\).
\(10\%\) of \(£40 = £4\)
\(5\%\) of \(£40 = £2\)
So, \(15\%\) of \(£40 = £4 + £2 = £6\).

Step 2: Add this increase to the original cost:
\(\text{New Price} = £40 + £6 = £46\).

Worked Example 2 (Decrease): A coat costs \(£80\) but is reduced by \(30\%\) in a summer sale. Find the sale price.

Step 1: Find \(30\%\) of \(£80\).
\(10\%\) of \(£80 = £8\)
\(30\%\) of \(£80 = 3 \times £8 = £24\).

Step 2: Subtract the discount from the original price:
\(\text{Sale Price} = £80 - £24 = £56\).

Method 2: The Single Multiplier Method

This method is super-fast, especially when using a calculator!

Instead of finding the change separately, we look at the whole picture using our baseline of \(100\%\):

• For an increase: Add the percentage to \(100\%\), convert to a decimal multiplier, and multiply.
Example: An increase of \(15\%\) means the new amount is \(100\% + 15\% = 115\%\). The multiplier is \(1.15\).
\(£40 \times 1.15 = £46\).

• For a decrease: Subtract the percentage from \(100\%\), convert to a decimal multiplier, and multiply.
Example: A decrease of \(30\%\) means the new amount is \(100\% - 30\% = 70\%\). The multiplier is \(0.70\) (or \(0.7\)).
\(£80 \times 0.7 = £56\).

Common Mistake to Avoid: Be careful with single-digit percentages! An increase of \(4\%\) gives \(100\% + 4\% = 104\%\), so the multiplier is \(1.04\), not \(1.4\) (which would be an increase of \(40\%\)).

Key Takeaway: Every increase has a multiplier greater than \(1\); every decrease has a multiplier less than \(1\).

3. Calculating Percentage Change

What if you know the old value and the new value, and you want to work out what percentage increase or decrease has occurred? We use the percentage change formula:

\(\text{Percentage Change} = \frac{\text{Change}}{\text{Original Value}} \times 100\%\)

Here, the Change is simply the difference between the new value and the original value (found by subtracting the smaller from the larger).

Worked Example 1: A runner's time for a \(5\text{ km}\) run improves from \(25\text{ minutes}\) down to \(22\text{ minutes}\). What is the percentage decrease in her time?

Step 1: Find the actual change (difference):
\(\text{Change} = 25 - 22 = 3\text{ minutes}\).

Step 2: Divide the change by the original value:
\(\frac{\text{Change}}{\text{Original}} = \frac{3}{25}\).

Step 3: Multiply by \(100\%\) to make it a percentage:
\(\frac{3}{25} \times 100\% = 12\%\).
Her time decreased by \(12\%\).

Worked Example 2: A small potted plant was \(16\text{ cm}\) tall. A month later, it measures \(20\text{ cm}\). Calculate the percentage increase.

Step 1: \(\text{Change} = 20 - 16 = 4\text{ cm}\).
Step 2: Put the change over the starting height: \(\frac{4}{16} = \frac{1}{4}\).
Step 3: Convert to a percentage: \(\frac{1}{4} \times 100\% = 25\%\).
The plant's height increased by \(25\%\).

Crucial Rule: Always divide by the original value (the starting amount), never by the new value!

Key Takeaway: Percentage Change compares the amount gained or lost back to where you started.

4. Original Value Problems (Reverse Percentages)

Sometimes you are given the final price after a sale or after a price hike, and asked: "What was the price before the change happened?"
These are known as original value problems or reverse percentage problems.

Don't worry if this seems tricky at first — there is one golden rule to keep you safe:

The Golden Rule: The original value is always \(100\%\). You CANNOT simply find the percentage of the new value and add or subtract it!

Method 1: The Unitary Method (Finding \(1\%\) First)

Step 1: Identify what percentage the new amount represents.
• If it was increased by \(20\%\), the new amount is \(120\%\).
• If it was decreased by \(20\%\), the new amount is \(80\%\).
Step 2: Divide by that percentage number to find what \(1\%\) is worth.
Step 3: Multiply by \(100\) to find \(100\%\) (the original value).

Worked Example 1 (Sale price): A jacket is on sale for \(£72\) after a \(10\%\) discount. What was its original price?

Step 1: The original price was \(100\%\). Since there was a \(10\%\) discount:
\(\text{Sale Price} = 100\% - 10\% = 90\%\).
So, \(90\% = £72\).

Step 2: Find \(1\%\):
\(1\% = £72 \div 90 = £0.80\).

Step 3: Find \(100\%\):
\(100\% = £0.80 \times 100 = £80\).
The original price was \(£80\).

Check: Let's test our answer. \(10\%\) of \(£80\) is \(£8\). Subtracting \(£8\) gives \(£80 - £8 = £72\). It matches!

Method 2: Dividing by the Multiplier

Remember that going forward, we multiply: \(\text{Original} \times \text{Multiplier} = \text{New Amount}\).
To work backwards, we use the inverse operation: divide!

\(\text{Original Amount} = \frac{\text{New Amount}}{\text{Multiplier}}\)

Worked Example 2 (Price rise): With a \(25\%\) pay rise, Samir now earns \(£250\) per week. How much did he earn before the rise?

Step 1: A \(25\%\) increase gives a multiplier of \(1 + 0.25 = 1.25\).
Step 2: Divide the new wage by the multiplier:
\(\text{Original Wage} = £250 \div 1.25 = £200\).
Samir originally earned \(£200\) per week.

Common Trap: If an item costs \(£90\) after a \(10\%\) discount, students often mistakenly calculate \(10\%\) of \(£90 = £9\), and add it to get \(£99\). That is incorrect! The discount was \(10\%\) of the original unknown price, not \(10\%\) of \(£90\). Always set the original price as \(100\%\).

Key Takeaway: To find the original amount, either find \(1\%\) and multiply by \(100\), or divide the new value by the decimal multiplier.

5. Simple Interest in Financial Mathematics

When you keep money in a savings account at a bank, the bank pays you an extra reward called interest. Similarly, if you borrow money, you have to pay back the amount you borrowed plus extra interest.

In simple interest, the interest paid or earned is calculated only on the original amount of money (called the principal). The amount of interest earned each year stays exactly the same!

Key Terms in Simple Interest

• Principal (\(P\)): The starting amount of money deposited or borrowed.
• Rate (\(R\)): The percentage of the principal added each year (given as a percentage per annum, or p.a.).
• Time (\(T\)): The number of years the money is kept or borrowed for.
• Total Balance: The final amount in the account, which is: \(\text{Principal} + \text{Total Interest}\).

Calculating Simple Interest Step-by-Step

Step 1: Work out the interest for one single year by calculating the given percentage of the principal.
Step 2: Multiply that one-year interest by the number of years.
Step 3: If the question asks for the total balance, add the total interest to the original principal.

Worked Example 1: Mia deposits \(£500\) into a savings account that pays \(3\%\) simple interest per year. How much interest does she earn in \(4\text{ years}\)? What is the total amount in her account?

Step 1: Find the interest for 1 year:
\(1\%\) of \(£500 = £500 \div 100 = £5\).
\(3\%\) of \(£500 = 3 \times £5 = £15\text{ per year}\).

Step 2: Find the interest for 4 years:
\(\text{Total Interest} = £15 \times 4 = £60\).

Step 3: Find the total balance:
\(\text{Total Balance} = \text{Principal} + \text{Interest} = £500 + £60 = £560\).

Worked Example 2: David borrows \(£1200\) from a lender at a simple interest rate of \(5\%\) per year. He repays the whole loan after \(3\text{ years}\). How much money does he pay back in total?

Step 1: Interest for 1 year: \(5\%\) of \(£1200 = 0.05 \times £1200 = £60\).
Step 2: Interest for 3 years: \(£60 \times 3 = £180\).
Step 3: Total amount to pay back: \(£1200 + £180 = £1380\).

The Simple Interest Formula

You can also summarise these steps using a handy formula:

\(\text{Interest} = \frac{P \times R \times T}{100}\)

Where:
• \(P =\) Principal (starting money)
• \(R =\) Rate of interest (as a percentage number, e.g. \(5\) for \(5\%\))
• \(T =\) Time in years

Using David's example above: \(\text{Interest} = \frac{1200 \times 5 \times 3}{100} = \frac{18000}{100} = £180\). Both methods give the exact same answer!

Key Takeaway: With simple interest, the interest amount never changes from year to year because it is always calculated from the original starting amount.

Summary Checklist: Are You Exam-Ready?

• Percentage Increase: Calculate the percentage and add it to the original, or multiply by \((1 + \text{decimal})\).
• Percentage Decrease: Calculate the percentage and subtract it from the original, or multiply by \((1 - \text{decimal})\).
• Percentage Change: Use \(\frac{\text{Change}}{\text{Original}} \times 100\%\). Always divide by the starting value!
• Original Value (Reverse): Treat the original amount as \(100\%\). Find \(1\%\) first and multiply by \(100\), or divide the new value by the multiplier.
• Simple Interest: Find the interest for 1 year, multiply by the number of years, and add to the principal if you need the final total balance.