Introduction: The Big Picture
Welcome to your study guide for Fractions, Decimals, and Percentages (often called FDP). In GCSE Mathematics, these three concepts are simply different ways of showing the exact same value. Think of them like three different languages describing the exact same thing: whether you say half a pizza, \(0.5\) of a pizza, or \(50\%\) of a pizza, you are eating the same amount of food!
Mastering FDP will boost your confidence across both your Gateway Unit (M1, M2, M3, or M4) and your Completion Unit (M5, M6, M7, or M8) papers. Don't worry if you have found these tricky in the past — we will break down every single process into simple, easy-to-follow steps.
1. Core Concepts & Definitions
Fractions
A fraction represents a part of a whole or a division. It is written in the form \(\frac{a}{b}\), where:
• Numerator (\(a\)): The top number — tells you how many parts you have.
• Denominator (\(b\)): The bottom number — tells you how many equal parts the whole is divided into (remember: \(b \neq 0\)).
There are three main types of fractions to know:
1. Proper Fraction: The numerator is smaller than the denominator (e.g. \(\frac{3}{4}\)).
2. Improper Fraction (Top-Heavy): The numerator is greater than or equal to the denominator (e.g. \(\frac{7}{4}\)).
3. Mixed Number: A whole number combined with a proper fraction (e.g. \(1\frac{3}{4}\)).
Decimals
A decimal is a number written using our base-10 place value system with a decimal point.
• Terminating Decimal: Has a finite number of digits after the decimal point and comes to an end (e.g. \(0.375 = \frac{3}{8}\)).
• Recurring Decimal: Has one or more repeating digits that go on forever. We show the repeating pattern using dot notation placed over the repeating digits (e.g. \(0.\dot{3} = \frac{1}{3}\), \(0.\dot{1}\dot{8} = \frac{2}{11}\)).
Percentages
The word percent means "per hundred" or "out of 100". The symbol \(\%\) simply means dividing by \(100\):
\(x\% = \frac{x}{100} = x \times 0.01\)
Key Takeaway: Fractions, decimals, and percentages are interchangeable representations of fractional amounts. \(\frac{1}{2} = 0.5 = 50\%\).
2. Conversions Between FDP
Converting Between Fractions and Decimals
• Fraction to Decimal: Divide the numerator by the denominator (\(a \div b\)).
Example: To convert \(\frac{3}{8}\), calculate \(3 \div 8 = 0.375\).
• Terminating Decimal to Fraction: Place the digits over the matching power of \(10\) (\(10\), \(100\), \(1000\), etc.) and simplify by dividing by the Highest Common Factor (HCF).
Example: \(0.45 = \frac{45}{100}\). Divide top and bottom by \(5\) to get \(\frac{9}{20}\).
Converting Between Decimals and Percentages
• Decimal to Percentage: Multiply by \(100\) (move the decimal point two places to the right) and add the \(\%\) sign.
Example: \(0.72 \times 100 = 72\%\).
• Percentage to Decimal: Divide by \(100\) (move the decimal point two places to the left) and drop the \(\%\) sign.
Example: \(6.5\% \div 100 = 0.065\).
Converting Between Percentages and Fractions
• Percentage to Fraction: Put the percentage value over \(100\) and simplify to lowest terms.
Example: \(35\% = \frac{35}{100} = \frac{7}{20}\) (after dividing numerator and denominator by \(5\)).
• Fraction to Percentage: Convert the fraction to a decimal first, then multiply by \(100\), or scale the denominator to \(100\).
Example: \(\frac{4}{25} = \frac{4 \times 4}{25 \times 4} = \frac{16}{100} = 16\%\).
Recurring Decimals to Fractions (Algebraic Method)
To convert a recurring decimal to an exact fraction, follow these algebraic steps:
Step 1: Set \(x\) equal to the recurring decimal: \(x = 0.\dot{1}\dot{8} = 0.181818...\)
Step 2: Multiply by a power of \(10\) (\(10\), \(100\), \(1000\)) matching the length of the repeating block. Here, two digits repeat, so multiply by \(100\):
\(100x = 18.181818...\)
Step 3: Subtract the original equation from the multiplied equation to cancel out the recurring decimal tail:
\(100x - x = 18.181818... - 0.181818...\)
\(99x = 18\)
Step 4: Solve for \(x\) and simplify the fraction:
\(x = \frac{18}{99} = \frac{2}{11}\) (dividing top and bottom by \(9\)).
Key Takeaway: For recurring decimals, multiplying by \(10^n\) aligns the repeating digits so they cancel out completely when subtracted.
3. Operations with Fractions
Addition and Subtraction of Fractions
You cannot add or subtract fractions unless they have a common denominator.
Step 1: If dealing with mixed numbers, convert them to improper fractions first.
Step 2: Find the Lowest Common Multiple (LCM) of the denominators.
Step 3: Create equivalent fractions with this common denominator:
\(\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}\)
Step 4: Add or subtract the numerators only. Keep the denominator the same, then simplify.
Example: Calculate \(\frac{2}{3} + \frac{1}{4}\)
• The LCM of \(3\) and \(4\) is \(12\).
• \(\frac{2 \times 4}{3 \times 4} + \frac{1 \times 3}{4 \times 3} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}\).
Multiplication of Fractions
Multiplying fractions is straightforward — multiply straight across:
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
Helpful Tip: Always convert mixed numbers to improper fractions first, and cancel common factors before multiplying to keep your numbers small!
Example: \(1\frac{1}{2} \times \frac{4}{5} = \frac{3}{2} \times \frac{4}{5} = \frac{12}{10} = \frac{6}{5} = 1\frac{1}{5}\).
Division of Fractions
To divide fractions, multiply by the reciprocal of the second fraction (turn the divisor upside down):
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\)
Memory Aid: Keep - Change - Flip (KCF)
• Keep the first fraction as it is.
• Change the \(\div\) sign to a \(\times\) sign.
• Flip the second fraction upside down.
Example: \(\frac{3}{5} \div \frac{9}{10} = \frac{3}{5} \times \frac{10}{9} = \frac{30}{45} = \frac{2}{3}\).
Key Takeaway: For addition and subtraction, get a common denominator. For multiplication, multiply straight across. For division, flip the second fraction and multiply.
4. Percentages & Financial Calculations
Finding a Percentage of an Amount
• Non-Calculator Method: Build up the target percentage using benchmark values:
- \(50\% = \text{divide by } 2\)
- \(10\% = \text{divide by } 10\)
- \(5\% = \text{half of } 10\%\)
- \(1\% = \text{divide by } 100\)
Example: Find \(15\%\) of \(£60\).
\(10\% = £6.00\), \(5\% = £3.00 \implies 15\% = £6 + £3 = £9.00\).
• Calculator / Multiplier Method: Convert the percentage into a decimal multiplier and multiply:
Example: \(34\%\) of \(£250 = 0.34 \times 250 = £85.00\).
Expressing One Quantity as a Percentage of Another
Write the part over the whole as a fraction and multiply by \(100\):
\(\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100\)
Example: A student scores \(36\) out of \(45\) in a test. What is their percentage mark?
\(\left(\frac{36}{45}\right) \times 100 = 0.8 \times 100 = 80\%\).
Percentage Change
To find the percentage increase or decrease:
\(\text{Percentage Change} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\)
Crucial Rule: Always divide by the Original Value, never the new value!
Percentage Multipliers for Increase and Decrease
• Increase by \(r\%\): Multiply by \(\left(1 + \frac{r}{100}\right)\)
Example: Increase \(£140\) by \(12\%\) \(\implies 140 \times 1.12 = £156.80\).
• Decrease by \(r\%\): Multiply by \(\left(1 - \frac{r}{100}\right)\)
Example: Decrease \(£80\) by \(15\%\) \(\implies 80 \times 0.85 = £68.00\).
Reverse Percentages (Finding the Original Value)
When you are given the final amount after a percentage increase or decrease, work backwards by dividing by the multiplier:
\(\text{Original Value} = \frac{\text{Final Value}}{\text{Multiplier}}\)
Example: A coat is on sale for \(£72\) after a \(20\%\) discount. Find its original price.
• A \(20\%\) discount means the sale price represents \(100\% - 20\% = 80\%\) of the original price.
• Multiplier \(= 0.80\).
• \(\text{Original Price} = \frac{72}{0.80} = £90.00\).
Compound Interest and Repeated Change
When an amount increases or decreases by a fixed percentage over regular intervals, use the compound formula:
\(\text{Total Amount} = P \left(1 \pm \frac{r}{100}\right)^n\)
Where:
• \(P\) = Principal (starting amount)
• \(r\) = Percentage rate per period
• \(n\) = Number of periods (years, months, etc.)
• Use \(+\) for growth/interest and \(-\) for decay/depreciation.
Example: \(£2000\) is invested at \(3\%\) compound interest per annum for \(4\) years.
\(\text{Total Amount} = 2000 \times (1 + 0.03)^4 = 2000 \times 1.03^4 = 2000 \times 1.12550881 = £2251.02\).
Key Takeaway: Always find the multiplier first. For reverse percentages, divide by the multiplier. For repeated change over time, raise the multiplier to the power of \(n\).
5. Common Pitfalls & Examiner Warnings
Avoid these common mistakes highlighted in CCEA examiner reports:
• Adding Denominators Together:
Incorrect: \(\frac{1}{3} + \frac{1}{4} = \frac{2}{7}\)
Correct: Find a common denominator first: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).
• Incorrect Reverse Percentages:
If an item costs \(£120\) after a \(20\%\) increase, do not subtract \(20\%\) of \(£120\). Instead, divide by the multiplier: \(120 \div 1.20 = £100\).
• Multiplying Mixed Numbers Directly:
Do not multiply whole numbers and fractions separately (e.g. \(2\frac{1}{2} \times 3\frac{1}{4} \neq 6\frac{1}{8}\)). Convert them to improper fractions first: \(\frac{5}{2} \times \frac{13}{4} = \frac{65}{8} = 8\frac{1}{8}\).
• Flipping the Wrong Fraction in Division:
Always flip the second fraction (the divisor), never the first.
• Writing Money Incorrectly:
Always write monetary amounts to exactly two decimal places unless whole pounds. Write \(£4.50\), never \(£4.5\).
• Confusing Recurring Decimals with Terminating Decimals:
Remember that \(0.\dot{3} = \frac{1}{3}\), whereas \(0.3 = \frac{3}{10}\). Do not round recurring decimals unless instructed.
Quick Review Checklist
Before sitting your exam, check that you can:
• Convert between any fraction, terminating decimal, and percentage.
• Use algebra to convert recurring decimals into fractions.
• Add, subtract, multiply, and divide proper fractions, improper fractions, and mixed numbers.
• Calculate percentage increases, decreases, and percentage changes.
• Solve reverse percentage problems using the division method.
• Calculate compound interest and depreciation using powers.