Welcome to Fractions, Decimals, and Percentages!
In this chapter, we are going to explore the three different ways we can represent "parts of a whole." Whether you are looking at a half-price sale, measuring \( \frac{3}{4} \) of a cup of flour, or checking a 0.5 liter water bottle, you are using these skills! By the end of these notes, you’ll see that fractions, decimals, and percentages are just different outfits worn by the same numbers.
1. Understanding Fractions
A fraction represents a part of a whole. It is made of two numbers: the numerator (the top number) and the denominator (the bottom number).
Types of Fractions
- Proper Fractions: The top is smaller than the bottom (e.g., \( \frac{1}{2} \)).
- Improper Fractions: The top is larger than or equal to the bottom (e.g., \( \frac{7}{4} \)). These are sometimes called "top-heavy" fractions.
- Mixed Numbers: A whole number and a fraction together (e.g., \( 1 \frac{3}{4} \)).
The Four Operations with Fractions
Adding and Subtracting: You must have a common denominator!
Example: \( \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \).
If the denominators are different, find the Lowest Common Multiple (LCM) to make them the same.
Multiplying: This is the easiest one! Just multiply the tops and multiply the bottoms.
\( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)
Dividing: Use the KFC method!
1. Keep the first fraction.
2. Flip the second fraction (this is called the reciprocal).
3. Change the sign to multiply.
Quick Review: Simplest Form
Always check if you can divide the top and bottom by the same number to make the fraction smaller. For example, \( \frac{10}{20} \) simplifies to \( \frac{1}{2} \).
Key Takeaway: Fractions are just divisions. \( \frac{3}{4} \) literally means "3 divided by 4."
2. Decimals and Place Value
Decimals use place value to show parts of a whole. Each place to the right of the decimal point is 10 times smaller than the one before it.
- The first digit is tenths (\( \frac{1}{10} \)).
- The second digit is hundredths (\( \frac{1}{100} \)).
- The third digit is thousandths (\( \frac{1}{1000} \)).
Common Mistake Alert!
Don't be fooled by the length of a decimal! Students often think \( 0.125 \) is bigger than \( 0.4 \) because 125 is bigger than 4. However, \( 0.4 \) is actually \( 0.400 \). Since 4 tenths is bigger than 1 tenth, \( 0.4 \) is the larger number.
Terminating vs Recurring:
A terminating decimal ends (like \( 0.5 \) or \( 0.375 \)).
A recurring decimal repeats forever (like \( 0.333... \), written as \( 0.\dot{3} \)).
Key Takeaway: Treat decimals like money to make them easier to visualize. \( 0.5 \) is like 50p, while \( 0.05 \) is only 5p!
3. Percentages: Out of 100
The word percentage comes from "per cent," which means "for every 100."
Finding a Percentage of an Amount
You can use a multiplier to make this fast. To find \( 15\% \) of \( 80 \):
1. Turn the percentage into a decimal: \( 15\% = 0.15 \).
2. Multiply: \( 0.15 \times 80 = 12 \).
Percentage Increase and Decrease
To increase an amount by \( 20\% \), you are finding \( 120\% \) of the original (Multiplier = \( 1.20 \)).
To decrease an amount by \( 20\% \), you are finding \( 80\% \) of the original (Multiplier = \( 0.80 \)).
Did you know?
Percentages are reversible! \( 8\% \) of \( 50 \) is the exact same as \( 50\% \) of \( 8 \). (Both are 4). This trick can make mental maths much easier!
Key Takeaway: A percentage is just a fraction with a denominator of 100.
4. Converting Between the Three
Being able to switch between these is a "superpower" in GCSE Maths. Don't worry if this seems tricky at first; with practice, it becomes second nature.
Fraction to Decimal
Divide the top by the bottom.
Example: To turn \( \frac{3}{8} \) into a decimal, calculate \( 3 \div 8 = 0.375 \).
Decimal to Percentage
Multiply by 100 (move the decimal point two places to the right).
Example: \( 0.45 \times 100 = 45\% \).
Percentage to Fraction
Put the percentage over 100 and simplify.
Example: \( 60\% = \frac{60}{100} = \frac{6}{10} = \frac{3}{5} \).
Quick Review Box: Common Equivalents
Try to memorize these!
\( \frac{1}{2} = 0.5 = 50\% \)
\( \frac{1}{4} = 0.25 = 25\% \)
\( \frac{3}{4} = 0.75 = 75\% \)
\( \frac{1}{10} = 0.1 = 10\% \)
\( \frac{1}{3} = 0.\dot{3} = 33.3\% \)
5. Ordering Numbers (N1)
When an exam question asks you to "order from smallest to largest," the best trick is to convert them all into decimals first. It’s much easier to compare \( 0.5 \), \( 0.45 \), and \( 0.6 \) than it is to compare \( \frac{1}{2} \), \( 45\% \), and \( \frac{3}{5} \).
Example: Order \( \frac{3}{4} \), \( 0.8 \), and \( 70\% \).
1. Convert all: \( 0.75 \), \( 0.8 \), \( 0.7 \).
2. Compare: \( 0.7 \) is smallest, then \( 0.75 \), then \( 0.8 \).
3. Write in original form: \( 70\% \), \( \frac{3}{4} \), \( 0.8 \).
6. Real-World Applications
Household Finance (N2/R9)
You will often see these in "money" contexts:
- Interest: Extra money paid on a loan or earned on savings. Simple Interest means you calculate the percentage once and add that same amount every year.
- Profit/Loss: If you buy something for \( \$10 \) and sell it for \( \$15 \), you made a \( 50\% \) profit.
- VAT: A tax added to the price of items (usually \( 20\% \) in the UK).
Fractions and Ratios (N11)
Fractions and ratios are closely related. If the ratio of boys to girls is \( 2:3 \), there are \( 5 \) parts in total.
The fraction of boys is \( \frac{2}{5} \) and the fraction of girls is \( \frac{3}{5} \).
Key Takeaway: Always find the total number of parts when moving between ratios and fractions.
7. Higher Tier Only: Recurring Decimals to Fractions
If you are taking the Higher Tier, you need to know how to turn a decimal like \( 0.\dot{7} \) into a fraction algebraically.
Step-by-step:
1. Let \( x = 0.7777... \)
2. Multiply by 10 to move the decimal: \( 10x = 7.7777... \)
3. Subtract the first equation from the second: \( 10x - x = 7.7777 - 0.7777 \)
4. This gives: \( 9x = 7 \)
5. Solve for \( x \): \( x = \frac{7}{9} \)
Memory Aid: If one digit repeats, the denominator is 9. If two digits repeat (like \( 0.\dot{1}\dot{2} \)), the denominator is 99!
Summary: The "Big Three" Checklist
- Fractions: Use KFC for division and find a common denominator for adding.
- Decimals: Line up the decimal points when adding or subtracting.
- Percentages: Think "parts per hundred." Use multipliers for fast calculations.
- Ordering: Convert everything to decimals to compare them easily.