Welcome to Fractions, Decimals, and Percentages (FDP)!
Hello future mathematician! This chapter is the foundation of the entire "Number" section. Don't worry if you've found these topics tricky before—we’re going to break them down into simple, manageable pieces.
Fractions, decimals, and percentages (FDP) are just three different ways of representing a part of a whole. You use them every day, whether you are calculating a discount in a shop, splitting a bill, or measuring ingredients!
Understanding how to move between these three forms is an essential skill, not just for your exams, but for real life. Let’s get started!
Section 1: Understanding Fractions
What is a Fraction?
A fraction represents a part of a whole. Think of it like slicing a pizza!
The standard fraction format is:
\[\frac{\text{Numerator}}{\text{Denominator}}\]
- The Numerator (the number on top) tells you how many parts you have.
- The Denominator (the number on the bottom) tells you how many equal parts the whole is divided into.
Example: In \(\frac{3}{4}\), you have 3 slices (Numerator) out of a pizza cut into 4 total slices (Denominator).
Simplifying Fractions (Finding Equivalent Fractions)
Often, fractions can be written in a simpler way without changing their value. This is called finding the simplest form or lowest terms.
Rule: To simplify, you must divide the numerator and the denominator by the same number.
The Trick: Find the Highest Common Factor (HCF) of the numerator and the denominator.
Step-by-Step Example: Simplifying \(\frac{12}{18}\)
- List the factors of 12: 1, 2, 3, 4, 6, 12.
- List the factors of 18: 1, 2, 3, 6, 9, 18.
- The HCF is 6.
- Divide both numbers by 6:
\[\frac{12 \div 6}{18 \div 6} = \frac{2}{3}\]
Key Takeaway: Simplifying fractions makes them easier to work with. If you can't divide the top and bottom by the same number anymore (except 1), the fraction is in its simplest form.
Section 2: Decimals and Percentages
Decimals
Decimals are another way to show parts of a whole, based on powers of ten (tenths, hundredths, thousandths, etc.).
Did you know? The word "decimal" comes from the Latin word "decimus," meaning "tenth."
Place Value Review:
0 . 4 5
- 4 is in the Tenths place (\(\frac{4}{10}\))
- 5 is in the Hundredths place (\(\frac{5}{100}\))
Percentages
A percentage is simply a fraction where the denominator is always 100. The term "per cent" literally means "out of 100".
The symbol \(\%\) replaces the words “out of 100”.
Example: 25% means \(\frac{25}{100}\).
Key Takeaway: Decimals use place value (tenths, hundredths), and percentages are fractions stuck over a denominator of 100.
Section 3: The Conversion Triangle (FDP)
Being able to switch quickly between Fractions, Decimals, and Percentages is vital for success in this chapter.
Conversion 1: Fraction to Decimal (F \(\rightarrow\) D)
Every fraction is a division problem waiting to happen!
Method: Divide the Numerator by the Denominator.
\[\text{Fraction} \rightarrow \text{Numerator} \div \text{Denominator}\]
Example 1: Convert \(\frac{3}{8}\) to a decimal.
\(3 \div 8 = 0.375\)
Example 2: Convert \(\frac{1}{3}\) to a decimal.
\(1 \div 3 = 0.3333... = 0.\dot{3}\) (This is a recurring decimal).
Conversion 2: Decimal to Percentage (D \(\rightarrow\) P)
Remember: Percentage means "out of 100". To go from a decimal (part of 1) to a percentage (part of 100), you multiply by 100.
Method: Multiply the decimal by 100 (move the decimal point two places to the right).
Memory Trick: Think P for Percentage, Push the decimal right!
Example: Convert 0.65 to a percentage.
\(0.65 \times 100 = 65\%\)
Conversion 3: Percentage to Decimal (P \(\rightarrow\) D)
This is the opposite of D \(\rightarrow\) P.
Method: Divide the percentage by 100 (move the decimal point two places to the left).
Memory Trick: Think D for Decimal, Depart the point left!
Example: Convert 12.5% to a decimal.
\(12.5 \div 100 = 0.125\)
Conversion 4: Decimal to Fraction (D \(\rightarrow\) F)
Use the decimal place value to create the fraction, then simplify.
Step 1: Look at the last digit's place value (tenths, hundredths, thousandths).
Step 2: Write the digits as the numerator, and the place value (10, 100, 1000) as the denominator.
Step 3: Simplify the resulting fraction.
Example: Convert 0.75 to a fraction.
Step 1 & 2: 75 is in the hundredths place, so we write \(\frac{75}{100}\).
Step 3: Simplify by dividing top and bottom by 25: \(\frac{75 \div 25}{100 \div 25} = \frac{3}{4}\).
Quick Review: Common FDP Equivalents
\(\frac{1}{2} = 0.5 = 50\%\)
\(\frac{1}{4} = 0.25 = 25\%\)
\(\frac{3}{4} = 0.75 = 75\%\)
\(\frac{1}{10} = 0.1 = 10\%\)
Key Takeaway: The decimal is the central hub. Most conversions require going through the decimal first (P \(\rightarrow\) D \(\rightarrow\) F, or F \(\rightarrow\) D \(\rightarrow\) P).
Section 4: Working with Fractions (Operations)
Now that you know what fractions are, let’s learn how to add, subtract, multiply, and divide them.
Adding and Subtracting Fractions
You can only add or subtract fractions if they have the same size pieces, meaning they must have the same denominator.
Analogy: You can't add \(\frac{1}{2}\) (a big slice) and \(\frac{1}{4}\) (a small slice) directly, unless you cut the big slice into quarters too!
Step-by-Step: \(\frac{1}{3} + \frac{1}{4}\)
- Find the Common Denominator: Find the Lowest Common Multiple (LCM) of 3 and 4. The LCM is 12.
- Change the Fractions: Convert both fractions so they have 12 as the denominator.
- \(\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}\)
- \(\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)
- Add (or Subtract) the Numerators:
\[\frac{4}{12} + \frac{3}{12} = \frac{4+3}{12} = \frac{7}{12}\]
- Simplify (if possible). \(\frac{7}{12}\) cannot be simplified further.
Common Mistake to Avoid: DO NOT add or subtract the denominators! The denominator just names the size of the slices; it doesn't change when you add them up.
Multiplying Fractions
Multiplication is actually the easiest operation!
Method: Multiply the numerators together, and multiply the denominators together.
\[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\]
Example: \(\frac{2}{5} \times \frac{1}{3}\)
\[\frac{2 \times 1}{5 \times 3} = \frac{2}{15}\]
Tip: Always simplify the answer at the end, or you can "cross-cancel" before multiplying if numbers share a common factor diagonally or vertically (this makes the numbers smaller and easier to handle!).
Dividing Fractions
Dividing fractions involves one simple extra step.
The KFC Rule (Mnemonic):
- Keep the first fraction the same.
- Flip (or invert) the second fraction (swap numerator and denominator).
- Change the division sign to multiplication.
Step-by-Step Example: \(\frac{1}{2} \div \frac{3}{4}\)
- Keep \(\frac{1}{2}\).
- Flip \(\frac{3}{4}\) to \(\frac{4}{3}\).
- Change \(\div\) to \(\times\).
- Now multiply: \(\frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6}\).
- Simplify: \(\frac{4 \div 2}{6 \div 2} = \frac{2}{3}\).
Key Takeaway: Addition/Subtraction need common denominators; Multiplication is straight across; Division uses the KFC trick.
Section 5: Working with Percentages
In this section, we focus on calculating a percentage of an amount, and expressing one amount as a percentage of another.
1. Finding the Percentage of an Amount
You have two reliable methods here. Choose the one that makes the most sense to you!
Method A: The Decimal Multiplier Method (Best for Calculator Use)
Step 1: Convert the percentage to a decimal (divide by 100).
Step 2: Multiply the amount by the decimal.
Example: Find 35% of 80.
Step 1: \(35\% \div 100 = 0.35\)
Step 2: \(0.35 \times 80 = 28\)
Method B: The Building Block Method (Best for Non-Calculator Use)
Use easy percentages (10%, 1%, 50%) to build up the required percentage.
Quick Rules for Building Blocks:
- To find 50%: Divide the amount by 2.
- To find 10%: Divide the amount by 10 (move decimal one place left).
- To find 1%: Divide the amount by 100 (move decimal two places left).
Example: Find 42% of 200 (Non-Calculator).
- Find 40%: \(10\%\) of \(200 = 20\). So, \(40\% = 4 \times 20 = 80\).
- Find 2%: \(1\%\) of \(200 = 2\). So, \(2\% = 2 \times 2 = 4\).
- Add the parts together: \(80 + 4 = 84\).
Encouraging Phrase: Don't worry if Method B feels slower at first. Practicing this method greatly improves your mental maths and understanding of number relationships!
2. Expressing One Quantity as a Percentage of Another
This is often used when calculating test scores, efficiency, or discounts.
The Formula: Put the part over the whole, and multiply the result by 100.
\[\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\]
Example: A student scores 45 marks out of a possible 60 marks. What percentage did they achieve?
\[\frac{45}{60} \times 100\]
Simplify the fraction first: \(\frac{45}{60} = \frac{3}{4}\) (dividing by 15).
\[\frac{3}{4} \times 100 = 0.75 \times 100 = 75\%\)
Key Takeaway: When finding a percentage of an amount, always convert the percentage to a decimal or use the building block method. When expressing A as a percentage of B, set up the fraction A/B first.