By Thread Academy · 16 September 2026 · Mathematics
Fractions appear in recipes, measurements, probabilities, and exam papers, yet they remain one of the topics students fear most. The fear is undeserved. Fractions follow a small set of consistent rules, and once you understand what a fraction means, every operation becomes a logical consequence of that meaning. This guide rebuilds fractions from the ground up.
What a fraction really means
A fraction describes a part of a whole. It has two numbers: the numerator on top, which counts how many parts you have, and the denominator on the bottom, which tells you how many equal parts the whole is divided into.
In the fraction 3/4, the denominator 4 says the whole is split into four equal parts, and the numerator 3 says you have three of those parts. Picture a pizza cut into four equal slices: 3/4 is three slices.
Fractions also describe division. The fraction 3/4 means 3 divided by 4, which is why 3/4 equals 0.75 as a decimal. Keeping both pictures in mind, parts of a whole and division, makes the rules below feel natural.
Equivalent fractions
Multiplying or dividing both the numerator and the denominator by the same number does not change the value of a fraction. The fractions you get are called equivalent fractions.
Think of the pizza again. Cutting each of the 4 slices in half gives 8 smaller slices, and your 3 original slices become 6 small ones. You still have the same amount of pizza:
1/2 = 2/4 = 3/6
Worked example 1: finding a missing numerator
Find the missing number: 3/5 = ?/20.
Step 1: Compare the denominators. To get from 5 to 20, multiply by 4.
Step 2: Do the same to the numerator to keep the value unchanged: 3 times 4 is 12.
Answer: 3/5 = 12/20.
Simplifying fractions
Simplifying, also called cancelling, means dividing the numerator and denominator by a common factor until no common factor remains except 1. A fraction in this state is in its lowest terms, and examiners expect answers in lowest terms.
Worked example 2: simplifying
Simplify 12/18.
Step 1: Find a common factor of 12 and 18. Both are divisible by 6.
Step 2: Divide top and bottom by 6: 12 divided by 6 is 2, and 18 divided by 6 is 3.
Answer: 2/3. Since 2 and 3 share no common factor, this is in lowest terms.
If you cannot spot the biggest factor immediately, simplify in stages. For 24/36, dividing by 2 gives 12/18, dividing by 2 again gives 6/9, and dividing by 3 gives 2/3. The result is the same.
Adding and subtracting fractions
You can only add or subtract fractions directly when they share the same denominator, because the parts must be the same size. If the denominators differ, convert to equivalent fractions with a common denominator first.
Worked example 3: adding with different denominators
Calculate 1/3 + 1/4.
Step 1: Find a common denominator. The smallest number both 3 and 4 divide into is 12.
Step 2: Convert each fraction: 1/3 = 4/12 and 1/4 = 3/12.
Step 3: Add the numerators, keeping the denominator: 4/12 + 3/12 = 7/12.
Answer: 7/12, which is already in lowest terms.
Worked example 4: subtracting
Calculate 5/6 - 1/3.
Step 1: Use 6 as the common denominator: 1/3 = 2/6.
Step 2: Subtract: 5/6 - 2/6 = 3/6.
Step 3: Simplify: 3/6 = 1/2.
Multiplying fractions
Multiplication is the simplest operation: multiply the numerators together, multiply the denominators together, then simplify. It often helps to cancel common factors before multiplying, which keeps the numbers small.
Worked example 5: straightforward multiplication
Calculate 2/3 x 3/5.
Step 1: Multiply numerators: 2 x 3 = 6. Multiply denominators: 3 x 5 = 15. This gives 6/15.
Step 2: Simplify by dividing top and bottom by 3: 2/5.
Worked example 6: cancelling first
Calculate 4/9 x 3/8.
Step 1: Look for cross-cancellations. The 4 and the 8 share a factor of 4, leaving 1 and 2. The 3 and the 9 share a factor of 3, leaving 1 and 3.
Step 2: Multiply what remains: 1/3 x 1/2 = 1/6.
Cancelling first avoids ever dealing with 12/72, which simplifies to the same answer with more effort.
Dividing fractions
To divide by a fraction, keep the first fraction, change the division to multiplication, and flip the second fraction upside down. This "keep, change, flip" rule works because dividing by a fraction is the same as multiplying by its reciprocal.
Worked example 7: dividing fractions
Calculate 2/3 divided by 4/5.
Step 1: Keep the first fraction as 2/3, change to multiplication, and flip the second to 5/4.
Step 2: Multiply: 2/3 x 5/4 = 10/12.
Step 3: Simplify: 10/12 = 5/6.
A common error is flipping the first fraction instead of the second. Division is not commutative, so only the divisor, the fraction after the division sign, gets flipped.
Mixed numbers
A mixed number combines a whole number with a fraction, such as 2 3/4. To use it in calculations, convert it to an improper fraction, where the numerator is larger than the denominator: multiply the whole number by the denominator, add the numerator, and keep the denominator.
Worked example 8: converting and adding
Calculate 1 1/2 + 2 1/3.
Step 1: Convert to improper fractions. 1 1/2 = 3/2, and 2 1/3 = 7/3.
Step 2: Find a common denominator of 6: 3/2 = 9/6 and 7/3 = 14/6.
Step 3: Add: 9/6 + 14/6 = 23/6.
Step 4: Convert back to a mixed number: 23 divided by 6 is 3 remainder 5, so 3 5/6.
Key takeaways
- A fraction is a part of a whole and also a division: the numerator counts parts, the denominator sets the part size.
- Multiplying or dividing top and bottom by the same number creates equivalent fractions without changing the value.
- Always give final answers in lowest terms by dividing out common factors.
- To add or subtract, first find a common denominator; to multiply, multiply straight across and cancel first when you can.
- To divide, keep the first fraction, change to multiplication, and flip the second fraction only.
- Convert mixed numbers to improper fractions before calculating, and convert back at the end.