A fraction represents a part of a whole. It is written as one number over another. The top number is the numerator — it tells you how many parts you have. The bottom number is the denominator — it tells you how many equal parts the whole has been divided into.
Example 1 — Identify the fraction shaded
Step 1
Count the total number of equal parts: 8 — this is the denominator.
Step 2
Count the shaded parts: 3 — this is the numerator.
Step 3
Write the fraction: \(\frac{3}{8}\)
Example 2 — Write \(\frac{7}{10}\) in words and place on a number line
Step 1
In words: seven tenths. Numerator = 7 (parts you have), denominator = 10 (equal parts in the whole).
Step 2
\(\frac{7}{10} = 0.7\) as a decimal — it sits between \(\frac{1}{2}\) (0.5) and \(\frac{3}{4}\) (0.75), closer to \(\frac{3}{4}\).
Step 3
On a 0–1 number line, place a dot 7 tenths of the way from 0 to 1 — just past the ¾ mark.
Memory tip — Denominator vs Numerator: Think of the Denominator as the number that Divides the whole up. The Numerator is the Number on top that you are counting.
✓ Quick Check 1 — Numerator and Denominator
Question 1 of 10
Proper Fractions, Improper Fractions and Mixed Numbers
Not all fractions represent a value less than 1. Once you understand the three types, you can confidently work with any fraction you encounter.
Type
Definition
Example
Value
Proper fraction
Numerator < denominator
\(\frac{3}{7}\)
Less than 1
Improper fraction
Numerator ≥ denominator
\(\frac{9}{4}\)
1 or more
Mixed number
Whole number + proper fraction
\(3\frac{1}{5}\)
More than 1
Example 3 — Identify the type of each: \(\frac{2}{7}\), \(\frac{9}{4}\), \(3\frac{1}{5}\)
\(3\frac{1}{5}\): whole number 3 plus a fraction part → mixed number
Improper fractions are not errors! They are a perfectly valid — and often more useful — way to write values greater than 1. Many calculations are actually easier with improper fractions than with mixed numbers.
Converting between improper fractions and mixed numbers gets its own full lesson — see Lesson 4.4 — Mixed Numbers and Improper Fractions.
✓ Quick Check 2 — Proper, Improper and Mixed Numbers
Question 1 of 10
Fractions of Quantities
Finding a fraction of a quantity is one of the most useful fraction skills in everyday life — discounts, recipes, distances, wages. The method is always the same: divide by the denominator first, then multiply by the numerator.
\(\frac{a}{b}\) of \(Q\;\) = \(\;Q \div b \times a\)
Divide the quantity by the denominator (bottom), then multiply by the numerator (top)
Example 5 — Find \(\frac{2}{5}\) of 40
Step 1
Divide by the denominator: \(40 \div 5 = 8\)
Step 2
Multiply by the numerator: \(8 \times 2 = 16\)
Step 3
Answer: \(\frac{2}{5}\) of \(40 = \mathbf{16}\)
Remember: divide first! Always divide the quantity by the denominator before multiplying by the numerator. This keeps the numbers small and avoids large multiplications.
Fractions of a quantity get their own full lesson with more real-life worked examples — see Lesson 4.8 — Finding a Fraction of an Amount.
✓ Quick Check 3 — Fractions of Quantities
Question 1 of 10
Practice: Fractions of Quantities
Work out each fraction of the given quantity. Type your numeric answer then click Check All.
#
Question
Answer
Result
1
\(\frac{1}{2}\) of 30
2
\(\frac{1}{4}\) of 48
3
\(\frac{3}{4}\) of 60
4
\(\frac{2}{5}\) of 35
5
\(\frac{3}{8}\) of 64
6
\(\frac{2}{3}\) of 90
7
\(\frac{5}{6}\) of 48
8
\(\frac{7}{10}\) of 120
9
\(\frac{4}{9}\) of 63
10
\(\frac{5}{8}\) of 72
⚠ Things to Watch Out For
Adding denominators: ½ + ½ ≠ 2/4. When adding fractions with the same denominator, add the numerators ONLY. The denominator stays the same: ½ + ½ = 2/2 = 1.
Confusing numerator and denominator: The numerator (top) is the number of parts you have. The denominator (bottom) is how many equal parts the whole is split into.
Thinking a larger denominator means a larger fraction: ⅛ is SMALLER than ½, even though 8 > 2. The larger the denominator, the smaller each individual piece.
Fractions of amounts — dividing by numerator: To find ¾ of 24: divide by the denominator (÷4=6), then multiply by the numerator (×3=18). Students sometimes divide by the numerator instead.
Improper fractions are not "wrong": 7/3 is a perfectly valid fraction — it just means more than one whole. It is equal to 2⅓.