A fraction is just a way to name parts of a whole. Once you can see the picture, fractions stop being scary.
What a fraction really is
Take a fraction like \( \frac{3}{4} \). It has two parts stacked over a bar. The number on top is the numerator — it tells you how many parts you have. The number on the bottom is the denominator — it tells you how many equal parts the whole was cut into. So \( \frac{3}{4} \) means: cut the whole into \( 4 \) equal parts, then take \( 3 \) of them.
That little bar is doing two jobs at once. It reads as "out of" — \( 3 \) out of \( 4 \) parts — and it also means divide: \( \frac{3}{4} \) is exactly \( 3 \div 4 \). Both readings describe the same amount.
The word equal is the part everyone rushes past, and it matters most. A pizza sliced into four ragged, lopsided bits is not cut into quarters — quarters means four pieces of the same size. Only when the parts are equal does the denominator honestly describe them.
Drag the slider below to change how many parts you take, and watch the bar fill up:
The same amount, written many ways
Here is something surprising the first time you meet it: the very same amount can wear many different names. Look at these three fractions:
\[ \frac{1}{2} = \frac{2}{4} = \frac{3}{6} \]They look different, but they all point to exactly half of the whole. Fractions that name the same amount are called equivalent fractions.
The trick that connects them is simple: if you multiply the top and bottom by the same number — or divide them both by the same number — the value never changes. Starting from \( \frac{1}{2} \), multiply top and bottom by \( 2 \) to get \( \frac{2}{4} \); by \( 3 \) to get \( \frac{3}{6} \). Picture cutting every slice of half a pie into two: now there are twice as many pieces, each half the size, but you still have exactly the same amount of pie.
Simplest form
If you can scale a fraction up by multiplying, you can also scale it down by dividing — and that is how we tidy a fraction into its simplest form. You divide the top and bottom by their greatest common factor: the biggest number that goes evenly into both.
Take \( \frac{6}{8} \). Both \( 6 \) and \( 8 \) are divisible by \( 2 \), so divide each by \( 2 \):
\[ \frac{6}{8} = \frac{3}{4} \]Now \( 3 \) and \( 4 \) share no common factor other than \( 1 \), so we can go no further. A fraction is in simplest form exactly when its numerator and denominator share no common factor but \( 1 \). It is the same amount as before — just wearing its cleanest, shortest name.
Comparing fractions
Which is bigger? It depends on whether the denominators match.
When two fractions have the same denominator, the pieces are the same size, so you just count them — the bigger numerator wins. Clearly \( \frac{5}{8} > \frac{3}{8} \), because five eighth-sized pieces beat three of them.
When the denominators are different, the pieces are different sizes, so counting them isn't fair yet. First rewrite both fractions with a common denominator, then compare the numerators. Compare \( \frac{2}{3} \) and \( \frac{3}{5} \). A denominator both \( 3 \) and \( 5 \) divide into is \( 15 \):
\[ \frac{2}{3} = \frac{10}{15}, \qquad \frac{3}{5} = \frac{9}{15} \]Now the pieces are the same size, so we just count: \( 10 > 9 \), which means \( \frac{2}{3} \) is the larger of the two.
In words The denominator names the size of each piece — a bigger bottom number means smaller pieces. The numerator simply counts how many of those pieces you take. Read the bottom for size, the top for how many.
Fractions on the number line
Fractions aren't only slices of pie — each one is also a single point sitting between two whole numbers. To find \( \frac{3}{4} \), split the gap from \( 0 \) to \( 1 \) into four equal steps and walk three of them: you land three-quarters of the way along, just short of \( 1 \).
This view makes equivalence vivid. Because \( \frac{3}{4} \), \( \frac{6}{8} \) and \( \frac{9}{12} \) all name the same amount, they all land on the same point on the line. Different names, one location — that is what "equivalent" really means.
Tip To compare or add fractions, give them a common denominator first. Once every piece is the same size, the fractions are speaking the same language — and you can simply compare or count the pieces.
- List the factors. The greatest common factor of \( 12 \) and \( 18 \) is \( 6 \).
- Divide top and bottom by \( 6 \): \( \frac{12 \div 6}{18 \div 6} \).
- That gives \( \frac{2}{3} \), and \( 2 \) and \( 3 \) share no common factor but \( 1 \).
So \( \frac{12}{18} = \mathbf{\frac{2}{3}} \) in simplest form.
- The denominators differ, so rewrite \( \frac{3}{4} \) with denominator \( 8 \): multiply top and bottom by \( 2 \) to get \( \frac{6}{8} \).
- Now both pieces are eighths, so just compare the numerators: \( 6 \) versus \( 5 \).
- Since \( 6 > 5 \), the first fraction is bigger.
So \( \mathbf{\frac{3}{4}} \) is larger than \( \frac{5}{8} \).
Practice
Try each one yourself, then reveal the full solution.
1. Simplify \( \frac{10}{15} \).
Find the greatest common factor of \( 10 \) and \( 15 \). Both divide evenly by \( 5 \), and nothing larger does, so the GCF is \( 5 \).
Divide top and bottom by \( 5 \): \( \frac{10 \div 5}{15 \div 5} = \frac{2}{3} \). Now \( 2 \) and \( 3 \) share no common factor but \( 1 \).
So \( \frac{10}{15} = \mathbf{\frac{2}{3}} \).
2. Fill in the blank: \( \frac{2}{5} = \frac{\square}{20} \).
The denominator went from \( 5 \) to \( 20 \), and \( 20 = 5 \times 4 \). To keep the value the same, multiply the top by the same number.
Multiply top and bottom by \( 4 \): \( \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \).
So \( \square = \mathbf{8} \).
3. Which is larger, \( \frac{2}{3} \) or \( \frac{5}{9} \)?
The denominators differ, so rewrite \( \frac{2}{3} \) with denominator \( 9 \). Multiply top and bottom by \( 3 \): \( \frac{2 \times 3}{3 \times 3} = \frac{6}{9} \).
Now both are ninths, so compare numerators: \( 6 \) versus \( 5 \). Since \( 6 > 5 \), the first is bigger.
So \( \mathbf{\frac{2}{3}} \) is larger.