Reference · module 2
Fractions
Everything this module explains, on one page.
What a fraction is
Read fractions, find equal ones and simplify them.
- In 3/4 the bottom number, the denominator, says how many equal parts the whole is cut into: four. The top number, the numerator, says how many of those parts you have: three.
- Multiplying the top and bottom by the same number gives an equal fraction: 1/2 = 2/4 = 3/6. You cut the pieces smaller, but you have more of them, so the amount stays the same.
- Simplifying goes the other way: divide both by a common factor. 6/8 becomes 3/4 by dividing both by 2. A fraction is fully simplified when top and bottom share no factor except 1.
The mistake you are about to make
1/3 > 1/2
1/2 > 1/3
3 is bigger than 2, so one third sounds like more than one half. A bigger denominator means the whole is cut into more pieces, and each piece is smaller.
Adding fractions
Add and subtract fractions with different denominators.
- Fractions with the same denominator are easy: add the tops and keep the bottom. 1/5 + 2/5 is 3/5, because one fifth plus two fifths is three fifths.
- With different denominators, first rewrite both with a common one. For 1/2 + 1/3, use 6: 1/2 is 3/6 and 1/3 is 2/6, so the sum is 5/6.
- A mixed number is a whole number and a fraction: 2 1/4 means 2 + 1/4. As a single fraction it is 9/4, because two wholes are eight quarters, plus one more.
The mistake you are about to make
1/2 + 1/3 = 2/5
1/2 + 1/3 = 5/6
Adding tops and bottoms separately feels natural, since that is how the numbers are written. It gives 2/5, which is less than the 1/2 you started with.
Multiplying fractions
Multiply and divide fractions, and take a fraction of an amount.
- Multiplying fractions needs no common denominator: multiply the tops and multiply the bottoms. 2/3 × 3/4 is 6/12, which simplifies to 1/2.
- To divide by a fraction, multiply by it flipped upside down. 1/2 ÷ 1/4 is 1/2 × 4/1, which is 2 — and indeed a half holds two quarters.
- A fraction of a number means multiplying: 3/4 of 20 is 20 ÷ 4 × 3. One quarter of 20 is 5, so three quarters are 15.
The mistake you are about to make
6 ÷ 1/2 = 3
6 ÷ 1/2 = 12
Dividing by two and dividing by a half sound alike, and dividing usually makes things smaller. Asking how many halves fit into 6 gives twelve.