Mathematics from zero
Equivalent fractions
Cut a pizza in half: that is 1/2. Now cut both halves again — same pizza, but each
piece is a quarter, and you still have two of them: 2/4. The amount on your plate
never changed. Two different fractions named one amount.
After this lesson you can say what equivalent fractions are, build an equivalent fraction by multiplying top and bottom by the same number, simplify a fraction by dividing them, and recognise a fraction written in its simplest form.
Equivalent fractions name the same amount with different numbers. 1/2 and 2/4
and 4/8 all describe exactly half of a whole. They are equivalent fractions: the
numbers on the page differ, but the amount they point to is identical.
Multiply the numerator and denominator by the same number to build an equivalent
fraction. Take 1/2. Multiply both the top and the bottom by 3: the top becomes
1 × 3 = 3, the bottom 2 × 3 = 6. So 1/2 = 3/6. You cut every part into 3 smaller
parts — three times as many pieces, each a third the size — so the total is unchanged.
Divide the numerator and denominator by the same number to simplify. This is the
same move in reverse. Take 6/8. Both 6 and 8 can be divided by 2: the top becomes
6 ÷ 2 = 3, the bottom 8 ÷ 2 = 4. So 6/8 = 3/4. Dividing top and bottom by the
same number is called simplifying — fewer, larger pieces, same amount.
A fraction is in simplest form when no number above 1 divides both parts. Keep
dividing top and bottom by common numbers until none is left. 4/8 simplifies to
2/4, then to 1/2 — and 1 and 2 share no divider above 1, so 1/2 is the
simplest form. The simplest form is the cleanest name for that amount.
Simplify 12/18.
Look for a number that divides both 12 and 18. Both are even, so try 2: 12 ÷ 2 = 6
and 18 ÷ 2 = 9, giving 6/9. Both 6 and 9 divide by 3: 6 ÷ 3 = 2 and 9 ÷ 3 = 3,
giving 2/3.
Now 2 and 3 share no divider above 1, so 2/3 is the simplest form. 12/18 = 2/3.
A shortcut: 6 divides both 12 and 18 at once — 12 ÷ 6 = 2, 18 ÷ 6 = 3 — reaching
2/3 in a single step. Either way, the amount is unchanged.
Why this works
Why does multiplying top and bottom by the same number leave the value alone? Because you are doing the same thing to both halves of the fraction. Multiplying the denominator by 3 cuts every part into 3, making each piece a third the size; multiplying the numerator by 3 gives you 3 times as many of them. Three times as many pieces, each a third the size — the total is exactly the same.
Common mistake
The most common mistake is changing only one number — turning 1/2 into 1/4 “to get
a bigger denominator”, or adding a number to top and bottom instead of multiplying.
Both break the value. You must multiply (or divide) both the numerator and the
denominator, and by the same number. Anything else makes a different fraction.
1/2 = ?/6. Type the missing numerator.
2/3 = 8/?. Type the missing denominator.
Simplify 6/8. Type the numerator of the simplest form.
Simplify 6/8. Type the denominator of the simplest form.
Simplify 10/10. How many whole units does it equal? Type the number.
Which move turns a fraction into an equivalent one — same amount, different numbers?
Equivalent fractions name the same amount with different numbers. Multiply the numerator and denominator by the same number to build an equivalent fraction; divide them by the same number to simplify. A fraction is in simplest form when no number above 1 divides both parts. The rule is always the same: do it to both numbers, and by the same number — never add, never change just one.