Equivalent Fractions
Two fractions with no digit in common can still be the same amount. That sounds like a trick until you see it once as a length, after which it stops being surprising at all.
Last updated
Ask someone whether 2/3 and 8/12 are the same number and you will usually get a pause. Nothing about them looks alike: not the top, not the bottom, not the size of the digits. And yet they are the same amount, exactly, with nothing rounded away.
The reason is easier to see than to say, so here it is as a length.
The dashed line never moves. Cutting the same length into more pieces changes both numbers and changes nothing about how much is shaded.
The shaded length is what does not change
The top bar is cut into 3 pieces with 2 of them shaded. The bottom bar is the same bar, cut into 6 pieces, with 4 shaded. Turn the multiplier up and the bottom bar is cut into 9, then 12, then 15 pieces — and the dashed line never moves.
That line is the whole idea. A fraction is a length, and the two numbers are only a description of how you cut it up. The bottom number says how many pieces the whole was cut into; the top number says how many of them you took. Cut every piece in half and you have twice as many pieces, each half the size, and you took twice as many of them. Two changes that cancel.
So 2/3 = 4/6 = 6/9 = 8/12, and the list does not stop — every fraction has infinitely many equivalent forms.
Making one: multiply top and bottom by the same number
To write a fraction a different way, multiply both parts by the same number:
2/3 = (2*4)/(3*4) = 8/12
The number you choose is up to you, and in real problems it is chosen for a reason — usually because you need a particular bottom number. If a question needs thirds written over 12, the multiplier is 4, because 3*4 = 12.
The one rule that matters: both parts, or neither. Multiplying only the top gives 8/3, which is four times too big. Move the top slider in the figure above and watch the dashed line jump — that is what changing one number on its own does.
Dividing works the same way, in reverse. 8/12 divided top and bottom by 4 is 2/3, and again nothing has moved.
Reading it backwards: reducing to lowest terms
Going down the list rather than up is called reducing, or simplifying, and it is what almost every answer is expected in. Divide the top and the bottom by their greatest common factor:
| fraction | common factor | lowest terms |
|---|---|---|
| 8/12 | 4 | 2/3 |
| 15/25 | 5 | 3/5 |
| 6/9 | 3 | 2/3 |
| 14/21 | 7 | 2/3 |
| 9/12 | 3 | 3/4 |
A fraction is in lowest terms when the only whole number dividing both parts is 1. Notice that three different-looking fractions in that table reduce to 2/3 — which is exactly the point: they were the same amount all along, written with different cuts.
If the common factor is not obvious, you can take it out in stages. 24/36 is even, so halve both to get 12/18; halve again for 6/9; then divide by 3 for 2/3. Getting there in three easy steps is no worse than getting there in one clever one.
Why they are needed at all
Equivalent fractions are not a topic on their own. They exist because adding fractions is impossible without them.
1/2 + 1/3 cannot be counted, because a half and a third are different-sized pieces. Rewrite both over 6 — 1/2 = 3/6 and 1/3 = 2/6 — and now they are pieces of one size, so they simply add: 3/6 + 2/6 = 5/6.
Every common-denominator step you will ever do is two equivalent fractions being written down. That is why this idea comes first.
Try one
If you got 9/12, you halved and stopped: 9 and 12 still share a factor of 3. Keep going until the only number dividing both is 1.
Common questions
- What are equivalent fractions?
- Two fractions are equivalent when they describe the same amount, even though the numbers look different. 2/3 and 8/12 are equivalent: cut a bar into 3 pieces and shade 2, or cut the same bar into 12 pieces and shade 8, and the shaded part reaches exactly the same place.
- How do I find an equivalent fraction?
- Multiply the top and the bottom by the same number. 2/3 becomes 4/6 (times 2), 6/9 (times 3), 8/12 (times 4), and so on forever. Multiplying both parts by the same number is the same as cutting every piece into that many smaller pieces, which changes the count of pieces but not how much you have.
- Why can I not just multiply the top?
- Because the top counts pieces and the bottom says how big a piece is. Changing only the top means taking more pieces of the same size, which really is more. You have to change the size of the piece at the same time, and multiplying the bottom by the same number is what does that.
- How do I reduce a fraction to lowest terms?
- Divide the top and the bottom by their greatest common factor. For 8/12 that factor is 4, so 8/12 becomes 2/3. A fraction is in lowest terms when the only number that divides both parts is 1, and that is the form almost every answer is expected in.
- What are equivalent fractions used for?
- Adding and subtracting. Pieces of different sizes cannot be counted together, so 1/2 + 1/3 goes nowhere until both are rewritten with the same bottom number: 3/6 + 2/6 = 5/6. Every one of those rewrites is an equivalent fraction, which makes this the idea the whole of fraction arithmetic is built on.
Want to run this on your own numbers?
Fraction Calculator Open the calculator