Equivalent Fractions

Look at these four strips. The first is cut into 2 pieces with 1 shaded, the next into 4 with 2 shaded, then 6 with 3, then 8 with 4. Every shaded region ends at exactly the same place down the middle, so all four show the same amount. That is what equivalent fractions are: different names for the same number. As the denominator grows the pieces get smaller, but you take more of them, and those two changes cancel out perfectly. So one half, two quarters, three sixths and four eighths are not four different amounts — they are one amount wearing four different labels. This matters because you get to choose whichever name makes a problem easiest.

Here is the rule that generates every equivalent fraction. Whatever you multiply the numerator by, multiply the denominator by the same thing. Start with two thirds and multiply both parts by 4: the top becomes 8, the bottom becomes 12, so two thirds equals eight twelfths. The reason this is allowed is that you are really multiplying by four over four, and four over four is just 1, so the value cannot change. Multiply two thirds by 2, 5 and 10 and you get four sixths, ten fifteenths and twenty thirtieths — all the same number. But be careful: adding the same number to top and bottom does not work. Two thirds plus 2 on each part gives four fifths, and four fifths is a different number entirely.

Going the other way is called simplifying, and it means dividing top and bottom by a common factor. Take twelve eighteenths. You can do it slowly: divide both by 2 to get six ninths, then both by 3 to get two thirds. Or do it in one move with the highest common factor. The factors of 12 are 1, 2, 3, 4, 6 and 12, and the factors of 18 are 1, 2, 3, 6, 9 and 18, so the highest they share is 6. Twelve divided by 6 is 2 and eighteen divided by 6 is 3, giving two thirds straight away. A fraction is in simplest form when the only factor top and bottom still share is 1, and unless a question says otherwise that is the form to give.

A very common question gives you one fraction and part of another, like three quarters equals question mark over twenty. Always start with the pair you can see completely — here the denominators, 4 and 20. Ask what you multiply 4 by to get 20. The answer is 5. Now do exactly the same to the top: 3 times 5 is 15, so three quarters equals fifteen twentieths. The same method works going down as well. For thirty forty-fifths equals question mark over 3, notice 45 divided by 15 is 3, so divide the top by 15 too, and 30 divided by 15 is 2, giving two thirds. The trap here is looking at the difference instead of the multiplier: 4 plus 16 is 20, but 3 plus 16 gives 19, which is wrong.

How do you check whether two fractions really are equivalent? Cross-multiply. For three quarters and nine twelfths, multiply 3 by 12 to get 36, and 4 by 9 to also get 36. The products match, so the fractions are equivalent. Try it on two thirds and five eighths: 2 times 8 is 16 but 3 times 5 is 15, and since those differ the fractions are not the same. The other check is to simplify both and see whether they land on the same fraction — nine twelfths divides down to three quarters, which agrees. This skill underpins everything that follows: you need equivalent fractions to compare sizes, you need them to add or subtract, and simplest form is what a final answer should look like.