Equivalent fractions
Different fractions can name exactly the same amount. Multiplying or dividing the top and bottom by the same number keeps the value unchanged.
The idea in one picture
Two fractions are equivalent when they describe the same portion of the same whole, even though the numbers look different.
Here is one half, two quarters and four eighths on bars of the same length. The pieces get smaller and there are more of them, but the shading always stops in the same place. That is what equivalent means.
How to make one
Cutting every piece in half does not change how much is shaded. It only changes how the shading is counted.
Three quarters becomes six eighths: twice as many pieces, twice as many of them shaded.
In numbers, multiply the top and the bottom by the same thing. Dividing both by the same thing works in reverse, and that is what simplifying is.
Where students slip
Adding the same number to the top and bottom feels like it should work. It does not.
Two thirds and three quarters look related — add one to each part of the first and you get the second — but the bars show they are different amounts. Only multiplying and dividing keep the value.
Finding the multiplier
Most questions ask you to rewrite a fraction with a particular denominator. There is one question to ask: what turns this bottom number into that one?
Whatever the answer is, do the same to the top. That is the whole method.
Simplifying is the same move backwards
Making an equivalent fraction multiplies the top and bottom by the same number. Simplifying divides them by the same number. It is one move, run in either direction.
Eight twelfths divided by four is two thirds. The bar does not change — you have regrouped the same shaded amount into fewer, larger pieces.
To simplify all the way in one go, divide by the largest number that goes into both. If you cannot spot it, divide by two twice, or by three, and keep going until nothing divides both any more.
Why the exam cares
Answers in a multiple-choice paper are almost always given in their simplest form. If you work out six ninths and the options list two thirds, you have not got it wrong — you have not finished.
The same is true in reverse when you compare or add fractions: you make the denominators match, do the work, then simplify the result before you look at the options.
Get into the habit of asking one question at the end of every fraction answer: can the top and bottom both be divided by something?
Worked examples
Write a fraction equivalent to three fifths with a denominator of twenty.
Ask what turns 5 into 20. It is multiplying by 4. So cut every fifth into four, which gives twenty pieces, and the three shaded fifths become twelve shaded twentieths. In numbers: 3 × 4 = 12.
Answer: Twelve twentieths
Simplify eight twelfths.
Simplifying is the same move backwards: divide instead of multiply. Both 8 and 12 divide by 4, giving two thirds. Check there is nothing left to divide by — 2 and 3 share no factor, so it is fully simplified.
Answer: Two thirds
Is six ninths the same as two thirds?
Look for the multiplier. 3 × 3 = 9 and 2 × 3 = 6, so two thirds multiplied top and bottom by three is six ninths. They are the same amount.
Answer: Yes
Aisha says one third and two fourths are equivalent because she added one to the top and one to the bottom. Is she right?
No. Adding does not preserve the value. One third is a little under a half and two fourths is exactly a half, so they are not the same amount. Only multiplying or dividing both parts works.
Answer: No
Simplify fifteen twenty-fifths.
Five divides both: fifteen becomes three, twenty-five becomes five. Nothing except one divides three and five, so that is as far as it goes.
Answer: Three fifths
Simplify twenty-four thirty-sixths.
Twelve divides both, giving two thirds. If you did not spot twelve, halving twice gives six ninths then three fourths — no, six ninths divides by three to give two thirds. Either route arrives in the same place.
Answer: Two thirds
A test answer is nine twelfths, but the options are one half, two thirds, three quarters and four fifths. What went wrong?
Nothing went wrong in the arithmetic. Nine twelfths has not been simplified: three divides both, giving three quarters.
Answer: The answer needed simplifying to three quarters
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