Equivalent fractions and simplifying by cancelling

Picture two friends comparing quiz scores. One scored eighteen out of thirty. The other scored three out of five. Different quizzes, different numbers. So who did better? Neither. Those two scores are exactly level. Eighteen out of thirty is the same share as three out of five. One fraction, wearing two different outfits. This video is about seeing through the disguise: how to grow a fraction into a different-looking equal one, and how to strip it back to its simplest form. First up: equivalent fractions.

Start with four tenths. On screen, that's a bar cut into ten equal pieces, with four of them shaded. Now multiply the top and the bottom by the same number. Let's use two. Four times two is eight. Ten times two is twenty. Four tenths becomes eight twentieths. Watch the bar. The shaded amount has not moved. All we did was cut every piece in half. More pieces, smaller pieces, exactly the same amount shaded. That is why the rule works: multiplying top and bottom by the same number changes the pieces, not the value.

And that is the one rule this whole chapter runs on. The top and the bottom travel together. Whatever happens to one happens to the other.

Quick check, and you can mark it yourself. Which of these is equivalent to three quarters? A: six eighths. B: four fifths. C: seven eighths.

Have a think. I'll wait. The answer is A. Multiply the top and bottom of three quarters by two, and you get six eighths. B is the trap: it adds one to the top and one to the bottom. Adding changes the value. Only multiplying or dividing keeps a fraction equal.

That is scaling up. Now run the same rule in reverse. This is called cancelling.

Back to four tenths. This time, divide the top and the bottom by the same number. Two divides exactly into both. Four divided by two is two. Ten divided by two is five. Four tenths becomes two fifths. On the bar, the pieces glue together in pairs. Fewer pieces, bigger pieces, same shaded amount. The top and the bottom travelled together again, just in the other direction. Dividing top and bottom by a shared factor is called cancelling. Keep cancelling until no number divides exactly into both. When nothing is left to cancel, the fraction is in its simplest form. For four tenths, that is two fifths. So one fraction sits in a whole family of equal fractions. From four tenths: multiply top and bottom by two, and you get eight twentieths, with a larger denominator. Divide top and bottom by two, and you get two fifths, with a smaller one. Same value every time.

Time to use it on a real exam-style question about a jar of sweets.

A jar contains thirty sweets. Eighteen of them are red, and the rest are blue. Write the fraction of the sweets that are red, giving your answer in its simplest form. The starting fraction is eighteen red sweets out of thirty sweets in total: eighteen over thirty. Your job is the second half. Pause, and cancel eighteen over thirty as far as it will go. Take your time. I'll wait right here. Here is one route. Two divides into both: eighteen over thirty becomes nine over fifteen. Check again: three divides into both of those. Nine over fifteen becomes three fifths. Nothing divides into both three and five, so we stop. The answer is three fifths. That middle step matters. Nine over fifteen looks tidy. The numbers got smaller, and it feels finished. It is not, because three still divides into both. Stopping at nine over fifteen is the classic slip: the value is right, but the fraction is not fully simplified. And if you had spotted a bigger shared factor, six, you could jump straight there. Eighteen divided by six is three. Thirty divided by six is five. Three fifths in one step. Any shared factor works. Bigger ones are just faster.

One more thing before the recap: what the exam does with this. When a question says simplest form, that instruction is part of the question. Nine over fifteen has the right value, and it still drops the final mark, because it can still be cancelled. The last mark sits on the last cancel. So before you move on, run the check. Look at your final answer. If any number still divides exactly into both the top and the bottom, you are not done yet.

Let's fold this down to three lines you can keep. The top and the bottom travel together: multiply or divide both by the same number, and the value does not change. Multiply, and you scale up: four tenths to eight twentieths. Divide, and you cancel: four tenths to two fifths. And an answer is only finished when nothing is left to cancel. If a number still divides into the top and the bottom, keep going.

Next in the chapter: mixed numbers and improper fractions. That video handles fractions worth more than one whole.

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