Multiplying and dividing fractions and mixed numbers

Take a flapjack recipe. It calls for two and a quarter cups of oats, and it feeds four. Tonight there are more of you coming round, so you decide to make one and a third times the recipe. Somewhere inside two and a quarter times one and a third is the amount of oats you actually need. Two awkward mixed numbers - and yet the answer turns out to be a perfectly clean whole number. Better still, you will not need a common denominator to find it. Multiplying and dividing fractions play by completely different rules from adding. This video is those two rules: what they are, why they work, and how to keep them apart.

First, multiplying. The whole rule fits in one line: multiply the tops together, multiply the bottoms together. Straight across. Try a half times three quarters. Tops: one times three is three. Bottoms: two times four is eight. A half times three quarters is three eighths. And you can see why. A half times three quarters means a half OF three quarters. Multiplying the bottoms makes the pieces smaller - half of a quarter is an eighth, so the answer lives in eighths. Multiplying the tops counts those pieces - three of them. Smaller pieces, counted up. That is all multiplying fractions is. Now, if you have watched the video on adding and subtracting fractions, alarm bells might be ringing. There, going straight across was the classic wrong move, and you had to match the denominators first. Here, that reverses. For multiplying, straight across is exactly right - and no common denominator is needed. Ever.

Quick check. Two thirds times one fifth. Is it A: two fifteenths, B: thirteen fifteenths, or C: ten thirds? Have a think. I'll wait. The answer is A: two fifteenths. Tops, two times one is two. Bottoms, three times five is fifteen. B, thirteen fifteenths, is what ADDING those fractions would give - wrong operation. And C, ten thirds, comes from flipping the one fifth. Flipping is a real move, but it belongs to division - and that is coming up.

Next, the move that keeps your numbers small: cancelling before you multiply. Back to the recipe: two and a quarter times one and a third. Mixed numbers go improper first - the video on mixed numbers and improper fractions covers the how, so here is the result: nine quarters times four thirds. Route one, the long way round: multiply straight across, tidy up later. Nine times four is thirty-six. Four times three is twelve. Thirty-six twelfths - which you then have to cancel all the way back down to three. Route two: cancel first. Once you multiply, everything lands in one fraction - nine times four on top, four times three on the bottom. And in one fraction, the top and the bottom travel together: a four up top and a four down below cancel, exactly like simplifying. Cross out both fours before you multiply anything. What is left is nine over one times one over three - nine thirds, which is three. Same answer, but the biggest thing you multiplied was nine by one. The ugly numbers never showed up. And that is the recipe solved: exactly three cups of oats. Two awkward mixed numbers in, one clean whole number out. Examiners watch both routes, and they are blunt about it. One report on this exact skill says: 'Cancellation when used made the multiplication easier; but multiplying without cancelling first is the most popular method.' The popular route is the slow route. Bigger numbers mean more arithmetic, and more arithmetic means more slips. Cancel first, and the hard sums never happen.

Now for dividing - the one place in this video where something genuinely does get flipped. Start with the simplest division there is: one divided by one fifth. That asks how many one-fifths fit into one whole - and five of them do. So dividing by one fifth did exactly the same job as multiplying by five. A unit fraction and its whole number are a flip apart. And that idea works for any fraction: dividing by two sevenths is the same as multiplying by seven halves. Every fraction division can be turned into a multiplication you already know how to do. So the method is three words: keep, flip, multiply. Keep the first fraction exactly as it is. Flip the second fraction upside down. Then multiply straight across, like before. Only the second fraction flips. Never the first. Your turn. Three fifths divided by two sevenths. Keep, flip, multiply - and give the answer as a mixed number. Take your time. I'll wait right here. Keep three fifths. Flip two sevenths into seven halves. Multiply straight across: three times seven is twenty-one, five times two is ten. Twenty-one tenths - and as a mixed number, the answer is two and one tenth.

Time for the exam angle: knowing which of the two rules the question is actually asking for. On a question that asked students to multiply mixed numbers, one examiner's report says: 'There were noticeable attempts to change the multiplication to division and inverting the second fraction; this rarely led to marks after the initial conversion.' The question said multiply - and students flipped anyway. So watch the same pair of numbers do both jobs, side by side. Nine quarters TIMES four thirds: nothing flips, cancel the fours, straight across - three. Nine quarters DIVIDED BY four thirds: keep, flip four thirds into three quarters, multiply - twenty-seven sixteenths. Same fractions, different operations, completely different answers. Two rules, one habit. Multiplying: never flip anything. Dividing: always flip the second fraction, and only the second. Before your pen moves, check which one the question actually says.

Time to gather the rules - four lines to carry into the exam. One: mixed numbers become improper fractions before anything else. Two: multiplying goes straight across - tops together, bottoms together - with no common denominator in sight. That rule belongs to adding and subtracting, not here. Three: cancel before you multiply. The top and the bottom travel together, so a matching factor up top and down below can go early - and the ugly arithmetic never happens. Four: dividing is keep, flip, multiply - and only the second fraction ever flips.

Next in the chapter: finding a fraction of a quantity, and writing one number as a fraction of another.

For more, visit scholafly.com, or watch the next video.