Mixed numbers and improper (vulgar) fractions

It's the end of a pizza night. Three whole pizzas left over, plus half of another. Ask the person stacking the boxes how much is left, and they count in wholes: three and a half. Ask the person cutting slices, and they count in slices: seven halves. Seven half-pizza pieces. Same amount of pizza. Two different ways of counting it: wholes with a bit left over, or slices only. Maths uses both counts, and it needs you to switch between them. There is exactly one correct order of steps for the switch. That order is what this video fixes.

First, the names. Three and a half is a whole number with a fraction sitting next to it. That is called a mixed number. Seven halves, written as seven over two, is called an improper fraction. Some books say vulgar fraction, and some say top-heavy. Top-heavy is the honest name: the top is bigger than the bottom. That is how you tell them apart. A mixed number shows you its wholes. An improper fraction hides the wholes inside a top-heavy top.

Quick check, and you can mark this one yourself. Which of these is the improper fraction? A: two and one third. B: seven thirds. C: three sevenths.

Look for the top-heavy one. Have a think. I'll wait. The answer is B. Seven thirds is top-heavy: seven on top, three underneath. A is a mixed number. C is an ordinary fraction, worth less than one whole.

So you can spot them. Now for the switch itself.

Now, the method: turning a mixed number into an improper fraction.

Back to three and a half. Each whole pizza is two halves. So three wholes make three times two: six halves. Add the extra half, and you have seven halves in total. That is the entire method, and it tells you why the order matters. Multiply the whole number by the bottom, because that turns the wholes into slices. Then add the top, because those slices were already cut. On the written fraction, it is a little arrow path. Start at the bottom number. Multiply up into the whole number. Then add across to the top number. That result becomes your new top. And the bottom does not move. In the video on equivalent fractions, the rule was that the top and the bottom travel together. Converting is a different job. The size of the slices never changes, so the bottom stays put. Only the top gets rebuilt.

Here is the handle to keep for the exam hall: MAD. Multiply, Add, Denominator stays.

Before you try one, a warning from the people who mark this. Students were asked to do exactly this conversion on a recent exam paper. The examiners reported: Muddled attempts at converting to improper fractions were seen and random manipulation of the figures given in the question; together with blank responses, this resulted in a little under half the students gaining no marks. Random manipulation means the right numbers in the wrong order: adding before multiplying, or changing the bottom. The arithmetic is easy. The order is the skill. Multiply, then add, and leave the bottom alone.

Now, your turn: running that order yourself on a fresh mixed number.

Convert five and three eighths into an improper fraction. Say the steps to yourself as you go: multiply, then add.

Pause here and work it out. I'll wait. The answer is forty-three over eight. Follow the arrows. Bottom times whole: eight times five is forty. Add the top: forty plus three is forty-three. The bottom stays as eight.

And the forty makes sense. Five whole ones cut into eighths give forty slices, eight from each whole. The three extra eighths were already there. Forty-three eighths.

Now the reverse direction: turning an improper fraction back into a mixed number.

Going forward, you multiplied. So coming back, you divide. Divide the top by the bottom. The whole-number part of the answer is your wholes. The remainder becomes the new top. And the bottom, once again, stays the same. Try this one. Convert forty-seven sixths into a mixed number. How many whole sixes fit inside forty-seven, and what is left over?

Have a go at this one yourself. I'll wait. The answer is seven and five sixths. Forty-seven divided by six: six sevens are forty-two, remainder five. So seven wholes, with five sixths left over. Nothing cancels in five sixths, so it is finished.

The two directions also check each other. Send seven and five sixths back the other way: six times seven is forty-two, add five, and you are at forty-seven sixths, exactly where you started. If your round trip lands somewhere new, one of the steps ran out of order.

Right. Let's shrink all of that down to a pocket version. A mixed number shows its wholes. An improper fraction is top-heavy and hides them. Mixed to improper is MAD: Multiply the whole by the bottom, Add the top, and the Denominator stays. Improper to mixed is the reverse: divide the top by the bottom, wholes out front, remainder on top, same bottom underneath.

Next in the chapter: finding a common denominator, which is how you get two fractions with different bottoms ready to work together.

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