Adding and subtracting fractions and mixed numbers

Here are two pizzas, the same size, cut differently. One is cut into quarters, and three quarters of it is left. The other is cut into sixths, and five sixths of that one is left. Put the leftovers together: clearly more than one whole pizza. Now watch a tempting shortcut wreck that. Add the tops: three plus five is eight. Add the bottoms: four plus six is ten. Answer: eight tenths of a pizza. Eight tenths is less than one pizza. Five sixths on its own is already more than eight tenths. Combining two piles of pizza left you with less than you started with. The shortcut did not just miss the answer. It broke the meaning of the question. Examiners see that exact shortcut again and again, and we will read their words later in the video. First: why it fails, the rule that fixes it, and how to add and subtract any fractions, mixed numbers included.

So, why it fails. The bottom of a fraction is not a number you add. It is a label. It tells you what size the pieces are: a four on the bottom means the whole was cut into four equal pieces. Quarters and sixths are different sizes of piece. Adding their counts is like adding three apples to five oranges and announcing eight bananas. A count only means something when every piece is the same size.

So here is the rule. To add or subtract fractions, the denominators must match first. Make the pieces match, then count the tops. And notice: this rule belongs to adding and subtracting specifically. Multiplying fractions plays by a completely different rule, and it gets its own video.

Try one where the pieces already match. Two ninths plus five ninths. Is it seven ninths, or seven eighteenths?

Have a think. I'll wait.

The answer is seven ninths. The pieces are all ninths, so count the tops: two plus five is seven. The bottom stays as ninths, because the piece size never changed. Seven eighteenths is the pizza shortcut again: adding the labels.

If you got that, you already own half the method. The other half is making the pieces match when they start out different. That is next.

Here is the pizza sum the way an exam would print it. Work out three quarters plus five sixths, giving your answer as a fraction in its simplest form.

Quarters and sixths do not match, so rewrite both over a common denominator. Finding that number is its own skill, covered in the video on finding a common denominator, so take the result: twelve, the smallest number both four and six divide into.

Rewrite three quarters in twelfths. This chapter's rule does the work: the top and the bottom travel together. To get from four to twelve, the bottom is multiplied by three, so the top is multiplied by three as well. Three quarters becomes nine twelfths.

Same treatment for five sixths. Six times two is twelve, so five times two makes ten. Five sixths becomes ten twelfths.

Now the pieces match, and the adding is the easy part. Count the tops: nine plus ten is nineteen. The bottom stays twelve. Nineteen twelfths.

Two finishing touches. Nineteen and twelve share no common factor, so nothing cancels. But nineteen twelfths is top-heavy, so write it as a mixed number: twelve twelfths makes one whole, with seven twelfths left over. The answer is one and seven twelfths.

And it passes the pizza test: one whole and a bit more, exactly what the picture promised. The broken shortcut said eight tenths. The real answer is nearly double that.

Now for mixed numbers, and a warning straight from the examiners.

One examiner report describes students doing this: many left their correct answer as eighteen eighths, however eight sixths was far more commonly seen. This came from correctly converting from mixed to improper fractions but then numerators and denominator were added separately, seven quarters plus one half equals seven plus one, over four plus two.

Look at what happened there. The hard step, the conversion, was done correctly. The marks vanished on the easy-looking step, when tops and bottoms were added separately. The habit that prevents it: check your denominators match before you touch the numerators.

Here is that situation for real. A recipe needs two and one third cups of flour for the base, and one and three quarters cups for the topping. How many more cups does the base use than the topping?

How many more means subtract: two and a third, take away one and three quarters. With mixed numbers, convert to improper fractions first, the multiply-and-add move from the video on mixed numbers and improper fractions. Quickly: three times two is six, plus one is seven, so seven thirds. Four times one is four, plus three is seven, so seven quarters.

Thirds and quarters do not match, and the common denominator here is twelve. That is everything you need. Pause the video and finish the subtraction yourself.

Take your time. I'll wait right here.

Here it is. The top and the bottom travel together: seven thirds becomes twenty-eight twelfths, and seven quarters becomes twenty-one twelfths. Now subtract the tops: twenty-eight take away twenty-one is seven. The base uses seven twelfths of a cup more. Nothing cancels, and it is not top-heavy, so that is the finished answer.

And watch the shortcut fail here. Seven thirds take away seven quarters, worked straight across: seven minus seven is zero on top. A difference of zero, for two amounts of flour that are plainly different. The denominators never matched, so the subtraction was never ready to happen.

And now the recap. Four moves, in order. One: check your denominators match. If they do not, you are not ready to add or subtract. Two: rewrite over a common denominator, and the top and the bottom travel together while you do it. Three: add or subtract the tops only. The bottom is a label, and it stays. Four: tidy up. Cancel what cancels, and turn a top-heavy answer into a mixed number. And when the question hands you mixed numbers, turn them into improper fractions before you start.

Next in the chapter: multiplying and dividing fractions and mixed numbers, where working straight across finally becomes the right move.

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