Adding and Subtracting Fractions

Teacher: Hey Jamie, want some birthday cake? It’s Sam’s birthday!

Student: Ooh, cake! Yes please!

Teacher: So I’ve got a whole cake. If I give you one eighth and give Sam a quarter, how much do I have left?

Student: Umm… I’m not really sure how to work that out.

Teacher: No worries, let’s learn how. By the end you’ll be able to solve it.

Teacher: So, how do we add fractions? Let’s try this one — one third plus one fifth.

Student: OK so… I times the bottoms, three times five is fifteen. Then add the tops… one plus one is two. So two fifteenths?

Teacher: That’s a really common mistake. You got the denominator right, but you can’t just add the numerators like that. They haven’t been converted yet. Let me show you.

Teacher: Let’s start simpler. What’s one fifth plus two fifths?

Student: That’s three fifths! One fifth plus two fifths — three fifths.

Student: So they need to have the same denominator, and then you just add the tops.

Teacher: Exactly. So for one third plus one fifth, we need to make the denominators the same.

Teacher: We can use equivalent fractions. If we multiply the top and bottom by the same number, the fraction stays equal — it’s just written differently. Watch.

Teacher: One third times five on top and bottom gives five fifteenths. One fifth times three on top and bottom gives three fifteenths. Now they’ve got the same denominator.

Student: So five fifteenths plus three fifteenths is… eight fifteenths!

Teacher: Exactly! Try this one.

Student: So… I can just times them by each other’s bottom number? So the quarter by three and the third by four?

Teacher: Exactly right!

Student: OK… I need twelfths. One quarter times three on top and bottom is three twelfths. One third times four is four twelfths. Three plus four is seven twelfths!

Teacher: Perfect.

Teacher: Now try this one — one third plus one sixth.

Student: So… times six on the first one gives six eighteenths, times three on the second gives three eighteenths. So nine eighteenths?

Teacher: This time, three and six have a common factor. Whenever they share a factor, we should try to take a shortcut. What number do both three and six go into?

Student: They both go into six.

Teacher: Perfect. This means we can change both fractions into sixths.

Student: Oh right.

Teacher: For the fraction on the left, what should we times by to make it into sixths?

Student: Times two on top and bottom. And the other one doesn’t need to be changed at all.

Student: So one third becomes two sixths. Two sixths plus one sixth is three sixths — that’s a half!

Teacher: Six is the LCM of three and six — the lowest common multiple. We really want to find the lowest number that both denominators go into. That’s the key.

Teacher: Now, back to the cake. I give you one eighth and Sam one quarter. How much have I given away?

Student: I notice eight and four have a common factor, so hmm… what do four and eight both go into? Ooo, I can times the one on the right by two to make it into eighths! So one eighth plus two eighths… you’ve given away three eighths!

Student: So you’ve got one whole minus three eighths… that’s five eighths of cake left!

Teacher: Perfect! See, not so hard.

Teacher: Right, try these questions then.

Teacher: Nice. Now what about subtracting fractions? What’s two thirds minus one sixth?

Student: Well it’s the same thing but subtracting, right? Three and six both go into six. So two thirds is four sixths… then four sixths take away one sixth is three sixths — that’s a half!

Teacher: Correct!

Teacher: What about mixed numbers? What’s one and a third plus two and a fifth?

Student: So the wholes are easy. One plus two is three. So we’ve got three wholes.

Student: Now what about the fractions? We have to add those too.

Student: One third plus one fifth. We can times the one on the left by five, and the one on the right by three.

Student: That gives us five fifteenths plus three fifteenths… eight fifteenths.

Student: So putting it all together we have three wholes and eight fifteenths!

Teacher: Great!

Teacher: Now a tricky one. What about one and two thirds plus three and four fifths?

Student: OK so one plus three is four wholes.

Student: Now for the fractions. Two thirds plus four fifths.

Student: Times five on the left, times three on the right.

Student: Ten fifteenths plus twelve fifteenths is… twenty-two fifteenths.

Student: Putting it all together we have four and twenty-two fifteenths?

Teacher: You’re on the right track, but we need to change twenty-two fifteenths to a mixed number first. Twenty-two fifteenths equals one and seven fifteenths.

Student: Now putting this together with the wholes, we get five and seven fifteenths!

Teacher: Right, try these last few.