Multiplying Fractions

Student: Hey Alex, what are we doing today?

Teacher: Multiplying fractions! And good news — it’s simpler than adding them.

Student: Oh… do I need common denominators again?

Teacher: Nope. With multiplying, you just multiply straight across. Numerator times numerator, denominator times denominator.

Student: Wait, really? That’s way easier!

Teacher: It is. But let’s make sure we understand why.

Teacher: Let’s start simple. What’s 2 times 2 fifths?

Student: So I times the fraction by 2… 2 times 2 is 4 on the top, 2 times 5 is 10 on the bottom… four tenths?

Teacher: Careful — does four tenths look right to you?

Student: Wait… four tenths simplifies to two fifths. That’s just what I started with!

Teacher: Exactly. Timesing the top and bottom by the same number just gives you an equivalent fraction — not double the amount. Let’s think about what 2 times 2 fifths actually means. What does “times” mean?

Student: Lots of?

Teacher: So 2 times 2 fifths is…

Student: 2 lots of 2 fifths! So 2 fifths plus 2 fifths… that’s four fifths!

Teacher: There you go. You just multiply the numerator — the bottom stays the same.

Teacher: Here’s another way to think about it. 2 times 30 is 2 “lots of” 30 — sixty. So what would a third times 30 be?

Student: A third lots of 30… a third of-a-lot-of 30… oh — a third OF 30! That’s 10!

Teacher: So times means “of.” A third times 30 is a third OF 30.

Student: So a fifth times 20 — that’s a fifth of 20, which is just 4! Way easier than 20 lots of a fifth.

Teacher: Right, try these questions.

Teacher: So now, what’s a third times 2 fifths?

Student: A third OF 2 fifths. So I want a third of 2 fifths.

Teacher: Right. And 2 fifths means 2 divided by 5 — the fraction line means divide.

Student: So 2 divided by 5… and I want a third of that, so I divide by 3 again… that’s just 2 divided by 15. Two fifteenths!

Teacher: And look at the denominators. 5 and 3 — five times three is fifteen.

Student: Oh! That’s why you times the denominators together!

Teacher: Now what about 2 thirds times 2 fifths?

Student: 2 thirds of 2 fifths. First, a third of 2 fifths — I triple the denominator — that’s 2 fifteenths.

Teacher: And you want 2 thirds, not just 1 third.

Student: So I double it! Double the numerator — four fifteenths!

Teacher: Check it against the rule.

Student: Tops: 2 times 2, four. Bottoms: 3 times 5, fifteen. Four fifteenths — it works!

Teacher: And that’s why multiplying straight across works.

Teacher: Right, try these last few.