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 Math Help > Basic Math > Word Problems and Basic Arithmetic > Multiplying and Dividing Fractions

Melissa wrote me this email:

 > hello I was wondering if you could help me a little bit with some of my  > math as I have some trouble sometimes. I am doing extra work to try and  > get better but so far its not really helping. I'm in grade 9, and were just  > doing review work right now. So we are working on multiplying and dividing  > fractions. Can you give me the right instructions on doing that, because  > every time I ask someone I always end up getting a different answer. Thank  > you so much!! bye bye

Melissa,

Welcome to high school. I have a daughter, Meghan, in 9th grade, so I know from her first few days that it can be overwhelming. Don't worry. I'll help you with multiplying and dividing fractions.

First of all, since this is email, it's not easy to show you the fractions they way your book shows them. I will show the numerator, then a slash, then the denominator. You will just have to imagine that one is above the other.

3
---
4

I will write it as 3/4.

And then, since the numerator and the denominator can be complicated, like for example they might be fractions themselves, I will put parentheses around them. Here is how I write one half divided by three fourths:

(1/2) / (3/4)

OK, so far? Good.

The first rule for dividing by a fraction is that it's the same thing as multiplying by the reciprocal of the fraction. Using the example, above, if you have one half divided by three fourths, that's the same as one half multiplied by four thirds. That's because four thirds is the reciprocal of three fourths. Do you get what I mean by reciprocal? If not, re-read this paragraph, and if you still don't get it, ask.

Writing it out in numbers, what I just said is this:

(1/2) / (3/4)

is the same as

(1/2) * (4/3)

(I use the asterisk to mean multiplication, because later, you will need the "x" as a variable.)

You can write it more the way your book would write it -- something like this:

1      4
---- * ----
2      3

If you do that, then you will see that you have 1 times 4 in the numerator and 2 times 3 in the denominator. So you can combine them like this:

(1*4) / (2*3)

You can always do this trick when you are multiplying two fractions together. So the answer is 4/6, which is the same as 2/3.

If you used variables instead of numbers, here's how you would do it. I'll use a, b, c, and d as the names of my variables.

(a/b) / (c/d)

is the same as

(a/b) * (d/c)

Do you remember that?  The reason is that dividing by a fraction is the same as multiplying by the reciprocal of the fraction.  Here, we were dividing by c/d, and then we changed it so we multiplied by d/c. It works out the same.

Now, (a/b) * (d/c) is the same as (a*d)/(b*c), do you remember that? Write it out on a piece of paper, and you'll see it more clearly.

When we multiply two variables together, we usually don't bother to write the "times sign" or asterisk. So this is (ad)/(bc). I still write parentheses around the numerator and the denominator so you can see clearly what variables are in the numerator, and what variables are in the denominator. If you wrote it on a piece of paper it would look more like this:

------
bc

So you wouldn't need to use parentheses, because it would be clearer.

I hope this was a good introduction to multiplying and dividing fractions.  If you have any questions about what I wrote, please don't hesitate to send me an email.  Good luck in 9th grade!

--Graeme

### Internet references

Math League: Fractions - Prime numbers, Greatest common factor, Least common multiple, Equivalent fractions, Comparing fractions, reducing fractions, Lowest terms, Improper fractions, Mixed numbers, Converting mixed numbers to/from improper fractions, Writing a fraction as a decimal, Rounding a fraction to the nearest hundredth, Adding and subtracting fractions, Adding and subtracting mixed numbers, Multiplying fractions and whole numbers, Multiplying fractions and fractions, Multiplying mixed numbers, Reciprocal, Dividing fractions, Dividing mixed numbers, Simplifying complex fractions, Repeating decimals

### Related pages in this website

The webmaster and author of this Math Help site is Graeme McRae.