Why Learn to Multiply Fractions?

After learning what fractions are and various ways to write them, you learned to find common denominators to add and subtract them.  Next, you will learn to multiply them.  Fractions can be multiplied by whole numbers like 3 x 1/6 or they can be multiplies by other fractions like 1/2 x 2/3.  Why would you want to learn how to multiply fractions?   In this post, we review fractions and then give two examples where fraction multiplication naturally arises.

Review of Fractions

Before you learned fractions, you learned whole numbers: 1, 2, 3, …
Then, you learned that 1 can be divided into parts such as
\begin{align} 1 &= \tfrac 12 + \tfrac{1}{2}\\ 1 &= \tfrac 13 +\tfrac 13+\tfrac 13 \\ 1 &= \tfrac 14 +\tfrac 14+\tfrac 14 +\tfrac 14 \\ &\vdots \end{align}

The fractions $\frac 12, \frac 13, \frac 14. \frac 15,...$ are called unit fractions (for a review, see Khan Academy’s Intro to Fractions). Unit fractions have the number 1 on top.  These unit fractions can be counted:
\begin{align} \tfrac 12, \tfrac 22,\tfrac 32, \tfrac 42, &\cdots\\ \tfrac 13, \tfrac 23,\tfrac 33, \tfrac 43, &\cdots\\ \tfrac 14, \tfrac 24,\tfrac 34, \tfrac 44, &\cdots \end{align}

The proper fractions have the numerator (top) smaller than the denominator (bottom). The improper fractions have a numerator that is bigger than or equal to the denominator.  For example, 1/4, 2/4, and 3/4 are proper fractions. 4/4, 5/4, 6/4, … are improper fractions.
Improper fractions can also be written as mixed number which are a combination of whole numbers and proper fractions. All improper fractions can be written equivalently as either a whole number or a mixed number:
\begin{align} \frac 44 &= 1 \text{ (a whole number)}\\ \frac 54 &= 1\tfrac 14 \text{ (a mixed number)}\\ \frac 64 &= 1\tfrac 24 \text{ (a mixed number)}\\ &\vdots \end{align}
See Pool Noodle Fractions for an example of a way to use physical objects to learn fractions.

After learning types of fractions, you learned to find common denominators in order to add and subtract fractions.  The next step is learning multiplication and division of fractions.  However, you may wonder why these skills are useful.

Multiply Fractions: Fair Sharing

Imagine that your pizza comes out of the oven smelling delicious.  You and your two friends can’t wait to start eating.  One of your friends cuts that pizza into eight equal pieces (each piece is 1/8 of the whole pizza).  You each quickly grab two pieces and enjoy them as fast as you can without burning your mouths.  There’s only two pieces of pizza left with three people eyeing the pizza hungrily.  Each one of you would love to eat the last pieces yourself, but you decide to be fair and split it evenly.  You each eat 1/3 of the remaining pizza.  In the end, you ate:

\begin{align} 2+ \frac 13 +\frac 13 &= 2 \frac 23 \text{ pieces of pizza}\\ 2 \frac 23 \times \frac 18 &\text{ of the whole pizza} \end{align}

Since you split the pizza evenly between you and your two friends, this multiplication comes out to be 1/3 of the whole pizza.  This is one example of multiplying fractions.

Multiply Fractions: Recipe Measurements

Imagine that you can’t find any sweets left in your house.  Your mom says you can eat brownies if you make them yourself.  You crave a sweet treat, so you find the recipe.

As you put each ingredient on the counter, you notice that the recipe calls for 1 cup of cocoa powder. “Uh oh,”  you think.  There isn’t much cocoa powder left.  You carefully measure your cocoa powder and find that you have 2/3 cup.  Do you give up on cooking the brownies?

You could maximize the amount of brownies you make by multiplying the original recipe by 2/3.  The problem with doing this is — if you make a mistake in your calculations, then your brownies might not come out right.  Can you imagine if you messed up the amount of salt — ewww, salty brownies.