分解分数

Fraction Decomposition

We know that fractions show parts of a whole, but they can be broken down further. For instance, the fraction $\frac56$ means we have $5$ parts out of a whole $6$ parts. These parts can be separated into even smaller parts!

$$\frac56 = \frac16 +\frac16 +\frac16 +\frac16 +\frac16$$

The fraction $\frac56$ has been broken down into $5$ pieces of $\frac16$ each. When we add these parts of $\frac16$ together, we get the full fraction $\frac56$. These parts don't need to be equally sized. Let's look at some other decompositions of $\frac56$.

$$\begin{align}\frac56&=\frac26+\frac26+\frac16\\ \frac56&=\frac36+\frac26\\ \frac56&=\frac46+\frac16\\ \frac56&=\frac16+\frac16+\frac16+\frac26 \end{align}$$

Let's try a practice problem to check our understanding.

All of the following add up to $\frac79$. $$\frac19 +\frac19 +\frac19 +\frac19 +\frac19 +\frac19 +\frac19 \\ \frac39 + \frac49\\ \frac29+ \frac59\\ \frac29 + \frac29+ \frac29+ \frac19$$

In particular, the numerators add up to $7$ in every row. The denominator remains a $9$.

Decomposing fractions means breaking fractions into individual pieces that can be added together. In the case of a pizza, it would mean looking at individual slices (or groups of individual slices) and adding them together to see how many there are total.

For example, if we have $7$ slices of pizza, and we know that each slice is $\frac{1}{8}$ of the whole thing, we'd have $\frac{7}{8}$ of a pie.

But what if we want to describe how many slices have different toppings on them? Then, we have to break up that $\frac{7}{8}$ into smaller pieces. Maybe our toppings are onion and pepperoni. Then we want to know how many slices have onion, how many have pepperoni, and how many are plain. Technically a slice can have onion and pepperoni. For now, though, we'll assume that doesn't happen.

For some of these questions we may get a fraction that can reduce. However, it's not always advantageous to write fractions in simplest form. For instance, two of the slices in our pizza have pepperoni on them. That means $\frac{2}{8}$ of the pizza has pepperoni. This fraction can be reduced to $\frac{1}{4}$.

The problem is, if we change the denominator, we forget how many slices there were to begin with. The denominator represents the number of parts.

$3$ slices have onion on them, so $\frac{3}{8}$ of the pizza has onion on it.

$2$ slices have pepperoni on them, so $\frac{2}{8}$ of the pizza has pepperoni on it.

$2$ slices have no toppings on them, so $\frac{2}{8}$ of the pizza has no toppings on it at all.

Great! We've now split the pizza up into $3$ chunks that are each $\frac{2}{8}$ of the pizza and one chunk that's $\frac{3}{8}$ of the pizza. Let's add them together to check our work.

$\frac{2}{8} + \frac{2}{8} + \frac{3}{8} = \frac{7}{8}$, which is the number of slices we know we have! Fantastic!

(Video) Decomposing Fractions

by Math Antics

Math Antics has a helpful video that explains how to decompose fractions into parts.

$\large{\text{Summary}}$

You can think of decomposing as breaking down a whole into parts. Since, the word fractions follows, it probably won't be a shock to know that we'll be doing this to fractions.

A unit fraction is a fraction that has $1$ in its numerator.

When decomposing a fraction, we want to break it down into its unit fractions. For example, if we want to decompose the fraction $\frac46$ we would do the following: $$\frac46=\frac16+\frac16+\frac16+\frac16$$

You don't always have to use unit fractions to decompose. There are many different ways to decompose a fraction. For example, let's look at the fraction $\frac58$. $$\begin{align} \frac58&=\frac18+\frac18+\frac18+\frac18+\frac18\\ &=\frac28+\frac28+\frac18\\ &=\frac48+\frac18\\ &\cdots \end{align}$$