# Category: 分数

## What is Fraction

So, what is fraction?

Suppose you divided a cake into 10 pieces and eat 3 pieces. How much of cake do you eat?

We write it as $$$\frac{{3}}{{10}}$$$.

3 is a number of pieces you eat and 10 is total number of pieces.

## Proper Fractions

Proper Fraction is a fraction in which numerator is less than denominator.

Examples of proper fractions are $$$\frac{{1}}{{5}}$$$, $$$\frac{{7}}{{19}}$$$, $$$\frac{{127}}{{128}}$$$.

Note, that $$$\frac{{5}}{{5}}$$$, $$$\frac{{8}}{{7}}$$$, $$$\frac{{125}}{{3}}$$$ are not proper fractions.

## Improper Fractions

Improper Fraction is a fraction in which numerator is greater or equal than denominator. In other words improper fractions are fractions that are not proper.

Examples of improper fractions are $$$\frac{{5}}{{5}}$$$, $$$\frac{{8}}{{7}}$$$, $$$\frac{{125}}{{3}}$$$.

## Mixed Numbers/Fractions

Mixed number (or mixed fraction) is a number that consists of integer and proper fraction.

In other words mixed numbers are "mix" of integer and proper fraction.

We write it in the following way: $$${4}\frac{{1}}{{5}}$$$. Here, 4 is integer and $$$\frac{{1}}{{5}}$$$ is proper fraction.

## Mixed Numbers on a Number Line

So, where are mixed numbers located on a number line?

For example, where is $$${2}\frac{{4}}{{5}}$$$ located? We see that this number consists of integer 2 and proper fraction $$$\frac{{4}}{{5}}$$$.

This means that it is greater than 2 (because there is 2 and proper fraction), but it is less than 2+1=3 because proper fraction is less than 1.

## Equivalent Fractions

Here we will learn about equivalent fractions.

Suppose you have a cake. You decide to divide it into 3 pieces and eat one piece. In this case you eat $$$\frac{{1}}{{3}}$$$ (1 out of 3) of the cake.

But you see that the piece is too big, so you divide each piece into two pieces, so there are $$${3}\cdot{2}={6}$$$ pieces now.

## Reducing Fractions

Let's learn how to reduce fractions.

We already learned about equivalent fractions. There are many fractions that are equivalent to the given one, but there is one special fraction.

Fraction is irreducible if numerator and denominator have no common factors.

## Adding Fractions with Like Denominators

To add fractions with like denominators we just add numerators (just like we add integers) and denominator leave the same: $$${\color{red}{{\frac{{a}}{{c}}+\frac{{b}}{{c}}=\frac{{{a}+{b}}}{{c}}}}}$$$.

## Subtracting Fractions with Like Denominators

To subtract fractions with like denominators we just subtract numerators (just like we subtract integers) and denominator leave the same: $$${\color{red}{{\frac{{a}}{{c}}-\frac{{b}}{{c}}=\frac{{{a}-{b}}}{{c}}}}}$$$.

## Adding Fractions with Unlike Denominators

It is a bit harder to add fractions with unlike denominators than with like denominators.

We saw that it is very simple to add fractions with like denominators.

But how to transform fractions that have different denominators into fractions that have same denominators? In fact, very easy. We use equivalence of fractions for this.

## Subtracting Fractions with Unlike Denominators

It is a bit harder to subtract fractions with unlike denominators than with like denominators.

We saw that it is very simple to subtract fractions with like denominators.

But how to transform fractions that have different denominators into fractions that have same denominators? In fact, very easy. We use equivalence of fractions for this.

## Converting Mixed Numbers to Improper Fractions

Let's see how to convert mixed numbers into improper fractions.

Actually, we convert them almost in the same way as we add fractions with unlike denominators.

Recall that every integer $$${m}$$$ can be expressed as fraction $$$\frac{{m}}{{1}}$$$.

## Converting Improper Fractions to Mixed Numbers

Converting improper fractions to mixed numbers is inverse of converting mixed number to improper fractions.

Suppose you want to convert $$${3}\frac{{4}}{{5}}$$$ to improper fraction.

We already know that $$${3}\frac{{4}}{{5}}=\frac{{3}}{{1}}+\frac{{4}}{{5}}=\frac{{{3}\cdot{5}}}{{5}}+\frac{{4}}{{5}}=\frac{{{3}\cdot{5}+{4}}}{{5}}=\frac{{19}}{{9}}$$$.

## Adding Fractions with Whole Numbers

Adding fractions with whole numbers is essentially the same as converting mixed number to improper fraction (just remember how to add integers correctly).

Indeed, suppose we want to add whole number $$${m}$$$ and fraction $$$\frac{{n}}{{q}}$$$.

## Subtracting Fractions with Whole Numbers

Subtracting fractions with whole numbers doesn't differ much from adding fractions with whole numbers (just remember how to subtract integers correctly).

Indeed, suppose we want to subtract whole number $$${m}$$$ from fraction $$$\frac{{n}}{{q}}$$$.

## Adding Mixed Numbers

Adding mixed numbers is quite easy.

We know that mixed number consists of integer part and fractional part.

To add mixed numbers three steps are needed:

- Convert each mixed number to improper fraction.
- Add improper fractions (using addition of fractions with unlike denominators)
- Convert improper fraction to mixed number if needed (and if possible).

Example 1. Find $$${1}\frac{{3}}{{5}}+{2}\frac{{4}}{{9}}$$$.

## Subtracting Mixed Numbers

Subtracting mixed numbers is quite easy.

We know that mixed number consists of integer part and fractional part.

To subtract mixed numbers three steps are needed:

- Convert each mixed number to improper fraction.
- Subtract improper fractions (using subtraction of fractions with unlike denominators)
- Convert improper fraction to mixed number if needed (and if possible).

Example 1. Find $$${1}\frac{{3}}{{5}}-{2}\frac{{4}}{{9}}$$$.

## Comparing Fractions

To compare two fractions we first need to make same denominators using equivalence of fractions. After this denominators are already equal, so we compare numerators just like we compared integers.

If we compare mixed numbers then we compare integer parts. If integer parts are equal then we need to compare fractional parts.

## Multiplying Fractions

To multiply fractions multiply separately numerators and separately denominators: $$${\color{green}{{\frac{{a}}{{b}}\cdot\frac{{c}}{{d}}=\frac{{{a}{c}}}{{{b}{d}}}}}}$$$.

After this you, possibly, need to reduce a fraction.

## Multiplying Mixed Numbers

To multiply mixed numbers

- convert mixed numbers to improper fractions
- multiply fractions
- reduce improper fraction (if possible) and convert to mixed number (if needed).

Note! Rules for determining sign of the result are same as when multiplying integers.

## Dividing Fractions by Whole Number

To divide fraction by whole number multiply denominator of the fraction by whole number, i.e. fraction $$$\frac{{n}}{{q}}$$$ divided by the whole number $$${m}$$$ is $$${\color{green}{{\frac{{n}}{{q}}\div{m}=\frac{{\frac{{n}}{{q}}}}{{m}}=\frac{{n}}{{{q}{m}}}}}}$$$.

## Dividing Fractions

To divide fraction by a fraction multiply numerator of the first fraction by the denominator of the second fraction and denominator of the first fraction by the numerator of the second fraction, i.e. $$${\color{green}{{\frac{{a}}{{b}}\div\frac{{c}}{{d}}=\frac{{\frac{{a}}{{b}}}}{{\frac{{c}}{{d}}}}=\frac{{{a}{d}}}{{{b}{c}}}}}}$$$.

## Dividing Mixed Numbers

To divide mixed numbers, convert them to improper fractions and then divide fractions.

After this you, possibly, need to reduce fraction.

Also, you can convert improper fraction back to mixed number.

## Reciprocals

Reciprocal of the fraction is fraction that is turned "upside down", i.e. reciprocal of the fraction $$${\color{green}{{\frac{{a}}{{b}}}}}$$$ is $$${\color{red}{{\frac{{b}}{{a}}}}}$$$.

There is very nice fact about reciprocals.

## Negative Exponents

Let's learn about negative integer exponents.

When we talked about exponents and integers, we assumed that exponent is positive integer.

But what if we want to raise number to negative integer exponent?

## Rational Numbers

Rational numbers are integers plus fractions.

In other words number is rational, if it can be written as fraction $$$\frac{{p}}{{q}}$$$ (we know that every integer $$${m}$$$ can be written as fraction $$$\frac{{m}}{{1}}$$$).