## Fractions

Fractions are two numbers written vertically around a horizontal bar. The upper number is called the *numerator, the lower number is called the *denominator*. Fractions signify the relationship between two numbers. In counting they show a portion of a whole.

*Proper fractions* have a numerator that is lesser than their denominator: $\frac 1 2$, $\frac 3 8$.

### Improper Fractions and Mixed Numbers

*Improper fractions* have a numerator that is greater or equal in value to their denominator: $\frac 7 4$, $\frac 5 5$. They can be written as mixed numbers, e.g. $\frac 7 3 = 2 \frac 1 3$. To find a *mixed number*, divide the numerator of an improper fraction by its denominator. The quotient becomes the whole number part of the mixed number and the remainder becomes the new numerator, while the denominator stays the same.

Therefore the remainder of a division can be written as a fraction. In reverse an improper fraction can be derived from a mixed number by multiplying the whole number with the denominator, adding the numerator to the resulting product and using the sum as the new numerator: $4 \frac 5 6 = \frac{29}{6}$. Again the denominator doesn't change.

### Simplest Form and Basic Simplification Sstrategies

Dividing the numerator and denominator of a fraction by their highest common factor results in an equal fraction that is in its simplest form: $\frac 4 8 \div 4 = \frac 1 2$, $\frac {21}{35} = \frac 3 5$. Fractions are *equal* when they share a common value although their denominators are not the same, e.g. $\frac 5 5 = \frac 9 9$, $\frac 1 2 = \frac 4 8$. When a fraction's parts are divided by a number that is not their highest common factor the resulting quotient will be further reducible.

Common factors in fractions cancel each other out: $\frac {12} {20} \div 4 = \frac 3 5$ => $\frac{4 \times 3}{4 \times 5} = \frac {12} {20} = \frac 3 5$.

Variables are left alone when reducing: $\frac {42x} {66} = \frac {7x} {11}$ (has the highest common factor $6$).

Prime factorization can be utilized to simplify fractions whose highest common factor is not easy to identify: $\frac {30} {108} = \frac {2 \times 3 \times 5} {2 \times 2 \times 3 \times 3 \times 3} = \frac {\cancel{2} \times \cancel{3} \times 5} {\cancel{2} \times 2 \times \cancel{3} \times 3 \times 3 } = \frac {5} {18}$. This also works with exponents: $\frac {6x^2} {2x^3} = \frac {3x^2} {x^3} = \frac {3 \times \cancel{x} \times \cancel{x}} {\cancel{x} \times \cancel{x} \times x} = \frac {3} {x}$ (simplify constants first).

### Fraction Representation of Whole Numbers

A whole number $n$ as a fraction is $\frac n 1$ ($n$ parts of $1$). So $\frac {6x^4} {60x} = \frac {x^3} {10} = \frac {1 \times x^3} {10 \times 1} = \frac {1}{10}x^3$.

### Multiplation of Fractions

To *multiply* fractions, just multiply their numerators and denominators individually (as above), e.g. $\frac 2 3 \times \frac 5 7 = \frac {10} {21}$. A product of fractions can be simplified before multiplying: $\frac 6 7 \times \frac {14} {27} = \frac {6 \times 14} {7 \times 27} = \frac {2 \times 2} {1 \times 9} = \frac 4 9$.

### Division of Fractions

To *divide* fractions *multiply* the dividend by the *reciprocal* of the divisor. The reciprocal in terms of fractions is a fraction with numerator and denominator switched. Examples:

$\frac a b \div \frac c d = \frac a b \times \frac d c$

$\frac a b \div c = \frac a b \div \frac c 1 = \frac a b \times \frac 1 c$

### Operations on Mixed Numbers

Always turn mixed numbers into improper fractions before working with them.

### Addition and Subtraction of Fractions

To *add* or *subtract* fractions they require to have the same denominator. In that case the numerators get added or subtracted, while the denominator *stays the same*: $\frac {20} {11} + \frac {6}{11} + \frac {7}{11} = \frac {33}{11} = 3$. If one fraction is negative the minus sign moves to its numerator! $- \frac {11}{8} + \frac 6 8 = \frac {-11 + 6} {8} = \frac {-5}{8}$

### Lowest Common Denominator

The *lowest common denominator* (LCD) of a set of fractions is the smallest number that has all of the denominators as a factor, e.g. the LCD of $\frac 7 8$ and $\frac {11}{16}$ is $16$ (16 divided by 16 and 8 all result in whole numbers). To find LCDs that are not obvious the highest denominator is increased in multiplies until it results in a number that has the lowest denominator as a factor.

The lowest common denominator can also be found by multiplying the denominators together: $\frac 3 4 \frac 2 7$ have an LCD of 28 ($3 \times 4$).

### Finding a Missing Numerator in Equal Fractions

Assuming one numerator in a set of equal fractions is missing $\frac 3 5 = \frac x {10}$, the missing value can be found by multiplying the factor that multiplies the lower denominator to the higher with the known numerator: $\frac 3 5 = \frac {3 \times 2}{10}$.

### Setting Up Fractions For Addition and Subtraction

In order to add or subtract fractions with differing denominators the first step is to convert them into fractions with the same denominators: $\frac 3 4 + \frac 1 6$

- Find the LCD (16)
- Multiply numerator and denominator of both fractions by the number required for both denominators to reach the LCD ($\frac {9}{12} + \frac {2}{12}$)
- Add/Subtract ($\frac {11}{12}$)

If one fraction is negative, problems can be solved easiest by moving the minus sign to the numerator first ($-\frac {1}{18} \frac {-1}{18} \frac {1}{-18}$ are all equal).

$\frac {3x} {4x} - \frac {20}{4x} = \frac {3x - 20} {4x}$ is in its simplest form because variables cannot be simplified when numerator or denominator are an expression.

### Complex Fractions

A *complex fraction* is a fraction with a fraction in either of its values, e.g. $\cfrac{\frac{7x}{10}}{\frac{1}{5}}$. Dealing with them simply involves multiplying the numerator with the reciprocal of the denominator of the main fraction.

$\cfrac{\frac{7x}{10}}{\frac{1}{5}} = \frac {7x}{10} \times \frac {5}{1}$

## Decimals and Place Value

Place values behind the decimal are called *tenths*, *hundredths*, *thousandths* ...

Decimals can be written as fractions, examples:

$\frac {43}{100} = 0.42$ (dividing by 100 makes decimal end in the hundreth)

$\frac {501}{1000} = 0.501$ (dividing by 1000 makes decimal end in thousandth).
$0.2193 = \frac {2193}{10000}$

$-2.12 = \frac {-212}{100}$

If a number's place value repeats infinitely it's denoted by a stroke above the number: $\frac 2 3 = 0.6\bar{6}$