## Friday, September 4, 2015

### Fractions

Many people struggle with fractions. It is important to know all the rules and how to apply them because the reason people struggle with this section is that they do not know these rules.

I have used numbers when explaining these rules for the first time. Please note that all of these rules still apply when using algebraic expressions such as x and y.

What is a fraction?

$$2 \div 3 = \frac{2}{3}$$

Writing '2 divided by 3' is the same thing as writing '2 over 3' in the form of a fraction.

$$\frac{\text{Numerator}}{\text{Denominator}}$$

The Numerator is the expression at the top of a fraction.

The Denominator is the expression at the bottom of a fraction.

Types of Fractions

A Proper Fraction is a fraction where the Numerator is smaller than the Denominator. It is called proper because it represents a fraction of a whole.

$$\frac{1}{2}$$

This represents 50% or 0.5 which makes 'proper' sense in terms of it being a 'fraction'. Therefore it is called a Proper Fraction.

An Improper Fraction is a fraction where the Numerator is greater than the Denominator. It is called improper because it represents more than just a fraction of a whole.

$$\frac{2}{1}$$

This represents 200% or 2 which does not make 'proper' sense in terms of this being a 'fraction' but it does make mathematical sense where we just have more than one whole. Therefore it is referred to as an Improper Fraction

A Mixed Fraction is an improper fraction written in a different way. We take the number of whole numbers separately and just show the remainder as a proper fraction.

$$\frac{5}{2} = \frac{4}{2} + \frac{1}{2} = 2 \frac{1}{2}$$

Please note that $$2 \frac{1}{2} \not = 2 \times \frac{1}{2}$$ although many people think it is because of the way that it is written!

Simplifying Fractions

The best way to simplify fractions (without a calculator) is to divide both the numerator and denominator (fraction must remain unchanged) by the biggest number that you can think of that will go into both the numerator and denominator without any remainder. If you are not in simplest form then you can divide by another number until you are.

$$\frac{27}{36} = \frac{27 \div 3}{36 \div 3} = \frac{9}{12}$$
$$= \frac{9 \div 3}{12 \div 3} = \frac{3}{4}$$

Now the fraction is in its simplest form.

When Adding Fractions, we need to find the LCD (Lowest Common Denominator) which is a denominator that can be common for all of the fractions we are adding together. You can the simply add all the numerators over this common denominator.

$$\frac{1}{2} + \frac{3}{8} = \frac{1 \times 4}{2 \times 4} + \frac{3}{8}$$
$$= \frac{4}{8} + \frac{3}{8} = \frac{4+3}{8} = \frac{7}{8}$$

Did you see how we multiplied the top and bottom of the fraction by 4 to get the denominator to be 8 without changing the fraction (since we multiply and divide by the same number).

When Subtracting Fractions, the approach is the same but replace the + signs with - signs.

$$\frac{1}{2} - \frac{3}{8} = \frac{1 \times 4}{2 \times 4} - \frac{3}{8}$$
$$= \frac{4}{8} - \frac{3}{8} = \frac{4-3}{8} = \frac{1}{8}$$

Multiplying Fractions

When Multiplying Fractions, we multiply the Numerators of the fractions and the Denominators of the fractions together.

$$\frac{2}{5} \times \frac{3}{2} = \frac{2 \times 3}{5 \times 2} = \frac{6}{10} = \frac{3}{5}$$

Note that we can also cancel / simplify numerators with denominators of other fractions when fractions are multiplied together.

$$\frac{\not 2}{5} \times \frac{3}{\not 2} = \frac{3}{5}$$

Dividing Fractions

When Dividing Fractions, it is always best to use the Tip and Times Rule (you swap the numerator and denominator of the second fraction around and change the sign from $$\div$$ to $$\times$$ ).

$$\frac{3}{2} \div \frac{5}{2} = \frac{3}{2} \times \frac{2}{5}$$
$$= \frac{3}{ \not 2} \times \frac{\not 2}{5} = \frac{3}{5}$$

Exponents / Roots

An Exponent / Root that is applied to an entire fraction can be applied to both the numerator and denominator separately.

$$\sqrt{\frac{4}{100}} = \frac{\sqrt{4}}{\sqrt{100}} = \frac{2}{10} = \frac{1}{5}$$

$$\sqrt{\frac{4}{100}}$$ means 'square root of 4 over 100'. This can be extended to cube roots, 5th roots, 12th roots, etc.

Note that $$(10)^2 = 10 \times 10 = 100$$ so $$\sqrt{100} = \sqrt{10 \times 10} = 10$$

$$10^2$$ means 'ten squared'. This can be extended to to cubed, to the power of 4, to the power of 5, to the power of a billion, etc.

$$(\frac{3}{10})^3 = \frac{3^3}{10^3} = \frac{3 \times 3 \times 3}{10 \times 10 \times 10}$$
$$= \frac{27}{1000}$$