### Fractions

The top line of a fraction is called the **numerator** and bottom line of a fraction is called the **denominator**.

**Types of Fractions**A

**proper**fraction is a fraction where the numerator is smaller than the denominator. eg

^{3}⁄

_{8}and

^{1}⁄

_{5}are

**proper**fractions.

An

**improper**fraction is a fraction where the numerator is bigger than the denominator. eg

^{8}⁄

_{7}and

^{15}⁄

_{4}are

**improper**fractions.

A

**mixed**number is an integer and a fraction written together. eg 3

^{1}⁄

_{2}and 1

^{1}⁄

_{4}

Equivalent fractions represent the same number. They can be simplified or cancelled down to fractions of equal value.

e.g. {

^{1}⁄

_{2},

^{4}⁄

_{8},

^{9}⁄

_{18},

^{27}⁄

_{54}} are equivalent fractions because they can all be simplified to

^{1}⁄

_{2}.

**Operations on Fractions**

Always begin by changing any mixed numbers into improper fractions.

If the calculation is very simple then the integers can be added together first, followed by the fractions. e.g. 3^{1}⁄_{4} + 7^{1}⁄_{2} = 10^{3}⁄_{4}

Addition and subtraction

Multiplication and Division

Step 1: Simplify by cancelling or dividing out common factors between top and bottom lines.Step 2: Multiply the numerators together and then multiply the denominators together.DivisionTurn the second fraction upside down (to make it into the reciprocal) and multiply.

### Decimals

The decimal system is based around the number 10. eg **34.98** and **0.375**The part of the number in front of the dot (called the decimal point) is an integer.

The part of the number after the dot is a decimal fraction.

**Types of Decimals**

- A
**terminating**decimal is one that has a finite number of digits. eg 0.5 and 0.875

- A
**recurring**or**repeating**decimal is one that has a repeating sequence of digits.

Recurring decimals are shown by a dot above the recurring digits or at the beginning and end of the repeating sequence.e.g. 0.3333... = 0.1666... = 0.207207... =

All fractions can be represented by terminating or recurring decimals.

- A
**non-repeating**decimal is a decimal that contains a non-repeating sequence of decimal digits.

eg 0.810675469... does not repeat.

**Operations with Decimals**

Calculators can be used to carry out the following operations involving decimals.

However, it is useful to be able to do these basic operations without a calculator.

Adding and subtractingMake sure that the decimal points are in line.

See

Multiplying

To multiply by multiples of 10, move the decimal point to theright.When multiplying two decimal numbers, carry out the calculation ignoring the decimal points. Place the decimal point in the answer so that the

answerhas thesamenumber of decimal places as thetotalnumber of places in the two numbers being multiplied.

See

DivisionWhen dividing by multiples of 10, move the decimal point to theleft.When dividing two decimal numbers, write the calculation as a fraction.

Move the decimal point in both the numerator and the denominator the same number of decimal places needed to make the bottom line into a whole number. Then carry out normal division.

See

### Percentages

Percentage means "out of 100" or "per hundred."

Calculations involving fractions often require multiplication or division by 100.

x percent, written x%, means ^{x}⁄_{100} (or x parts out of 100)

e.g. 40% = ^{40}⁄_{100}

= ^{20}⁄_{50} = ^{10}⁄_{25} = ^{2}⁄_{5}

**Calculations involving percentages**

Below are some of the types of problem that use percentages.

CalculationExamplesSolutions

To find x percent of a quantity.

Calculate:^{x}⁄_{100}× (the quantity)Find 8% of 300

300 × ^{8}⁄_{100}= 24

To find a out of b as a percentage.

Calculate:⁄^{a}×_{b}^{100}⁄_{1}Write 13 out of 20 as a percentage

^{13}⁄_{20}×^{100}⁄_{1}= 65%

To increase a quantity by x percent.

Calculate:Increase 50 by 8%

To decrease a quantity by x percent.Calculate:Decrease 50 by 8%

**Calculations involving percentages and money**

Below are some of the types of problems involving money.

CalculationExamplesSolutions

To add on GST (at 15%) to a price.Calculate: 0.15 × (the price)

How much GST should be added to $80? GST = 80 × 0.15 = $12

To find the percentage profit of a sale.Calculate:

A boy buys a skateboard for $50 and sells it for $60.

What is the percentage profit for the sale?

Amount of profit = $60 − $50 = $10

Percentage profit =

^{10}⁄_{50}× 100 = 20%

To find the percentage loss of a sale.Calculate:

A shopkeeper buys a walkman for $250 and sells it for $150.

What is the percentage loss for the sale?

Amount of loss = $250 − $150 = $100

Percentage loss =

^{100}⁄_{250}× 100 = 40%

To find a discount percentage.Calculate:

A shirt originally costs $40. It is sold at a discounted price of $30.

What percentage discount is this?

Discount = $40 − $30 = $10

Percentage discount =

^{10}⁄_{40}× 100 = 25%