Fractions, Decimals and Percentages

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 31⁄2 and 11⁄4
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. 31⁄4 + 71⁄2 = 103⁄4

Addition and subtraction

    • If the denominators are different, change them to equivalent fractions with the same denominator.
    • Add or subtract the top lines only − never add or subtract the bottom lines!
    • Simplify where possible.
      See examples.gif
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.
  • Division Turn the second fraction upside down (to make it into the reciprocal) and multiply. 

See examples.gif

 

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... =
    Y9_Fractions_Decimals_and_Percentages_26.gif
    0.1666... =
    Y9_Fractions_Decimals_and_Percentages_27.gif
    0.207207... =
    Y9_Fractions_Decimals_and_Percentages_28.gif
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 subtracting
Make sure that the decimal points are in line. 
See examples.gif

Multiplying
To multiply by multiples of 10, move the decimal point to the right.

When multiplying two decimal numbers, carry out the calculation ignoring the decimal points. Place the decimal point in the answer so that the answer has the same number of decimal places as the total number of places in the two numbers being multiplied. 
See examples.gif

Division
When dividing by multiples of 10, move the decimal point to the left.

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 examples.gif

 

Percentage_man.jpgPercentages

A percentage is a way of writing a fraction with a denominator of 100.
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.

Calculation
Examples
Solutions

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: Y9_Fractions_Decimals_and_Percentages_15.gif

Increase 50 by 8%

 

 Y9_Fractions_Decimals_and_Percentages_16.gif

To decrease a quantity by x percent.
Calculate:Y9_Fractions_Decimals_and_Percentages_17.gif

Decrease 50 by 8%

 

 Y9_Fractions_Decimals_and_Percentages_18.gif

 

Calculations involving percentages and money

Below are some of the types of problems involving money.

Calculation
Examples
Solutions

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:   Y9_Fractions_Decimals_and_Percentages_20.gif

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:   Y9_Fractions_Decimals_and_Percentages_22.gif

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:   Y9_Fractions_Decimals_and_Percentages_24.gif

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%