01Afractions

=** Fractions **= toc

As you move into senior maths it is vital that your fraction skills are as strong as you can get them. Just to be sure, to be sure, we will quickly review them here.

A fraction should always be written as math \dfrac{3}{5} \quad \textit{ and not as } \quad {}^{3}\!\!\diagup\!\!{}_{5} math {The line in a fraction is called the __vinculum__ - the vinculum should always be horizontal}

Literacy
In any fraction,
 * the **numerator** is the value on the top of the fraction
 * the **denominator** is the value on the bottom of the fraction.

math \textit{In } \dfrac{3}{5} \; \textit{ the numerator is }3 \; \textit{ and the denominator is } 5 math
 * For example: **


 * A ** proper fraction ** has the numerator smaller than the denominator (like the example above)
 * An ** improper fraction ** has the numerator bigger than the denominator. It can be simplified to a mixed number.
 * A ** mixed number ** is formed with a whole number plus a proper fraction.

math \\ \textit{A proper fraction : } \frac{3}{5} \\ \\ \textit{An improper fraction : } \frac{8}{3} \\ \\ \textit{A mixed number : } 4\frac{1}{2} math
 * For example: **

Simplifying
Fractions should always be written in simplest form.

Divide both the numerator (top line) and denominator(bottom line) by the highest common factor (if there is one).

Mixed Numbers
To convert mixed numbers into improper fractions,
 * 1) multiply the whole number part by the denominator
 * 2) add the numerator to the result
 * 3) your answer is the numerator of the improper fraction (the denominator stays the same)

Adding & Subtracting

 * 1) Change any mixed numbers into improper fractions.
 * 2) Find the Lowest Common Denominator (LCD) {the smallest value that has both denominators as factors}
 * 3) Change the fractions so they both have the LCD.
 * 4) Add (or subtract) the numerators and keep the same denominator.
 * 5) Simplify by dividing numerator and denominator by the highest common factor (if there is one).

Multiplying

 * 1) Change any mixed numbers to improper fractions.
 * Multiply across the numerators __and__ multiply across the denominators, __then__ simplify the result (see Eg 3ai)
 * OR
 * Simplify first, __then__ multiply across the numerators __and__ multiply across the denominators (see Eg 3aii)


 * Note: ** The second method is usually easier because it makes the numbers to multiply smaller. You can cancel anything from the top row (numerators) with anything from the bottom row (denominators).

Dividing

 * 1) Change any mixed numbers to improper fractions.
 * 2) Flip the __second__ fraction upside down (ie write the reciprocal of the second fraction - the divisor)
 * 3) Multiply the resulting fractions.

Why this Works
Notice that if we start with any division, and multiply both values by the same amount, the result stays the same. Eg: math \\ 10 \div 5 = 2 \qquad \textit{so} \\ \\ \big( 10 \times 3 \big) \div \big( 5 \times 3 \big) = 2 \qquad \textit{and} \\ \\ \big( 10 \times 8 \big) \div \big( 5 \times 8 \big) = 2 math

This is also true if we multiply both values by a fraction, so: math \big( 10 \times \frac{1}{2} \big) \div \big( 5 \times \frac{1}{2} \big) = 2 math

If we deliberately multiply both values by the reciprocal of the divisor, the divisor simplifies to 1: math \big( 10 \times \frac{1}{5} \big) \div \big( 5 \times \frac{1}{5} \big) \\ \\ = \;\; \big( 10 \times \frac{1}{5} \big) \div \big( 1 \big) math

and since dividing by 1 has no effect, this simplifies to: math \\ = 10 \times \frac{1}{5} \\ = 2 math

We use that idea to simplify the division of 2 fractions: math \\ \dfrac{2}{5} \div \dfrac{3}{4} \;\; = \left( \dfrac{2}{5} \times \dfrac{4}{3} \right) \div \left( \dfrac{3}{4} \times \dfrac{4}{3} \right) \\ \\ . \qquad \quad = \left( \dfrac{2}{5} \times \dfrac{4}{3} \right) \div \big( 1 \big) \\ \\ . \qquad \quad = \dfrac{2}{5} \times \dfrac{4}{3} \\ \\ . \qquad \quad = \dfrac{8}{15} math

Scientific Calculator
Most scientific calculators have a fraction button that looks like this: From now on, I will use **[a b/c]** to indicate where you should press the fraction button.

math \text{To enter } \frac{3}{4} \;\; \text{type:} \;\; 3 \textbf{ [a b/c] } 4 math If you now press "=", the fraction will be displayed as: **3 r 4**

math \text{To enter } 1\frac{2}{3} \;\; \text{type:} \;\; 1 \textbf{ [a b/c] } 2 \textbf{ [a b/c] } 3 math If you now press "=", the fraction will be displayed as: **1 r 2 r 3**

Pressing **[a b/c]** again after pressing "=" swaps between fraction and decimal form.

Now proceed to enter your fraction problem on the calculator: math \frac{3}{4} \div 1\frac{2}{3} \qquad \text{type: } \;\; 3 \textbf{ [a b/c] } 4 \div 1 \textbf{ [a b/c] } 2 \textbf{ [a b/c] } 3 = math

You should get the result: **9 r 20** math \frac{3}{4} \div 1\frac{2}{3} = \frac{9}{20} math

Casio Classpad
To enter fractions on the Classpad, go to the Main calculation screen.

First set your calculator to STANDARD mode by clicking on the word DECIMAL at the bottom of the screen (see screenshot).

Then open up the virtual keyboard {by pressing the blue **Keyboard** button on the keypad.

Click on the 2D tab and select the top left icon in that tab ( it looks like a fraction) (see screen shot).

The form for a fraction should appear on your screen. Now click on the empty boxes and type in the numbers.

Now proceed to enter your fraction problem.

To enter a __mixed number__ you will have to use brackets and put "+" between the whole number part and the fraction part. {the bottom left icon of the 2D tab creates brackets - or just use the keypad}


 * [[image:01Aclasspadfracbutton.GIF align="left"]]Note: ** If you highlight any decimal or fraction on the screen, the icon on the left of the row at the top of the screen will swap between decimal and fraction form.

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