# Question #27055

Dec 4, 2015

$\textcolor{b l u e}{\text{Another way of looking at it!}}$
A bit long but it will be worth your while reading through it

#### Explanation:

I am about to show you something that you may not thought of before so please do read through it.

$\textcolor{b l u e}{\text{Introduction}}$
Consider when you divide whole numbers, for example $4 \div 2$. The response is normally immediate. But have you ever considered beyond the 'just doing it' and think about the question: Why can I divide these numbers directly but not some others?

$\textcolor{b l u e}{\text{Expansion of concept and reasoning}}$
There is very large set of number that are given the name 'Rational Numbers'. Remembering the name is not important but remembering the logic behind them is!
A rational number is any number that may be expressed in a special form using whole numbers. That is $\left(\text{Some whole number")/("Another whole number}\right)$

The top number (numerator) is the count of what you have. The bottom number (Denominator) is an indication of the size of what you are counting. The size is how many of what you are counting it takes to make a whole of something.

To $\underline{\text{directly}}$ add, subtract or divide the counts of what you have, the bottom numbers must be the same.

Back to $\textcolor{w h i t e}{.} 4 \div 2$. Although people don't do it, these may be written as $\frac{4}{1} \div \frac{2}{1}$. They are rational numbers. The bottom numbers are the same so you can $\underline{\text{directly}}$ add, subtract, or divide only using the $\underline{\text{top}}$numbers.

$\textcolor{b l u e}{\text{Application of concept}}$

Suppose you have two fractions, say , $\frac{1}{2} \text{ and } \frac{1}{4}$

If you change them so that they both have the same number 'underneath' (denominator) you can then apply your operations just using the top numbers (Numerators). This approach
fails for multiplication. Consider:

$\frac{1}{2} \div \frac{1}{4}$

Multiply $\frac{1}{2}$ by 1 but in the form of $\frac{2}{2}$ giving:

$\left[\left(\frac{1}{2} \times \frac{2}{2}\right) \div \frac{1}{4}\right] = \left[\frac{1 \times 2}{2 \times 2} \div \frac{1}{4}\right] = \left[\frac{2}{4} \div \frac{1}{4}\right]$

$\textcolor{red}{\text{Now just divide the top numbers and you have your answer.}}$

$\textcolor{b l u e}{\text{You are dividing counts of things that have the same unit size}}$
This is the same method that every one else uses but it just looks diverent