# What is the value of? 1/3÷4

Apr 1, 2018

$\frac{1}{12}$ is the value.

#### Explanation:

What you do is the KCF method. Keep, Change, Flip. You would keep the $\frac{1}{3}$. Then you change the divide sign to a multiply sign. Then you flip the $4$ to $\frac{1}{4}$. You do that since $\frac{1}{4}$ is the reciprocal of $4$.

$\frac{1}{3} \div 4 = \frac{1}{3} \times \frac{1}{4}$

Apr 2, 2018

$\frac{1}{12}$

#### Explanation:

You can work it out using the usual fraction division process, or just through what is happening...

If you take one third and cut it in half ( same as dividing by $2$), then each piece will be $\frac{1}{6}$. (More pieces, therefore they get smaller)

If you take $\frac{1}{6}$ and cut it in half, the pieces get smaller again. Each piece will be $\frac{1}{12}$

$\frac{1}{3} \div 4 = \frac{1}{3} \div 2 \div 2 = \frac{1}{12}$

A nifty short cut: To divide a fraction in half, either halve the top (if it is even) or double the bottom:

$\frac{2}{3} \div 2 = \frac{1}{3}$

$\frac{4}{11} \div 2 = \frac{2}{11} \text{ } \leftarrow$ pretty obvious if you think about it!!

$\frac{5}{9} \div 2 = \frac{5}{18}$

$\frac{7}{8} \div 2 = \frac{7}{16}$

In the same way: To divide a fraction by $3$ in half, either divide the by $3$ (if possible) or treble the bottom:

$\frac{6}{11} \div 3 = \frac{2}{11} \text{ } \leftarrow$ share out $6$ portions equally.

$\frac{5}{8} \div 3 = \frac{5}{24}$

Apr 7, 2018

This is why the 'turn upside down and multiply' works.

#### Explanation:

$\textcolor{b l u e}{\text{Answering the question using the shortcut method}}$

Write as $\frac{1}{3} \div \frac{4}{1}$

giving: $\frac{1}{3} \times \frac{1}{4} = \frac{1 \times 1}{3 \times 4} = \frac{1}{12}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{w h i t e}{}$

$\textcolor{b l u e}{\text{The teaching bit}}$

A fraction structure is such that we have:

$\left(\text{numerator")/("denominator") ->("count")/("size indicator of what you are counting}\right)$

YOU CAN NOT $\textcolor{red}{\underline{\text{DIRECTLY}}}$ ADD, SUBTRACT OR DIVIDE ONLY THE COUNTS UNLESS THE SIZE INDICATORS ARE THE SAME.

You have been applying this rule for years without realising it!
Consider the numbers: 1,2,3,4,5 and so on. Did you know that it mathematically correct to write them as: $\frac{1}{1} , \frac{2}{1} , \frac{3}{1} , \frac{4}{1} , \frac{5}{1}$ and so on. So their SIZE INDICATORS ARE THE SAME.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Explaining the principle using a different example}}$

$\textcolor{b r o w n}{\text{I have chosen to use a different example as I wished}}$$\textcolor{b r o w n}{\text{to avoid using 1's. In avoiding 1's the behaviour is more obvious.}}$

Consider the example color(green)(3/color(red)(4)-:2/color(red)(8)")

Turn upside down and change the sign to multiply

color(green)(3/color(red)(4)xxcolor(red)(8)/2 larr" as per the method"

Note that: $4 \times 2 = 8 = 2 \times 4.$ This is commutative.

Using the principle of being commutative swap the 4 and 2 round the other way giving:

color(green)(color(white)("ddd")ubrace(3/2)color(white)("ddd")xxcolor(white)("ddd")color(red)(ubrace(8/4))

$\textcolor{g r e e n}{\text{directly dividing ") color(red)("Converting the}}$
color(green)(color(white)("dd")"the counts")color(white)("ddddddd") color(red)("counts")

Now split them up like this:

$\left(\textcolor{g r e e n}{3} \times \textcolor{red}{\frac{8}{4}}\right) \div \textcolor{g r e e n}{2}$

$\textcolor{m a \ge n t a}{\textcolor{w h i t e}{\text{ddd") 6 color(white)("dddd}} \div 2}$

And compare to the original of color(green)([3/color(red)(4)]-:2/color(red)(8)")

$\textcolor{w h i t e}{}$

$\textcolor{g r e e n}{\left[\frac{3}{\textcolor{red}{4}} \textcolor{b l a c k}{\times \frac{2}{2}}\right] \textcolor{g r e e n}{\div} \frac{2}{\textcolor{red}{8}}} \textcolor{w h i t e}{\text{dddd")->color(white)("dddd}} \frac{\textcolor{m a \ge n t a}{6}}{8} \div \frac{\textcolor{m a \ge n t a}{2}}{8}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So the $\textcolor{red}{\frac{8}{4}}$ is the equivalent action of making the size indicators the same and adjusting the counts to suit.

$\textcolor{red}{\text{IT IS A CONVERSION FACTOR}}$
So by turning upside down' and multiplying you are applying a conversion and directly dividing the counts all at once.