# Cassie has a wood board that is 3/4 foot long. She wants to cut the board into pieces that are 1/8 foot long. Which expression will help Cassie find how many pieces she can cut?

Oct 23, 2016

The expression that will help Cassie find out how many pieces she can cut is $\frac{3}{4} \div \frac{1}{8}$.

#### Explanation:

This question requires division. The expression used to find how many pieces Cassie can cut is:

$\frac{3}{4} \div \frac{1}{8}$

Oct 23, 2016

This is not building the expression required but explains the principle behind it. It demonstrates the principle upon which the shortcut is built.

#### Explanation:

A fraction consists of two parts the top number and the bottom one.

Think of the top number as the count and the bottom as the size indicator. So you have:

$\left(\text{count")/("size indicator")" "vec("proper names")" "("numerator")/("denominator}\right)$

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Consider the action example of dividing 3 into 6.

We can do this division directly because their size indicators are the same. That is, we have: $\frac{6}{1} \div \frac{3}{1}$. People do not normally write it this way but it is correct.

$\textcolor{b r o w n}{\text{To directly divide the counts the size indicator must be the same}}$

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So we need to see how many of $\frac{1}{8}$ will fit into $\frac{3}{4}$

We can not do this directly in one operation as the size indicators are not the same.

$\textcolor{g r e e n}{\text{Lets make the size indicators (denominators) the same}}$

$\textcolor{b r o w n}{\text{Multiply by 1 and you do not change the value but}}$
$\textcolor{b r o w n}{\text{1 comes in many forms.}}$

$\left[\frac{3}{4} \textcolor{m a \ge n t a}{\times 1}\right] \div \frac{1}{8} \text{ "->" } \left[\frac{3}{4} \textcolor{m a \ge n t a}{\times \frac{2}{2}}\right] \div \frac{1}{8}$

$\left[\frac{3 \times 2}{4 \times 2}\right] \div \frac{1}{8} \text{ "->" } \frac{6}{8} \div \frac{1}{8}$

Now that the size indicators (denominators) are the same we can just divide the counts (numerators)

$\left(\text{count")/("size indicator}\right) \to \left[\frac{6}{8} \div \frac{1}{8}\right] = 6 \div 1 = 6$

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I could explain more showing the connection between this and the shortcut method but what I have shown you will do for now.

If you wish further explanation leave me a message on https://socratic.org/users/abr-1