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I'm trying to understand fractions more by using a visual representation, and I having a hard time making a visual representation of the 1/6 of 4/5 and getting a visual understandable representation of the answer 4/30.

I'd appreciate If somebody could represent it for me.

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Look at the following grid:enter image description here

Suppose you want to take $\frac{1}{6}$ of it, then just take one of the rows, as shown in the following drawing:

enter image description here

Now, suppose we only want $\frac{4}{5}$ of that $\frac{1}{6}$. Then just take $4$ of the $5$ squares that make up that row:

enter image description here

At the end we ended up with $4$ squares, while the entire grid has a total of $30$. therefore $\frac{1}{6}\cdot\frac{4}{5}=\frac{4}{30}$

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    $\begingroup$ How can I make the images smaller? $\endgroup$ – Jorge Fernández Hidalgo Jun 3 '14 at 21:35
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    $\begingroup$ You could do something without graphics, like $$\begin{array}{ccccc} X&X&X&X&O\\O&O&O&O&O\\O&O&O&O&O\\O&O&O&O&O\\O&O&O&O&O\\O&O&O&O&O\end{array}$$ $\endgroup$ – MPW Jun 3 '14 at 21:46
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    $\begingroup$ Or, even better, $$\begin{array}{ccccc} \blacksquare&\blacksquare&\blacksquare&\blacksquare&\square\\ \square&\square&\square&\square&\square\\ \square&\square&\square&\square&\square\\ \square&\square&\square&\square&\square\\ \square&\square&\square&\square&\square\\ \square&\square&\square&\square&\square\end{array}$$ $\endgroup$ – MPW Jun 3 '14 at 21:50
  • $\begingroup$ Now all that's left is to explain why the word "of" translates into multiplication. $\endgroup$ – Mehrdad Jun 4 '14 at 11:04
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    $\begingroup$ Right-click and select "Show as LaTeX commands", you can see it. Its basically an array filled with \square and \blacksquare items $\endgroup$ – MPW Jun 5 '14 at 1:11
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Divide a cake into $30$ equal slices. $\frac45$ of the cake is represented by $\frac45\times 30=24$ slices. $\frac16$ of these $24$ slices is $\frac16\times 24=4$ slices, which is also $\frac{4}{30}$ of the whole cake.

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    $\begingroup$ Jasper Loy is a curious individual. In every 30-day month, he eats a hamburger every day, except when the day is divisible by 5. Every 6th time he eats a hamburger, he adds cheese. At the end of the month, he will have eaten 4 hamburgers with cheese. $\endgroup$ – Emily Jun 3 '14 at 21:36
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    $\begingroup$ Good job people - now you have me hungry :( $\endgroup$ – Dason Jun 4 '14 at 1:47
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Take 15 squares and shade 12 of them. Then you have shaded 4/5 of the squares. Now divide the 12 shaded squares into 6 equal pieces. Then each piece consists of two squares. So two squares represents 1/6 of 4/5 and two squares out of 15 squares equals 4/30.

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