# Alice baked a certain number of pies. She gave 1/8 of the pies to her friends and 1/4 of the remainder to her neighbor She was left with 63 pies. How many pies did Alice bake at first?

Apr 17, 2018

Original count of pies is 96

#### Explanation:

Let the original count of pies be represented by $x$

She gave $\frac{1}{8}$ of the pies to her friend
Thus for the remainder 'remove' $\frac{1}{8}$ of the count $\to x - \frac{x}{8}$

A quarter of this given to the neighbour

So the remainder after this is:

$\textcolor{g r e e n}{\left(x - \frac{x}{8}\right) - \frac{x - \frac{x}{8}}{4}}$

We are told that this count is 63. So we have:

$\textcolor{g r e e n}{\left(x - \frac{x}{8}\right) - \frac{x - \frac{x}{8}}{4} = 63}$
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$\textcolor{g r e e n}{\left(x - \frac{x}{8}\right) - \left(x - \frac{x}{8}\right) \frac{1}{4} = 63}$

$\textcolor{g r e e n}{\frac{8 x - x}{8} - \left(\frac{8 x - x}{8}\right) \frac{1}{4} = 63}$

$\textcolor{g r e e n}{\frac{7 x}{8} - \frac{7 x}{32} = 63}$

$\textcolor{g r e e n}{\left[\frac{7 x}{8} \textcolor{red}{\times 1}\right] - \frac{7 x}{32} = 63 \textcolor{w h i t e}{\text{dddd") -> color(white)("dddd}} \left[\frac{7 x}{8} \textcolor{red}{\times \frac{4}{4}}\right] - \frac{7 x}{32} = 63}$

color(green)((28x)/32-(7x)/32=63

color(green)((21x)/32=63

Multiply both sides by $\frac{32}{21}$

$\textcolor{g r e e n}{x = \frac{32}{\cancel{21}} ^ 1 \times {\cancel{63}}^{3}}$

$\textcolor{g r e e n}{x = 96}$