# Rafael is going to have a party. Three times as many girls as boys told Rafael they would come. If nine out of ten girls said they would come, and six boys said they could not come, how many people did Rafael INVITE to the party?

May 10, 2018

$19$ people were invited to the party.

#### Explanation:

I'll start by assigning a few variables:

$b = \text{ boys invited}$
$b y = \text{ boys that said yes}$
$b n = \text{ boys that said no}$
$g = \text{ girls invited}$
$g y = \text{ girls that said yes}$
$g n = \text{ girls that said no}$

We can make a few equations:

$b = b y + b n$
$g = g y + g n$

And plug in what we know ($g y = 9$, $g n = 1$, $b n = 6$)

$b = b y + 6$
$10 = 9 + 1$

Use "Three times as many girls as boys told Rafael they would come" to make another equation:

$b y \times 3 = g y$

Get $b y$ by itself:

$\frac{b y \times \textcolor{red}{\cancel{3}}}{\textcolor{red}{\cancel{3}}} = \frac{g y}{3}$

$b y = \frac{g y}{3}$

Plug in $9$ for $g y$:

$b y = \frac{9}{3} = \frac{9 \div 3}{3 \div 3} = \frac{3}{1} = 3$

$b y = 3$

So now we know:

$b = \text{ boys invited} = 3 + 6 = \textcolor{red}{9}$
$b y = \text{ boys that said yes} = 3$
$b n = \text{ boys that said no} = 6$

$g = \text{ girls invited} = 9 + 1 = \textcolor{red}{10}$
$g y = \text{ girls that said yes} = 9$
$g n = \text{ girls that said no} = 1$

Our last step, now that we know that $9$ boys and $10$ girls were invited, is to add these numbers to get our final answer.

$b + g = \text{ total invitees}$
$9 + 10 = \textcolor{red}{19}$