# How many seats did party Q have after the change in government as describe below?

## A certain country is ruled by a government with 500 seats divided between 2 political parties: P and Q. After a recent change in government, party Q gained control with a number of seats more than party P which was twice as many as the number of seats party P had more than party Q before the change. Party Q's number of seats increased by 30%

Oct 11, 2016

Q has $300$ seats in the new government.

#### Explanation:

Let
$\textcolor{w h i t e}{\text{XXX}} p$ be the number of seats party P had in the old government.
$\textcolor{w h i t e}{\text{XXX}} q$ be the number of seats party Q had in the old government.
$\textcolor{w h i t e}{\text{XXX}} \hat{p}$ be the number of seats party P has in the new government
$\textcolor{w h i t e}{\text{XXX}} \hat{q}$ be the number of seats party Q has in the new government.

We are told
$\textcolor{w h i t e}{\text{XXX}} \hat{q} - \textcolor{m a \ge n t a}{\hat{p}} = 2 \left(\textcolor{red}{p} - \textcolor{b l u e}{q}\right)$
To solve for $\hat{q}$ will will need to find expressions for $\textcolor{m a \ge n t a}{\hat{p}} , \textcolor{red}{p} , \mathmr{and} \textcolor{b l u e}{q}$ in terms of $\hat{q}$

$\hat{q}$ is a 30% increase over $\textcolor{b l u e}{q}$
$\rightarrow \hat{q} = \frac{4}{3} \textcolor{b l u e}{q} \textcolor{w h i t e}{\text{XX}} \rightarrow \textcolor{b l u e}{q} = \frac{3}{4} \hat{q}$

$\textcolor{red}{p} + \textcolor{b l u e}{q} = 500$
$\rightarrow \textcolor{red}{p} = 500 - \textcolor{b l u e}{q}$
$\rightarrow \textcolor{red}{p} = 500 - \frac{3}{4} \hat{q}$

$\textcolor{m a \ge n t a}{\hat{p}} + \hat{q} = 500$
$\rightarrow \textcolor{m a \ge n t a}{\hat{p}} = 500 - \hat{q}$

Therefore $\hat{q} - \textcolor{m a \ge n t a}{\hat{p}} = 2 \left(\textcolor{red}{p} - \textcolor{b l u e}{q}\right)$
can be written as
$\textcolor{w h i t e}{\text{XXX}} \hat{q} - \left(500 - \hat{q}\right) = 2 \left(\left(500 - \frac{3}{4} \hat{q}\right) - \frac{3}{4} \hat{q}\right)$

$\textcolor{w h i t e}{\text{XXX}} - 500 + 2 \hat{q} = 1000 - 3 \hat{q}$

$\textcolor{w h i t e}{\text{XXX}} 5 \hat{q} = 1500$

$\textcolor{w h i t e}{\text{XXX}} \hat{q} = 300$