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

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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 $30%$

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Answer 1 · 2016-10-11T12:05:45

Q has $300$ seats in the new government.

Explanation:

Let
$color(white)("XXX")p$ be the number of seats party P had in the old government.
$color(white)("XXX")q$ be the number of seats party Q had in the old government.
$color(white)("XXX")hatp$ be the number of seats party P has in the new government
$color(white)("XXX")hatq$ be the number of seats party Q has in the new government.

We are told
$color(white)("XXX")hatq-color(magenta)hatp=2(color(red)p-color(blue)q)$
To solve for $hatq$ will will need to find expressions for $color(magenta)hatp, color(red)p, and color(blue)q$ in terms of $hatq$

$hatq$ is a $30%$ increase over $color(blue)q$
$rarr hatq=4/3color(blue)qcolor(white)("XX")rarr color(blue)(q)=3/4hatq$

$color(red)p+color(blue)q = 500$
$rarr color(red)p=500-color(blue)q$
$rarr color(red)p=500-3/4hatq$

$color(magenta)hatp+hatq=500$
$rarr color(magenta)hatp=500-hatq$

Therefore $hatq-color(magenta)hatp=2(color(red)p-color(blue)q)$
can be written as
$color(white)("XXX")hatq-(500-hatq)=2((500-3/4hatq)-3/4hatq)$

$color(white)("XXX")-500+2hatq= 1000 -3hatq$

$color(white)("XXX")5hatq=1500$

$color(white)("XXX")hatq=300$

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

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