Question

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%`

Answer 1

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`