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%#

1 Answer
Oct 11, 2016

Answer:

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#