Let first house be x
Let the second house be y
Let unknown portion of ratio be #z#
Then year 1 #-> x_1 ; y_1#
Then year 2#-> x_2; y_2#
#color(blue)("Expressing ratios in fractional format")#
#color(red)("Do not confuse this format with fractions of the whole!")#
Year 1 #x_1/y_1 = 16/23# ....................( 1 )
Year 2# x_2/y_2=9/11#........................( 2 )
#color(blue)("Ratios expressed as fractions of the whole")#
#x_1 -> 16/(16+23)#
#y_1 -> 23/(16+23)#
.==============================================
#color(blue)("Note that keeping fractional values reduces rounding error")#
Let year 1 be #t_1#
Let year 2 be #t_2#
Where house 1 and 2 are #x " and "y#
Let #t_1 -> x_1/y_1 -= 16/23# as a ratio
Let #t_2 -> x_2/y_2 -= 9/11# as a ratio
.==============================================
#color(green)("Taking it a step at a time and expressing mathematically:")#
#color(green)("It is given that " x_2 = x_1 + 25/100 x_1)#
#color(green)(x_2 =x_1(1+1/4))#
#color(green)("That is: "x_2 =5/4x_1)#
#color(green)("It is also given that "y_2= y_1 + 5200)#
#color(green)("So " x_2/y_2 =(5/4 x_1)/(y_1+5200) = 9/11)#
#color(green)(11(5/4x_1) = 9(y_1+5200))# .....................( 3 )
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Changing two unknowns to just 1, making it solvable.")#
#color(brown)("But from (1) "y_1 = 23/16 x_1)# ............( 4 )
#color(brown)("Substitute (4) into (3) giving:")#
#color(brown)(55/4x_1 = 9(23/16x_1) + 9(5200))#
#color(brown)("Collecting like terms")#
#color(brown)(55/4x_1 -9(23/16x_1)= 9(5200))#
#color(brown)(x_1(55/4 - 207/16)=46800)#
#color(brown)(13/16x_1= 46800)#
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(red)(x_1 =16/13 times 46800 =Rs57600)#...... (5)
so at #t_1 x_1/y_1 = 16/23 = 57600/y_1#
#color(red)(y_1 = (23 times 57600)/16= Rs82800) #
Check: #57600/82800 = 16/23#
