Question 316218
The ratio A:B is 2:3. We can rewrite this as {{{a/b=2/3}}}. 

Multiplying both sides by b gives {{{a=(2/3) b}}}. This is our first equation. 


If you add 100 to each, we get {{{(100+a)/(100+b)=3/4}}}. 

Multiplying both sides by 4 gives {{{ ((100+a)/(100+b)) 4 = 3 }}}. 

Multiplying both sides by (100+b) gives {{{ (100+a)4 = 3(100+b) }}}. 

Distributing gives {{{400 + 4a = 300 + 3b}}}. 

Now we can substitute {{{a = (2/3) b}}} into this equation to get {{{400 + 4((2/3) b) = 300 + 3b}}}. 

Simplifying this gives {{{400 + (8/3) b = 300 + 3b}}}. 

Subtracting 300 from both sides gives {{{100 + (8/3) b = 3b}}}. 

Since 3 = 9/3, we can rewrite this as {{{100 + (8/3) b = (9/3) b }}}. 

Subtracting {{{(8/3) b}}} from both sides gives {{{100 = (1/3) b}}}. 

Multiplying both sides by 3 gives {{{ 300 = b }}}. 

Since {{{a=(2/3) b}}}, we can plug in b=300 to get {{{a = (2/3) b = (2/3) (300) = 200}}}. 

Now we check: the ratio of a=200 to b=300 is indeed 2:3. If we add 100 to both, we get the ratio (200+100)=300 to (300+100)=400, which is 300:400 which is 3:4 just as we wanted.