Susan is half the age of Amy. Five years ago, Susan was three-fourths as old as Amy. How old is each?

Let's call Susan's currrent age: #s#

And, let's call Amy's current age: #a#

Right now we can say:

#s = 1/2a#

But, fives years ago when Susan was #(s - 5)# years old and Amy was #a - 5# years old we can write:

#(s - 5) = 3/4(a - 5)#

From the first equation we can substitute #1/2a# for #s# into the second equation and solve for #a#:

#(s - 5) = 3/4(a - 5)# becomes:

#(1/2a - 5) = 3/4(a - 5)#

#1/2a - 5 = (3/4 xx a) - (3/4 xx 5)#

#1/2a - 5 = 3/4a - 15/4#

#-color(red)(1/2a) + 1/2a - 5 + color(blue)(15/4) = -color(red)(1/2a) + 3/4a - 15/4 + color(blue)(15/4)#

#0 - 5 + color(blue)(15/4) = -color(red)(1/2a) + 3/4a - 0#

#-5 + color(blue)(15/4) = -color(red)(1/2a) + 3/4a#

#(4/4 xx -5) + color(blue)(15/4) = (2/2 xx -color(red)(1/2a)) + 3/4a#

#-20/4 + color(blue)(15/4) = -color(red)(2/4a) + 3/4a#

#-5/4 = (-color(red)(2/4) + 3/4)a#

#-5/4 = 1/4a#

#color(red)(4) xx -5/4 = color(red)(4) xx 1/4a#

#cancel(color(red)(4)) xx -5/color(red)(cancel(color(black)(4))) = cancel(color(red)(4)) xx 1/color(red)(cancel(color(black)(4)))a#

#-5 = a#

#a = -5#

We can now substitute #-5# for #a# in the first equation and calculate Susan's age.

#s = 1/2a# becomes:

#s = (1/2 xx -5)#

#s = -5/2#

#s = -2.5#

This problem does not make sense unless you state:

Amy is not born yet and will be born in 5 years

Susan is also not born yet but will be born in #2 1/2# years.