2 rational numbers are added. The sum is 1/4 more than the product. What are the rational numbers?

2 Answers
Sep 23, 2017

If one is #alpha != 1#, then the other is: #1+3/(4(alpha-1))#

Explanation:

Given:

#alpha+beta-alphabeta-1/4 = 0#

Then:

#beta(alpha-1)=alpha-1/4#

So:

#beta = (alpha-1/4)/(alpha-1) = (alpha-1+3/4)/(alpha-1) = 1+3/(4(alpha-1))#

Sep 24, 2017

There are any number of pairs than can be calculated from the given relationship. Two are derived below.
#(A,B) = (1, 4)# and
#(A,B) = (2, 1.333)#

Explanation:

Let A and B be the numbers. Then from the relationships we have this equation.
#A + B = 1.25 xx (A xx B)#

#(A + B)/1.25 = (A xx B)#
#0.8A + 0.8B = A xx B#
#0.8 + 0.8B/A = B#
#0.8B/A = B - 0.8# let #A = 1#

#0.8B = B - 0.8#
#0 = B - 0.8B - 0.8#
#0 = 0.2B - 0.8# ; # 0.2B = 0.8# ; #B = 4#
CHECK: #A + B = 1.25 xx (A xx B)#; #1 + 4 = 1.25 xx (1 xx 4)#
#5 = 1.25 xx 4# ; #5 = 5# Correct!

Let's try it with A = 2
#0.8B/2 = B - 0.8# ; #0 = B - 0.4B -0.8# ; #0.6B = 0.8#
#B = 1.333#
CHECK: #A + B = 1.25 xx (A xx B)#; #2 + 1.333 = 1.25 xx (2 xx 1.333)#
#3.333 = 1.25 xx 2.666# ; #3.333 = 3.333# Correct!