SOLUTION: the difference between two numbers is eight. If the smaller number exceeds half of the larger number by fourteen, what are the numbers?

Question 758461: the difference between two numbers is eight. If the smaller number exceeds half of the larger number by fourteen, what are the numbers?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let a represent the larger number
let b represent the smaller number

formulas become:

a - b = 8 (difference between the numbers is 8).
b > (a/2) + 14 (smaller numbrer exceeds 1/2 * bigger number by 14)
solve for b in the first equationj to get b = a - 8
substitute for b in the second equation to get:
a - 8 > (a/2) + 14
multiply both sides of this equationm by 2 to get:
2a - 16 > a + 28
subtract a from both sides of this equation and add 16 to both sides of this equation to get:
2a - a > 28 + 16
simplify to get:
a > 44
when a > 44, both equations will be satisfied.
as an example, let a = 50
when a = 50, 50 - b = 8 which makes b = 42
you have a 50 and b = 42
first equation is satisfied because 50 - 42 = 8
second equation becomes:
42 > (50/2) + 14 which becomes:
42 > 25 + 14 which becomes:
42 > 39 which is true so the second equation is satisfied as well.
since both equations are satisfied, your answer is a > 44.
you can also confirm by making a = 44 and a < 44 and seeing that the equation is not satisfied.
for example:
let a = 44
this makes b = 36
first equation is satisfied because 44 - 36 = 8
second equation becomes:
36 > (44/2) + 14 which becomes:
36 > 22 + 14 which becomes:
36 > 36 which is NOT true because 36 = 36.
when a < 44, you will get similar results.
for example:
let a = 40
this makes b = 32
first equation is satisfied because 40 - 32 = 8
second equation becomes:
32 > (40/2) + 14 which becomes:
32 > 20 + 14 which becomes:
32 > 34 which is not true because 32 < 34.
the solution of a > 44 is good because:
when a > 44 both equations are satisfied.
when a <= 44 both equations are not satisfied.

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