Compute the maximum value of x+y for all positive integer pairs (x,y) that satisfy 15x+55y=2000
if you could explain to me how you got it that would be great!
2007-10-01 11:32:53 · 3 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
128
I like to analyse maths problems by looking at extremes.
You can purchase an 'x' for 15 bucks.
You can purchase a y for 55 bucks.
You have 2000 bucks.
How can you end up with the most stuff?
By buying as many 'x' as you can.
What is the maximum?
You have to spend exactly 2000.
2000/15 = 133.333...
(you can't buy 0.333)
So, go down the list:
133*15 = 1995, leaving 5 bucks.
132*15 = 1980, leaving 20 bucks
How long must we go until the "change" is a multiple of 55?
5, 20, 35, 50....
Keep going until you have a multiple of 55 (55, 110, 165, 220), making sure to stop at the first one you find.
find out how many 'y' you can buy. divide what is left by 15 to find how many 'x' you get.
Verify the calculation 15 x + 55 y = 2000 just to make sure
Add x + y to get the answer.
---
Do the same thing with the maximum number of y, to convince yourself that the answer is not at the other end.
2007-10-01 11:44:12 · answer #1 · answered by Raymond 7 · 0⤊ 0⤋
First solve for general integer solution. Mod 15:
10y = 5 mod 15
so y=5mod 15 set y= 5 + 15t then x= 126 -55t and then
x + y = 128 -40t which maxes out at (t = 0) 128.
2007-10-01 19:53:27 · answer #2 · answered by knashha 5 · 0⤊ 0⤋
szxdcfvgbhnj
2007-10-01 18:36:45 · answer #3 · answered by flower 1 · 0⤊ 0⤋