Find the number of pairs of integers (x,y)?

Find the number of integers (x,y) with 0<x, y<10 that satisfy \frac{1}{1-\frac{10}{x}} > 1 - \frac{5}{y}

1 Answer
May 20, 2017

16

Explanation:

Given:

1/(1-10/x) > 1-5/y

As is typical when solving inequalities, it may be helpful to look at the corresponding equation first:

1/(1-10/x) = 1-5/y

Adding 5/y - 1/(1-10/x) to both sides we get:

5/y = 1-1/(1-10/x)

color(white)(5/y) = ((color(red)(cancel(color(black)(1)))-10/x)-color(red)(cancel(color(black)(1))))/(1-10/x)

color(white)(5/y) = (-10/x)/(1-10/x)

color(white)(5/y) = (10/x)/(10/x-1)

color(white)(5/y) = 10/(10-x)

Taking the reciprocal of both ends, we find:

y/5 = (10-x)/10 = 1-x/10

Multiplying both sides by 5 we get:

y = 5-x/2

Hence, in order to satisfy the original inequality, we need one (which one to be determined) of these:

y < 5-x/2" " or " "y > 5-x/2

To find out which one we need we could carefully analyse what has gone before, but it is probably easier to just try.

Consider x=1 and look at y=4 and y=5...

With x=1, we have:

1/(1-10/x) = 1/(1-10) = -1/9

With y=4, we have:

1-5/y = 1-5/4 = -1/4

With y=5, we have:

1-5/y = 1-5/5 = 0

So we need the condition that gives us y=4, namely:

y < 5-x/2

So for each value of x we find the possible values of y are as follows:

x=1:" "y = 1,2,3,4

x=2, 3:" "y = 1,2,3

x=4, 5:" "y = 1,2

x=6, 7:" "y = 1

giving a total of 4+6+4+2 = 16 possible pairs.