# Thread: The difference between two positive integers is 4 and ......

1. ## The difference between two positive integers is 4 and ......

Hi guys, I have a question from my workbook which i don't understand and know how to do.

Question: The difference between two positive integers is 4 and the difference between this reciprocals is (1)/(48). Find the integers.

Ok so I have form this two equations,

Let x be first integer
Let y be second integer

x - y = 4
(1)/(x) - (1)/(y) = (1)/(48)

But I don't know how to continue from there, I don't even know whether my equation is the right one.

2. Your equations are correct.

From equation 1, it is clear that x = y + 4.

So substitute this into equation 2 to get

1/(y + 4) - 1/y = 1/48

y/[y(y + 4)] - (y + 4)/[y(y + 4)] = 1/48

[y - (y + 4)]/[y(y + 4)] = 1/48

-4/[y(y + 4)] = 1/48

-4 = [y(y + 4)]/48

-192 = y(y + 4)

Now expand and set the equation equal to 0 and solve the resulting quadratic.

3. I managed to get to here but, how do i solve "-192 = y^2 + 4y"

4. Originally Posted by FailInMaths
I managed to get to here but, how do i solve "-192 = y^2 + 4y"
y^2 + 4y + 192 = 0 is a quadratic equation. When all else fails, use the quadratic formula.

-Dan

Edit: Interesting. I was about to say that this factors, but I'm getting only complex zeros.

Edit II: Ah. I have it now. There was an understandable mistake. Notice that if x is the larger number then 1/x - 1/y is negative. So your two equations are
x - y = 4
1/x - 1/y = -1/48

This gives a reasonable solution set.

5. Originally Posted by FailInMaths
Question: The difference between two positive integers is 4 and the difference between this reciprocals is (1)/(48). Find the integers.
Ok so I have form this two equations,
Let x be first integer Let y be second integer
x - y = 4
(1)/(x) - (1)/(y) = (1)/(48)
Originally Posted by Prove It
Your equations are correct.
-192 = y(y + 4)
The solutions to the above are two complex numbers.
The reason being is that your equations are not correct.
If you have $x-y=4$ where $x~\&~y$ are positive integers then that implies that $x>y$.
Thus you need to have $\frac{1}{y}-\frac{1}{x} =\frac{1}{48}$ because we need $\frac{1}{y}>\frac{1}{x}.$

Now use the same ideas in post #2.