Question

The sum of two number is 8 and 15 times the sum of their reciprocal is also 8. How do you find the numbers?

Answer 1

3, 5

Explanation:

Let's call the two numbers `x` and `y`.

We're told that `x+y=8`

We're also told that 15 times the sum of their reciprocal is also 8. I'll interpret what this says this way:

`15(1/x+1/y)=8`

We have two equations and two variables, so we should be able to solve this. Let's first solve the first equation for `x`:

`x=8-y`

And now substitute into the second equation:

`15(1/(8-y)+1/y)=8`

`1/(8-y)+1/y=8/15`

`1/(8-y)(y/y)+1/y((8-y)/(8-y))=8/15`

`y/(y(8-y))+(8-y)/(y(8-y))=8/15`

`8/(y(8-y))=8/15`

Notice that with the numerators equal, we can say:

`y(8-y)=15`

`8y-y^2=15`

`y^2-8y+15=0`

`(y-3)(y-5)=0=>y=3,5`

And by substituting these values back into our first equation, we get that `x=5,3`

Now let's check our answer:

`15(1/x+1/y)=8`

`15(1/3+1/5)=8`

`15(5/15+3/15)=8`

`15(8/15)=8`

`8=8color(white)(000)color(green)root`