The sum of the reciprocal of a number and the reciprocal of 8 less than the number is 6 times the reciprocal of the original number. Find the original number

Question

I've been trying to solve this for several hours now and I seem to have no idea how, I'm sorry but i need help 

Answer 1

Hello Charlene,

 

I think the hardest part of this problem is the setting it up. So let's begin with the problem and take it step by step. The sum of means you are going to be adding 2 things right? So we will have ______+______. Now what are those things? The reciprocal of a number...since you don't know what the number is, let's call it x so the reciprocal would be 1/x. So at this point we have 1/x + ______. Now the second part of the addition comes after the word "and". The reciprocal of 8 less than the number.  Well 8 less than the number would be x-8 so the reciprocal of that would be 1/(x-8). At this point we have 1/x+1/(x-8). Now, the word "is" means equals so now we have 1/x+1/(x-8)=. Then 6 times the reciprocal of the original number would be 6*(1/x) or in other words 6/x. Now our equation looks like 1/x+1/(x-8)=6/x. 

 

At this point it is a solving rational equations problem so we need to find the common denominator which would be x(x-8) and multiply each term by it.

x(x-8)(1/x) + x(x-8)(1/(x-8)) = x(x-8)(6/x)

In the first term the x's will cancel because you have one on top and one on bottom. In the second term the x-8 will cancel, and on the right side the x's will cancel leaving you with

x-8+x=6(x-8)

Now we can distribute the 6 to get

x-8+x=6x-48

And combining like terms we get

2x-8=6x-48

And moving the 6x to the left

-4x-8=-48

Then moving the -8 to the right

-4x=-40

Finally dividing both sides by -4 to get

x=10.

 

The last thing you always need to do with rationals is to plug the number back in and make sure 1) it works and 2) it doesn't create a 0 denominator.

 

So 1/10 + 1(10-8) = 6/10

1/10 + 1/2 = 6/10

1/10 + 5/10 = 6/10

6/10 = 6/10

 

And plugging 10 in did not create any 0 denominators so the original number is 10. 

 

I hope this has helped you and if you need more tutoring please look me up on Wyzant. I would be happy to help you.