The reciprocal of 9 less than a number is four time the reciprocal of the number itself. What is the number?

Question

The reciprocal of 9 less than a number is four times the reciprocal of the number itself. What is the number?

Answer 1

Hi Dylan--You can set this up as a proportion--but be careful to understand the language of the question:

 

"reciprocal of 9 less than a number..." so, take the reciprocal of n - 9, which would be

 

1/(n - 9), and have it equal to "the reciprocal of the number itself." The number is n, so its reciprocal is 1/n.

 

So your proportion is

 

1/(n - 9) = 4 times (1/n), or 4/n

 

cross-mutiply 1/(n - 9) x 4/n, and you get

 

1n = 4n - 36  (because you use the distributive property.)

 

Combine terms to get 0 = 3n - 36, so that means 3n = 36, and...

 

n = 12.

 

Check it by plugging 12 into your two fractions:

 

1/ (12 - 9) = 4/12, which is correct, because it becomes

 

1/3 = 4/12, which are equivalent fractions. Check!

 

Hope this helps.

 

Matt in NY