The reciprocal of half a number increased by half the reciprocal of the number is one half. Find the number

Question

It's for a college algebra class. I need your help. 

Answer 1

Hi Eden,

 

You need to translate the English into math. . .

 

Let X = the unknown number.

 

Now it says to take the reciprocal of half the number. . . half the number is X/2 so the reciprocal is 2/X.

 

Now it says to increase that by half the reciprocal . . . half of 1/X is 1/X times 1/2 which equals 1/2X

 

And that sum equals 1/2 so 2/X + 1/2X = 1/2

 

Now you need to find a common denominator for the left side of the equation to add the fractions . . . 2X looks good.

 

So multiply the first fraction by 2/2 . . . since 2/2 equals 1 you aren't changing the value of the fraction and you'll end up with 2X in the denominator.  The fraction becomes 4/2X

 

 

4/2X + 1/2X = 1/2

 

Add the numerators on the left side --> 5/2X = 1/2

 

Cross multiply (multiply the top of one fraction times the bottom of other):

 

5 x 2 = 2X x 1

 

10 = 2X

 

X = 5

 

Check the answer by plugging 5 into the question and see if you get 1/2.

 

Have fun.

 

Dan

Answer 2

Eden,

 

This is what I think is going on:

 

Use x for the given number.  Half of the given number is x/2.  The reciprocal is then 2/x.

 

 

2/x + 2/x = 1/2 per the criteria of the problem.  Solve for x.

 

4/x = 1/2

8 = x

 

So checking, half of 8 is 4.  The reciprocal is 1/4.  1/4 + 1/4 = 1/2

 

What do you think??

 

Frank