It has been asserted that 35% of the persons belonging to a certain
race have blood type O. Results of a recent study of 100 persons of
this race show that 29 have type O blood. Suppose the study wishes to
estimate a 95% confidence interval so that its width is 0.1. What
sample size is required?

A normal confidence interval for these types of problems comes in the general form:

p + or - z*sqrt(p*(1-p)/N)

In this case, we want the width of the confidence interval to be .1,
which means that the positive and negative halves each need to be .05.
 This gives:

.05=z*sqrt(p*(1-p)/N)

Since you asked for a 95% confidence interval, z=1.96, and p here is
.35.  This gives:

.05=1.96*sqrt(.35*.65/N)  You can solve this equation by the following steps:

Divide by 1.96 --> .0255 = sqrt(.35*.65/N)
Square both sides --> .00065077 = .35*.65/N
Invert each side --> 1536.64 = N/.35*.65
Multiply by .35 and .65 --> 349.58 = N

So, you'd want a sample size of 350 people.  (The information about
the actual study done with 100 people is meaningless, as far as I can
see.)

Let me know if this is unclear/ if you have any questions.

I'm sorry - I am still confused. 
1. Why do we halve the confidence interval of .1 to then become 0.5?
2. Is the 'N' - the sample size we are trying to determine?
3. When yo divide 1.96 to get 0.255 - how do you get this answer?
4. What do you mean when you say to invert each side - what does this look like?

Thanks for your help so far