Question

if the denominator of 2/3 is increased by number and the numerator is doubled the result is 1 find the number

Answer 1

2/3
denominator is increased by a number so 3+a
numerator is doubled so 2(2)= 4
in order for it to be a whole (or 1) then it would have to be 4/4
 so 4-3=1 
the number is 1

Answer 2

Remark
The way I'm reading this is that the numerator becomes 4. 

Equation
4/(3 + x) = 1

Solution
For convenience and ease of seeing, make the right hand side = 1/1

4/(3 + x) = 1/1 Cross Multiply
4*1 = (3 + x)*1
4 = (3 + x) Remove the brackets.
4 = 3 + x   Subtract 3 from both sides.
x = 4 - 3
x = 1

check
4/(3 + 1) = 4/4 = 1

Note
this is kind of an interesting problem. If you have choices and 1 is not one of them, leave a note.