Let the certain number be \( x \).
According to the problem, we have the equation:
\[ \frac{35 + x}{4} = 2x \]
To eliminate the fraction, we can multiply both sides by 4:
\[ 35 + x = 8x \]
Now, we can rearrange the equation by isolating \( x \):
\[ 35 = 8x - x \]
\[ 35 = 7x \]
Next, we can solve for \( x \) by dividing both sides by 7:
\[ x = \frac{35}{7} = 5 \]
Thus, the certain number is \( \boxed{5} \).
To verify, we can substitute \( x = 5 \) back into the original condition:
The sum of 35 and the number is:
\[ 35 + 5 = 40 \]
Now dividing by 4 gives:
\[ \frac{40}{4} = 10 \]
And double the number \( x \) is:
\[ 2 \times 5 = 10 \]
Since both sides equal 10, our solution is confirmed. The certain number is indeed \( \boxed{5} \).