Let the certain number be \( x \).
According to the problem, half of the sum of the number and 7 is equal to one quarter of the difference of the number from 3. We can set up the equation as follows:
\[ \frac{1}{2}(x + 7) = \frac{1}{4}(3 - x) \]
To eliminate the fractions, we can multiply both sides of the equation by 4:
\[ 4 \cdot \frac{1}{2}(x + 7) = 4 \cdot \frac{1}{4}(3 - x) \]
This simplifies to:
\[ 2(x + 7) = 3 - x \]
Next, we distribute the 2 on the left side:
\[ 2x + 14 = 3 - x \]
Now, we will add \( x \) to both sides to combine the \( x \) terms:
\[ 2x + x + 14 = 3 \]
This simplifies to:
\[ 3x + 14 = 3 \]
Next, we subtract 14 from both sides:
\[ 3x = 3 - 14 \]
This simplifies to:
\[ 3x = -11 \]
Now, we divide both sides by 3:
\[ x = -\frac{11}{3} \]
Thus, the certain number is \( -\frac{11}{3} \).