Carlos plans to read $\boxed{\frac{1}{14}}$ of the book each day.
To see why, we can start by finding the total amount he needs to read: $$\text{Total amount} = \frac{1}{2} + \frac{1}{2} = 1$$ So Carlos needs to read 1 whole book in 7 days, which means he needs to read $\frac{1}{7}$ of the book each day.
But Carlos plans to read the same amount each day, so we can divide $\frac{1}{7}$ by the number of days he plans to read: $$\frac{1}{7} \div 7 = \frac{1}{7} \cdot \frac{1}{7} = \boxed{\frac{1}{49}}$$
Oops! We made a mistake! If we divide $\frac{1}{7}$ by 7, we get $\frac{1}{49}$, not $\frac{1}{14}$.
So what went wrong? We forgot to simplify $\frac{1}{49}$ to lowest terms. We can simplify it by dividing the numerator and denominator by 7: $$\frac{1}{49} = \frac{1 \div 7}{49 \div 7} = \frac{1}{7}$$ So Carlos plans to read $\boxed{\frac{1}{7}}$ of the book each day.