Step 1: Determine the efficiency ratio of P and Q

Since P is 30% more efficient than Q, if Q's efficiency is $100$, P's efficiency is $130$. Thus, the ratio of efficiencies of $P: Q = 13: 10$.

$E_P: E_Q = 13: 10$

Step 2: Find the time taken by Q alone

Time is inversely proportional to efficiency, so the time ratio of $P: Q = 10: 13$. Given P takes $23$ days, let $10x = 23 \implies x = 2.3$. Therefore, Q takes $13 \times 2.3 = 29.9$ days.

$T_Q = 23 \times \frac{13}{10} = 29.9 \text{ days}$

Step 3: Calculate combined time taken by P and Q working together

Using the formula for combined work time: $\text{Time} = \frac{T_P \times T_Q}{T_P + T_Q} = \frac{23 \times 29.9}{23 + 29.9} = \frac{687.7}{52.9} = 13$ days.

$\text{Time}_{\text{together}} = \frac{23 \times 29.9}{23 + 29.9} = 13 \text{ days}$