1. **State the problem:** A factory needs to produce a batch of pens using 12 machines in 20 days. Due to a fault, only 3 machines worked for the first 8 days, then all 12 machines worked from day 9 onwards. We need to find the total number of days taken to complete the batch. 2. **Formula and concept:** The total work is constant. Work done = number of machines \( \times \) number of days. 3. **Calculate total work:** Using 12 machines for 20 days, total work \( W = 12 \times 20 = 240 \) machine-days. 4. **Work done in first 8 days:** Only 3 machines worked, so work done \( W_1 = 3 \times 8 = 24 \) machine-days. 5. **Remaining work:** \( W_2 = W - W_1 = 240 - 24 = 216 \) machine-days. 6. **Machines working after day 8:** 12 machines. 7. **Days needed to finish remaining work:** Let \( d \) be the days needed. \[ 12 \times d = 216 \implies d = \frac{216}{12} = 18 \] 8. **Total days to complete batch:** First 8 days + \( d \) days = \( 8 + 18 = 26 \) days. **Final answer:** The total number of days taken to make the batch of pens is **26 days**.