The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
1. **Calculate the original rate of processing items:**
   \[
   \text{Original rate} = \frac{1250 \text{ items}}{10 \text{ hours}} = 125 \text{ items per hour}
   \]

2. **Determine the number of items processed in the first 6 hours:**
   \[
   \text{Items processed in 6 hours} = 125 \text{ items/hour} \times 6 \text{ hours} = 750 \text{ items}
   \]

3. **Calculate the number of items remaining after 6 hours:**
   \[
   \text{Remaining items} = 1250 \text{ items} - 750 \text{ items} = 500 \text{ items}
   \]

4. **Include the additional items to be processed:**
   \[
   \text{Total remaining items} = 500 \text{ items} + 150 \text{ items} = 650 \text{ items}
   \]

5. **Determine the time left to process the remaining items:**
   \[
   \text{Time left} = 10 \text{ hours} - 6 \text{ hours} = 4 \text{ hours}
   \]

6. **Calculate the new required rate to process the remaining items in the remaining time:**
   \[
   \text{New rate} = \frac{650 \text{ items}}{4 \text{ hours}} = 162.5 \text{ items per hour}
   \]

7. **Calculate the percent increase in the rate:**
   \[
   \text{Percent increase} = \frac{\text{New rate} - \text{Original rate}}{\text{Original rate}} \times 100\%
   \]
   \[
   \text{Percent increase} = \frac{162.5 \text{ items/hour} - 125 \text{ items/hour}}{125 \text{ items/hour}} \times 100\%
   \]
   \[
   \text{Percent increase} = \frac{37.5}{125} \times 100\% = 0.3 \times 100\% = 30\%
   \]

The final answer is $\boxed{30\%}$