1. **State the problem:** We are given the value of $\sqrt{14} \approx 3.74165738677...$ and need to find the difference between rounding this value to 3 decimal places (d.p.) and truncating it to 3 decimal places. 2. **Recall definitions:** - Rounding to 3 d.p. means looking at the 4th decimal digit to decide whether to round the 3rd decimal digit up or keep it the same. - Truncating to 3 d.p. means simply cutting off all digits after the 3rd decimal place without rounding. 3. **Identify digits:** The first 4 decimal digits of $\sqrt{14}$ after the decimal point are 7, 4, 1, and 6. 4. **Calculate Ayan's rounded value:** Since the 4th decimal digit is 6 (which is $\geq 5$), we round the 3rd decimal digit (1) up by 1. $$\text{Rounded value} = 3.742$$ 5. **Calculate Tiana's truncated value:** Simply cut off after the 3rd decimal digit: $$\text{Truncated value} = 3.741$$ 6. **Find the difference:** $$\text{Difference} = 3.742 - 3.741 = 0.001$$ **Final answer:** $$\boxed{0.001}$$