How many perfect squares lie between 120 and 300?

### Concept & Logic To find the number of perfect squares within a range, determine the integers whose squares fall strictly within the lower and upper bounds. ### Step-by-Step Solution * **Given range:** Greater than $120$ and less than $300$. * **Find lower limit:** $$10^2 = 100 \quad (\text{too small})$$ $$11^2 = 121 \quad (\text{greater than } 120)$$ So, the first perfect square is $11^2$. * **Find upper limit:** $$17^2 = 289 \quad (\text{less than } 300)$$ $$18^2 = 324 \quad (\text{greater than } 300)$$ So, the last perfect square is $17^2$. * **Count the numbers:** The integers are $11, 12, 13, 14, 15, 16, 17$. $$\text{Count} = 17 - 11 + 1 = 7$$ ### Exam Strategy & Shortcut Memorize squares up to $30$. Knowing instantly that $11^2 = 121$ and $17^2 = 289$ allows you to find the count using the simple formula: $\text{Last Number} - \text{First Number} + 1$. $$17 - 11 + 1 = 7$$ ### Common Pitfall * **Mistake:** Off-by-one errors when subtracting limits ($17 - 11 = 6$) without adding $1$ to include both endpoints. * **Tip:** Always use $\text{Upper} - \text{Lower} + 1$ when counting inclusive continuous integer sequences. ### Final Answer **Therefore, the correct answer is 7.**