Representing real numbers on the number line

Correct answer: B. -3.43

The governing concept is representation and comparison of real numbers on a number line, especially the order of negative numbers. Estimate the endpoints: √12 is approximately 3.4641, so −√12 is approximately −3.4641. Also, −17/5 = −3.4 exactly. Therefore a number between them must be greater than −3.4641 but less than −3.4. Option B, −3.43, satisfies −3.4641 < −3.43 < −3.4. Option A, −3.30, and option D, −3.20, are to the right of −3.4, while option C, −3.60, is to the left of −3.4641. The comparison may seem counterintuitive because a negative number closer to zero is larger.

Correct answer: A. 14.10

Estimate the square root by locating 199 between nearby perfect squares. Since 14² = 196 and 15² = 225, √199 lies just above 14. A more precise estimate is √199 ≈ 14.1067, because 14.1² = 198.81, which is very close to 199. The distance from 14.10 is about 0.0067, much smaller than the distances from 14.35, 13.90, or 14.90. Therefore option A is closest. The other choices are respectively about 0.24, 0.21, and 0.79 away from the root. The key concept is numerical approximation: compare the candidate’s distance from the irrational value rather than merely choosing a nearby-looking number.