Decision and conditions
Use an interval or an exact power before reaching for decimal approximations. Rounding uses the stated half-up convention for positive values; endpoints are not interchangeable.
Worked model
A positive number x rounds to 0.53 to the nearest hundredth, with half values rounded up. What is the smallest possible x? The half-step is 0.005. The lower endpoint is 53/100 − 1/200. It is included under half-up rounding, giving 21/40. Before calculating: Find the lower halfway point, and decide whether it rounds upward into the displayed value.
Contrast: a different decision
What is (6 × 10^6) divided by (3 × 10^2)? The coefficient ratio is 6/3 = 2. The exponent difference is (6) − (2) = 4. The quotient is 2 × 10⁴ = 20000. Before calculating: Divide the coefficients and subtract the exponents separately.
Transfer with solution
What is the greatest integer less than √51? 49 < 51 < 64. Taking nonnegative roots gives 7 < √51 < 8. The greatest integer below the root is 7. Before calculating: Locate the integer between two consecutive perfect squares.
Check the boundary
A decimal rounded to two places is not necessarily equal to the displayed decimal. What is (2/3)^(−3)? Rewrite the expression as (3/2) to a positive power. Raise numerator and denominator separately to power 3. The result is 27/8.
A positive number x rounds to 0.53 to the nearest hundredth, with half values rounded up. What is the smallest possible x?
- The half-step is 0.005.
- The lower endpoint is 53/100 − 1/200.
- It is included under half-up rounding, giving 21/40.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overview