Decision and conditions
Propagate rounding intervals, eliminate repeating tails and compare prime-power representations. Rounding intervals use the stated convention; terminating-decimal places depend on the reduced denominator.
Worked example
Positive real measurements x and y round to 30 and 60, respectively, to the nearest 10. Exact halfway values round upward. What is the least possible product xy? The possible intervals are [25,35) and [55,65). Both quantities are positive, so their product is smallest at both included lower endpoints. The least product is 25×55=1375. Method choice: Translate each rounded measurement to an interval before choosing the product’s extremal endpoints.
Contrast: change the reasoning
The decimal x=0.313131… repeats the two-digit block 31 forever. What is x as an exact fraction? Multiplying by 100 shifts exactly one repeating block: 100x=31.3131… . Subtract x to cancel the identical tails: 99x=31. x=31/99. Method choice: Shift by one whole repeating block and subtract the original decimal to cancel its infinite tail.
Transfer with solution
What is the fewest number of digits after the decimal point needed to write 12/(3 × 2^5 × 5²) exactly as a terminating decimal? Cancel 12=3×2²; the reduced denominator is 2^3×5². A denominator 10^d supplies d factors of both 2 and 5, so d must be at least 3 and at least 2. The smallest such d is 3. Method choice: Reduce the fraction first, then compare the remaining powers of two and five with those in a power of ten.
Boundary check
For real x, 9^x=27^(x−3). What is x? Write both bases as powers of 3. Equality gives 2x=3(x−3)=3x−9. Therefore x=9. Method choice: Express both bases using the same prime and retain the multiplier on the entire parenthesized exponent.
Misconception to reject
A displayed rounded value is not an exact original value. Treats both rounded displays as exact measurements.
Positive real measurements x and y round to 30 and 60, respectively, to the nearest 10. Exact halfway values round upward. What is the least possible product xy?
- The possible intervals are [25,35) and [55,65).
- Both quantities are positive, so their product is smallest at both included lower endpoints.
- The least product is 25×55=1375.
Compare the other answer choices
Subtracts a full rounding unit from each display instead of a half-unit.
Allows the first measurement to decrease but incorrectly fixes the second at its displayed value.
Treats both rounded displays as exact measurements.
Uses both excluded upper half-step endpoints instead of the lower endpoints.
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.
GMAT exam content