A reliable method
Use a common denominator for addition, cancellation for multiplication, and sign analysis before cross multiplication. Convert a decimal to an integer over a power of ten before reducing.
The common trap
Taking the reciprocal reverses order for nonzero quantities of the same sign; crossing zero needs separate analysis. Squaring a number between zero and one makes it smaller, not larger.
Make the connection
Use benchmark numbers such as zero, one half and one. Check whether an exact fractional answer makes sense before converting it to a decimal.
Contrast the cases
Compare 2/3 with 3/5 using common denominator 15: 10/15 exceeds 9/15. Cross products 2×5 and 3×3 give the same conclusion because both denominators are positive. However, a negative denominator requires sign care; first rewrite −2/−3 as 2/3. Squaring 2/3 gives 4/9, which is smaller, while squaring 3/2 gives 9/4, which is larger. The interval containing the input determines which rule applies.
Transfer and explain
Transfer: compare 4/7 and 5/9 without decimals, then compare the larger fraction with its square. Since 36>35, 4/7 is larger; because it lies between 0 and 1, its square is smaller.
What is the exact value of 1/3 + 1/4?
- Use common denominator 12.
- The numerator becomes 4 + 3 = 7.
- The reduced result is 7/12.
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 overviewSiyavula Grade 10 mathematics foundation reference