2.3 Comparing Two or More Ratios
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To compare a : b and c : d, cross-multiply: if ad > bc, then a/b > c/d; if ad < bc, then a/b < c/d; if ad = bc, the ratios are equal. Alternatively, convert each ratio to a decimal.
Example 4: Which is greater, 3 : 4 or 5 : 7? Cross-multiply: 3 × 7 = 21, 4 × 5 = 20. Since 21 > 20, 3 : 4 > 5 : 7.
Example 5: Arrange 2 : 3, 3 : 4, and 5 : 6 in ascending order. Convert to decimals: 2/3 = 0.667, 3/4 = 0.75, 5/6 = 0.833. Ascending order: 2 : 3 < 3 : 4 < 5 : 6.
Example 6: Arrange 4 : 5, 7 : 9, 11 : 13 in descending order (LCM method — useful when decimals are messy). LCM of denominators 5, 9, 13 = 585. 4/5 = 468/585; 7/9 = 455/585; 11/13 = 495/585. Descending order by numerator: 495 > 468 > 455 → 11 : 13 > 4 : 5 > 7 : 9.