Percentages
Comparing Ratios
Once you understand what a ratio is, a natural next question is: given two ratios, which one represents a “bigger” comparison? The most reliable method is cross-multiplication – multiply diagonally across the two ratios and compare the results, avoiding the need to convert to decimals or find a common denominator by hand.
Cross-Multiplication Method
To compare a:b and c:d, calculate a × d and c × b. Whichever product is larger tells you which ratio is larger.
Cross-multiply: 3 × 8 = 24, and 5 × 5 = 25. Since 25 > 24, 5:8 is the larger ratio.
Checking for Equivalent Ratios
Two ratios are equivalent if they represent the exact same comparison, just scaled up or down – like 2:3 and 4:6. Cross-multiplication confirms this: if a × d = c × b exactly, the ratios are equivalent.
Seeing Equivalent Ratios as Scaled Bars
2:3, 4:6 and 6:9 are all the same ratio – only the number of pieces changes, not the overall proportion. Below are three bars, all exactly the same total length, each split into blue and orange sections. Notice that the blue and orange portions take up exactly the same amount of space in every bar – only the number of dividing lines changes.
top: 2:3 (5 parts) · middle: 4:6 (10 parts) · bottom: 6:9 (15 parts) — same blue:orange split every time
Cross-multiply: 4 × 9 = 36, and 6 × 6 = 36. These are equal, so yes, the ratios are equivalent.
Real-Life Application
- Recipe scaling: checking if a doubled recipe keeps the same taste balance.
- Sports statistics: comparing win ratios between two seasons.
- Shopping: comparing ratios of active ingredient to total volume in two products.
Key Takeaways
- Cross-multiplication compares two ratios without needing a common denominator.
- For a:b vs c:d, compare a×d to c×b – the larger product marks the larger ratio.
- Equal cross-products mean the two ratios are equivalent.