Question
Are the ratios 2:3 and 4:6 the same? Justify your answer.
Solution — Step by Step
Convert each ratio to fraction form so we can compare them directly.
We simplify 4/6 by dividing both numerator and denominator by their HCF, which is 2.
Both fractions reduce to 2/3. Since their simplest forms are identical, the two ratios are equal.
Cross-multiply to double-check: 2 × 6 = 12 and 3 × 4 = 12. The products match, confirming the ratios are equivalent.
This cross-product check is the fastest method in an exam — two multiplications and you’re done.
Yes, 2:3 and 4:6 are the same ratio.
Why This Works
A ratio compares relative sizes, not absolute values. When you multiply (or divide) both terms of a ratio by the same non-zero number, the relationship between them doesn’t change — only the scale does.
Think of it like a recipe: a batter with 2 cups flour and 3 cups milk tastes identical to one made with 4 cups flour and 6 cups milk. You’ve doubled everything, so the proportion is preserved.
This is the same principle behind equivalent fractions from Class 5. A ratio a:b and ka:kb (for any k ≠ 0) always represent the same comparison.
Alternative Method
Use the unitary approach — bring the first term to 1 in both ratios and see if the second terms match.
For 2:3: divide both by 2 → 1 : 1.5
For 4:6: divide both by 4 → 1 : 1.5
Both give 1 : 1.5, so they’re equivalent. This method is slower but builds good intuition for proportional reasoning — useful when you hit direct/inverse proportion in Class 7.
Common Mistake
Many students subtract instead of divide when comparing ratios. They check 4 − 2 = 2 and 6 − 3 = 3, see that the differences aren’t equal, and conclude the ratios are different. That logic is wrong. Ratios are multiplicative relationships, not additive ones. Always divide (or cross-multiply) — never subtract — when checking equivalence.
Quick exam check: for any two ratios a:b and c:d, just verify a × d = b × c. If yes, they’re equivalent. This cross-product trick saves 30 seconds per question in MCQ papers.