When we say that two ratios are equivalent, we mean that they express the same relationship between two quantities, even if the numbers used in the ratios are different. In mathematical terms, two ratios a:b and c:d are considered equivalent if:
- The cross products of the ratios are equal, which mathematically can be expressed as: a × d = b × c.
- This means that if you take the first term of the first ratio and multiply it by the second term of the second ratio, it will equal the first term of the second ratio multiplied by the second term of the first ratio.
For example, consider the ratios 1:2 and 3:6. We can check their equivalence by calculating:
- 1 × 6 = 6
- 2 × 3 = 6
Since both calculations yield the same result (6), we conclude that 1:2 is equivalent to 3:6.
Equivalent ratios are fundamental in various areas of mathematics, particularly in proportions and when solving problems related to scaling, such as cooking recipes or converting unit measurements. Understanding equivalent ratios can also enhance one’s ability to analyze data and make comparisons effectively.