Verified Answer :

  • Define ratio equivalence as \(\frac{a}{b} = \frac{c}{d}\) if and only if \(ad = bc\).
  • Utilize simplification by reducing both ratios to their lowest terms using the greatest common divisor.
  • Apply cross-multiplication (\(a \times d = b \times c\)) to rigorously test for equivalence.
  • Verify equivalence by confirming that both ratios result in the same numerical quotient when calculated.
  • The two ratios are equivalent if \(\boxed{\frac{a}{b} = \frac{c}{d}}\).

Explanation

  1. Understanding Ratio Equivalence

Two ratios, \(\frac{a}{b}\) and \(\frac{c}{d}\), are considered equivalent if they represent the same numerical value. To show that they are equivalent, we essentially need to prove the equality \(\frac{a}{b} = \frac{c}{d}\), where \(b, d eq 0\). This is a fundamental concept in proportionality, often arising in geometry (like similar triangles), scale factors, or algebraic expressions. Understanding why these ratios are equivalent requires checking if their relationship remains constant under division or multiplication.

  1. Methods for Verifying Equivalence

There are three primary mathematical ways to confirm that two ratios \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent:

  1. Simplification to a Common Form: By reducing both fractions to their simplest form by dividing the numerator and denominator by their greatest common divisor (GCD), we can see if they result in the same base fraction. For instance, if \(\frac{a}{b} = \frac{k \times x}{k \times y}\) and \(\frac{c}{d} = \frac{m \times x}{m \times y}\), both simplify to \(\frac{x}{y}\).
  2. Cross-Multiplication: This is perhaps the most robust method. If \(\frac{a}{b} = \frac{c}{d}\), then the product of the means must equal the product of the extremes, which mathematically means \(a \times d = b \times c\). If this equality holds, the ratios are undeniably equivalent.
  3. Decimal Conversion: By performing the division, one can check if the quotient of the first ratio \(\frac{a}{b}\) matches the quotient of the second ratio \(\frac{c}{d}\). If both equate to the same decimal value, they are equivalent.
  1. Logical Justification of Equivalence

To justify that the two ratios in 'part B' are equivalent, one must apply the most appropriate method mentioned above based on the specific numerical or algebraic components of the ratios. For example, if the ratios involve variables, cross-multiplication is usually the preferred method to demonstrate that the equality is a universal truth regardless of the variable values. By showing that the equality \(ad = bc\) simplifies into an identity or a known property from earlier in the problem, you provide a rigorous proof of their equivalence. If numerical values are provided, showing they simplify to the same fraction is often the most intuitive and clear explanation. Since we are proving equivalence, consistently using one of these methods will verify that the relationship is maintained. The final conclusion rests on showing that the ratio of the first pair consistently scales by the same factor as the second pair.

Examples

Imagine you are baking a cake where the recipe calls for a ratio of 2 cups of flour to 1 cup of sugar, written as \(\frac{2}{1}\). If you decide to double the recipe, you now need 4 cups of flour and 2 cups of sugar, represented by the ratio \(\frac{4}{2}\). You know these ratios are equivalent because \(\frac{4}{2} = \frac{2 \times 2}{1 \times 2} = \frac{2}{1}\), or by cross-multiplying: \(2 \times 2 = 4 \times 1\), confirming that the proportion of ingredients remains consistent. This ensures the cake tastes exactly the same regardless of how much you scale the batch!

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