To know when to use cross-multiplication in ratio problems, it's important to understand what equivalent ratios are. Cross-multiplication is a handy tool. It helps us check if two ratios are equal or to find unknown values in proportional relationships.

When to Use Cross-Multiplication

1. Comparing Two Ratios:
   When you have two ratios, like (\frac{a}{b}) and (\frac{c}{d}), you can compare them easily with cross-multiplication. Here’s how:

   • First, multiply (a) by (d) (this gives you (ad)).
   • Next, multiply (b) by (c) (this gives you (bc)).
   • Now, look at the results: If (ad = bc), then the ratios are equal. If (ad \neq bc), then they are not equal.

2. Finding Unknowns:
   If one of the ratios has an unknown number, like (\frac{a}{b} = \frac{c}{x}), you can still use cross-multiplication:

   • Cross-multiply to get (a \cdot x = b \cdot c).
   • Then, solve for (x) by rearranging the equation: (x = \frac{b \cdot c}{a}).

Practical Application

• Example: Let’s look at the ratios (\frac{2}{3}) and (\frac{4}{6}).

  • Cross-multiply: (2 \cdot 6) and (3 \cdot 4) both equal (12) (so (12 = 12)). This tells us that the ratios are equal.

• Statistical Insight: Imagine a survey where students prefer orange juice more than apple juice, showing a ratio of 3:2. Another group has a preference ratio of 6:4.

  • Using cross-multiplication, we can check: (3 \cdot 4 = 12) and (2 \cdot 6 = 12). This shows both ratios are equal.

By getting good at using cross-multiplication, students can tackle ratio problems in math with confidence.