Part B How can you use multiplication and division to check your equivalent fractions in Part A? Why does this work?

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To check equivalent fractions using multiplication and division, you can follow these steps:

1. **Understanding Equivalent Fractions**: Two fractions are equivalent if they represent the same value or proportion. For example, $$\frac{1}{2}$$ and $$\frac{2}{4}$$ are equivalent because they both represent the same part of a whole.

2. **Using Multiplication**: To check if two fractions $$\frac{a}{b}$$ and $$\frac{c}{d}$$ are equivalent, you can cross-multiply:
   $$
   a \times d \quad \text{and} \quad b \times c
   $$
   If $$a \times d = b \times c$$, then the fractions are equivalent.

3. **Using Division**: Alternatively, you can simplify both fractions to their lowest terms. If both fractions simplify to the same value, they are equivalent. For example, if you divide the numerator and denominator of $$\frac{c}{d}$$ by their greatest common divisor (GCD) and it equals $$\frac{a}{b}$$, then they are equivalent.

4. **Why This Works**: 
   - **Multiplication**: Cross-multiplication works because it is based on the property of proportions. If two fractions are equal, then the product of the means (the inner terms) is equal to the product of the extremes (the outer terms).
   - **Division**: Simplifying fractions involves dividing both the numerator and denominator by the same non-zero number, which does not change the value of the fraction. Thus, if both fractions reduce to the same simplest form, they are equivalent.

In summary, you can use multiplication (cross-multiplication) and division (simplifying) to check for equivalent fractions because these methods rely on the fundamental properties of equality and proportions.
To check if fractions are equivalent, use cross-multiplication by multiplying the numerator of one fraction by the denominator of the other. If the products are equal, the fractions are equivalent. Alternatively, simplify both fractions to their lowest terms; if they match, they are equivalent. This works because cross-multiplication relies on the properties of proportions, and simplifying ensures both fractions represent the same value.

Part B How can you use multiplication and division to check your equivalent fractions in Part A? Why does this work?

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