Which fractions are equivalent to \( \frac{2}{5} \) ? Hint: There may be more than one.

Equivalent fractions are different fractions that represent the same value or proportion. To find fractions equivalent to \( \frac{2}{5} \), you can multiply or divide both the numerator (top number) and the denominator (bottom number) by the same non-zero integer. This process maintains the value of the fraction because you're scaling both parts equally. **Method to Find Equivalent Fractions:** 1. **Multiplication:** Multiply both the numerator and the denominator by the same integer. 2. **Division:** Divide both the numerator and the denominator by the same integer (as long as the number divides both evenly). Since there are infinitely many integers you can multiply by, there are infinitely many equivalent fractions to \( \frac{2}{5} \). **Examples of Equivalent Fractions to \( \frac{2}{5} \):** 1. **Multiplying by 2:** \[ \frac{2 \times 2}{5 \times 2} = \frac{4}{10} \] 2. **Multiplying by 3:** \[ \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \] 3. **Multiplying by 4:** \[ \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \] 4. **Multiplying by 5:** \[ \frac{2 \times 5}{5 \times 5} = \frac{10}{25} \] 5. **Multiplying by 6:** \[ \frac{2 \times 6}{5 \times 6} = \frac{12}{30} \] **General Form:** For any integer \( k \neq 0 \), the fraction \( \frac{2k}{5k} \) is equivalent to \( \frac{2}{5} \). **Example with \( k = 7 \):** \[ \frac{2 \times 7}{5 \times 7} = \frac{14}{35} \] **Visualization:** All these fractions represent the same point on the number line. Whether you have \( \frac{2}{5} \), \( \frac{4}{10} \), or \( \frac{6}{15} \), they all simplify to the same value when reduced to their simplest form. **Simplifying Equivalent Fractions:** If you have a fraction and you want to check if it's equivalent to \( \frac{2}{5} \), you can simplify it by dividing both the numerator and denominator by their greatest common divisor (GCD). If the simplified form is \( \frac{2}{5} \), then the original fraction is equivalent. **Example:** Simplify \( \frac{12}{30} \): - GCD of 12 and 30 is 6. - Divide both numerator and denominator by 6: \[ \frac{12 \div 6}{30 \div 6} = \frac{2}{5} \] Since the simplified form is \( \frac{2}{5} \), the original fraction \( \frac{12}{30} \) is equivalent to \( \frac{2}{5} \). **Conclusion:** There are countless fractions equivalent to \( \frac{2}{5} \). By multiplying or dividing the numerator and denominator by the same non-zero integer, you can generate as many equivalent fractions as needed.