Which fractions are equivalent to \( \frac{6}{10} \) ? Hint: There may be more than one.

Fractions equivalent to \( \frac{6}{10} \) are those that represent the same value when simplified or scaled by multiplying both the numerator and the denominator by the same non-zero integer. Here are several examples: 1. **Simplified Form:** \[ \frac{6}{10} = \frac{3}{5} \] (Both numerator and denominator are divided by 2.) 2. **Multiples:** \[ \frac{6 \times 2}{10 \times 2} = \frac{12}{20} \] \[ \frac{6 \times 3}{10 \times 3} = \frac{18}{30} \] \[ \frac{6 \times 4}{10 \times 4} = \frac{24}{40} \] \[ \frac{6 \times 5}{10 \times 5} = \frac{30}{50} \] \[ \frac{6 \times 6}{10 \times 6} = \frac{36}{60} \] \[ \frac{6 \times 7}{10 \times 7} = \frac{42}{70} \] \[ \frac{6 \times 8}{10 \times 8} = \frac{48}{80} \] \[ \frac{6 \times 9}{10 \times 9} = \frac{54}{90} \] \[ \frac{6 \times 10}{10 \times 10} = \frac{60}{100} \] 3. **Negative Equivalents:** \[ \frac{-6}{-10} = \frac{6}{10} = \frac{3}{5} \] 4. **Non-Integer Multiples (for completeness):** While less common in basic fractions, you can also have equivalent fractions by using fractions to multiply: \[ \frac{6 \times \frac{1}{2}}{10 \times \frac{1}{2}} = \frac{3}{5} \] \[ \frac{6 \times \frac{3}{2}}{10 \times \frac{3}{2}} = \frac{9}{15} \] **General Rule:** To find equivalent fractions, multiply or divide both the numerator and the denominator by the same non-zero integer. This maintains the value of the fraction while changing its appearance. **Summary of Examples:** - \( \frac{3}{5} \) - \( \frac{12}{20} \) - \( \frac{18}{30} \) - \( \frac{24}{40} \) - \( \frac{30}{50} \) - \( \frac{36}{60} \) - \( \frac{42}{70} \) - \( \frac{48}{80} \) - \( \frac{54}{90} \) - \( \frac{60}{100} \) - \( \frac{-6}{-10} \) These are just a few examples; there are infinitely many fractions equivalent to \( \frac{6}{10} \).