Solution:
1. Given:
- Total trail mix batch: 4 \frac{1}{2} cups
- Fraction of raisins: \frac{1}{4}
2. Convert the total trail mix batch to an improper fraction:
- 4 \frac{1}{2} = \frac{9}{2} cups
3. Find the amount of raisins:
- Multiply the total trail mix batch by the fraction of raisins:
\frac{9}{2} \times \frac{1}{4} = \frac{9}{8} cups
4. Simplify the fractions:
- \frac{9}{8} = 1 \frac{1}{8} cups
5. Determine if the proportion holds:
- For every 1 cup of trail mix, the amount of raisins should be:
\frac{1}{4} \text{ cups}
- Check if 1 \frac{1}{8} cups of raisins is proportional to 4 \frac{1}{2} cups of trail mix:
- Simplify the ratio:
\frac{1 \frac{1}{8}}{4 \frac{1}{2}} = \frac{\frac{9}{8}}{\frac{9}{2}} = \frac{9}{8} \div \frac{9}{2} = \frac{9}{8} \times \frac{2}{9} = \frac{1}{4}
6. Conclusion:
- The ratio of 1 \frac{1}{8} cups of raisins to 4 \frac{1}{2} cups of trail mix is indeed \frac{1}{4}, which means the mixture is proportional.