The image presents a problem involving the mixing of cranberry juice and apple juice. Lin’s favorite blend is represented by the ratio of cranberry juice to apple juice, shown on the double number line: 9 fluid ounces of cranberry juice to 21 fluid ounces of apple juice.

To find smaller or larger batches that maintain the same ratio:

1. Determine the ratio: The ratio of cranberry juice to apple juice is: $\frac{9}{21} = \frac{3}{7}$ So, Lin's favorite juice blend has a ratio of 3 parts cranberry juice to 7 parts apple juice.

2. Smaller Batches: To find smaller batches, scale the ratio down:

   • One possible smaller batch would be multiplying the ratio by $1$, so the next point is $\frac{3 \times 1}{7 \times 1} = 3$ fluid ounces cranberry juice and $7$ fluid ounces apple juice.

3. Larger Batches: To find larger batches, scale the ratio up:

   • Multiplying the ratio by $2$, for example, results in $\frac{3 \times 2}{7 \times 2} = 6$ fluid ounces cranberry juice and $14$ fluid ounces apple juice.

Thus, the diagram can be filled with these points:

• Smaller batch: $3$ fluid ounces cranberry juice and $7$ fluid ounces apple juice.
• Larger batch: $6$ fluid ounces cranberry juice and $14$ fluid ounces apple juice.

Would you like further details or assistance on this problem?

Related Questions:

1. What other ratios would maintain the same proportion?
2. How would you verify the consistency of the ratios in real-world mixing?
3. Can this ratio be used to determine amounts for much larger blends, such as 45 ounces of cranberry juice?
4. What is the simplest way to scale a recipe up or down while preserving the ratio?
5. How can ratios be applied in different contexts, such as cooking or chemistry?

Tip: When working with ratios, multiplying both parts of the ratio by the same number will maintain the relationship!