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### Problem Details:
1. A smoothie recipe involves a mixture of \( \frac{1}{6} \) cup of bananas to \( \frac{1}{3} \) cup of apples.
2. The questions are:
   - **For A:** How many cups of bananas are used for 1 cup of apples?
   - **For B:** How many cups of smoothie are made with 1 cup of apples?

### Solution:
#### For A:
We are given that \( \frac{1}{6} \) cup of bananas corresponds to \( \frac{1}{3} \) cup of apples. We need to find out how many cups of bananas are used for 1 cup of apples.

We can set up the proportion:
\[
\frac{1}{6} \text{ cups of bananas} = \frac{1}{3} \text{ cups of apples}
\]
To find how many cups of bananas are used for 1 cup of apples, we solve:
\[
\frac{1}{6} \times \frac{1}{\frac{1}{3}} = \frac{1}{6} \times 3 = \frac{3}{6} = \frac{1}{2} \text{ cups of bananas}.
\]
Thus, **A = \( \frac{1}{2} \) cups of bananas**.

#### For B:
A full smoothie consists of \( \frac{1}{6} \) cup of bananas and \( \frac{1}{3} \) cup of apples. So, the total smoothie mixture for one unit is:
\[
\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \text{ cups of smoothie}.
\]
Now, we need to determine how many cups of smoothie can be made with 1 cup of apples. Since \( \frac{1}{3} \) cup of apples is used in a total of \( \frac{1}{2} \) cup of smoothie, we find:
\[
\frac{1}{2} \text{ cups of smoothie} = \frac{1}{3} \text{ cups of apples}.
\]
Thus, for 1 cup of apples:
\[
\frac{1}{2} \times \frac{1}{\frac{1}{3}} = \frac{1}{2} \times 3 = \frac{3}{2} \text{ cups of smoothie}.
\]
Thus, **B = \( \frac{3}{2} \) cups of smoothie**.

### Final Answers:
- **A:** \( \frac{1}{2} \) cups of bananas.
- **B:** \( \frac{3}{2} \) cups of smoothie.

Would you like more details or have any other questions?

Here are some related questions for further practice:
1. If the ratio was \( \frac{1}{4} \) cup of bananas to \( \frac{1}{2} \) cup of apples, how many bananas would be used for 1 cup of apples?
2. How much smoothie can be made with \( \frac{2}{3} \) cup of apples?
3. What if the ratio of bananas to apples was reversed (i.e., more bananas than apples)? How would that affect the smoothie volume?
4. How much total mixture is needed to make 2 cups of smoothie?
5. If you increase the apple quantity, what effect does it have on the ratio of bananas?

**Tip:** Understanding proportions is crucial for solving recipes and mixtures like these. Always consider scaling both parts equally!

A smoothie recipe takes a mixture of 1/6 cup of bananas to 1/3 cup of apples. For A, how many cups of bananas are used for 1 cup of apples? For B, how many cups of smoothie are made with 1 cup of apples?

Learn how to solve proportions in a smoothie recipe using fractions. This problem involves calculating how many cups of bananas are needed for 1 cup of apples and determining how many cups of smoothie are made with 1 cup of apples. Suitable for Grades 6-8.

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