Lani makes punch by mixing  5 3/4 cups of orange juice, 3 1/3 of pineapple juice, and 2 5/6 cups of seltzer water. Write an expression using part-whole reasoning to determine how many cups of punch Lani makes

To determine how many cups of punch Lani makes, we need to add the amounts of each ingredient together. Here's the expression using part-whole reasoning:

1. **Orange juice:** \( 5 \frac{3}{4} \) cups
2. **Pineapple juice:** \( 3 \frac{1}{3} \) cups
3. **Seltzer water:** \( 2 \frac{5}{6} \) cups

The total amount of punch Lani makes is the sum of these three amounts. The expression is:

\[
5 \frac{3}{4} + 3 \frac{1}{3} + 2 \frac{5}{6}
\]

Now, let's add them step by step:

1. **Convert mixed numbers to improper fractions:**

   \[
   5 \frac{3}{4} = \frac{4 \times 5 + 3}{4} = \frac{23}{4}
   \]
   \[
   3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}
   \]
   \[
   2 \frac{5}{6} = \frac{6 \times 2 + 5}{6} = \frac{17}{6}
   \]

2. **Find a common denominator and add the fractions:**

   The least common denominator for 4, 3, and 6 is 12.
   \[
   \frac{23}{4} = \frac{23 \times 3}{4 \times 3} = \frac{69}{12}
   \]
   \[
   \frac{10}{3} = \frac{10 \times 4}{3 \times 4} = \frac{40}{12}
   \]
   \[
   \frac{17}{6} = \frac{17 \times 2}{6 \times 2} = \frac{34}{12}
   \]

3. **Add the fractions:**
   \[
   \frac{69}{12} + \frac{40}{12} + \frac{34}{12} = \frac{143}{12}
   \]

4. **Convert the improper fraction back to a mixed number:**
   \[
   \frac{143}{12} = 11 \frac{11}{12} \text{ cups}
   \]

Thus, Lani makes a total of \( 11 \frac{11}{12} \) cups of punch.

Would you like more details on any part of this solution? 

Here are some related questions:

1. What if Lani used \( 4 \frac{1}{2} \) cups of seltzer water instead?
2. How much punch would Lani make if she doubled the amount of orange juice?
3. What if the pineapple juice was increased by \( \frac{2}{3} \) cup? How much punch would she make then?
4. How many cups of punch would Lani make if she used \( 1 \frac{1}{4} \) cups of cranberry juice in addition?
5. What if Lani used 1 cup less of orange juice? How much punch would she then have?

**Tip:** When adding mixed numbers, it can be easier to first convert them to improper fractions, especially when working with different denominators.

Learn how to calculate the total amount of punch Lani makes by combining 5 3/4 cups of orange juice, 3 1/3 cups of pineapple juice, and 2 5/6 cups of seltzer water. Detailed step-by-step solution provided with conversions of mixed numbers to improper fractions and addition of fractions. Suitable for Grades 5-6.

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