Monique made several batches of soup for a potluck supper. Each batch required Three-fourths of a pound of potatoes, and she used a total of 6 and one-half pounds of potatoes. How many batches of soup did Monique make?

Question

Which division and multiplication problems could represent this scenario? Check all the apply.
Three-fourths divided by StartFraction 13 over 2 EndFraction
StartFraction 13 over 2 EndFraction divided by three-fourths
Three-fourths (StartFraction 13 over 2 EndFraction)
StartFraction 13 over 2 EndFraction (three-fourths)
StartFraction 13 over 2 EndFraction (four-thirds)

Answer 1

To solve the problem, we first need to determine how many batches of soup Monique made. Each batch requires \( \frac{3}{4} \) pounds of potatoes, and she used a total of \( 6 \frac{1}{2} \) pounds of potatoes.

1. Convert \( 6 \frac{1}{2} \) to an improper fraction: \[ 6 \frac{1}{2} = \frac{12}{2} + \frac{1}{2} = \frac{13}{2} \]

2. Let \( x \) represent the number of batches Monique made. Since each batch requires \( \frac{3}{4} \) pounds of potatoes, we can express the total amount used as: \[ x \times \frac{3}{4} = \frac{13}{2} \]

3. To solve for \( x \), we can rearrange the equation: \[ x = \frac{13}{2} \div \frac{3}{4} \]

4. Dividing by a fraction is the same as multiplying by its reciprocal: \[ x = \frac{13}{2} \times \frac{4}{3} = \frac{13 \times 4}{2 \times 3} = \frac{52}{6} = \frac{26}{3} \] This means Monique made \( \frac{26}{3} \) batches of soup, which is approximately \( 8.67 \) batches.

Now, let's identify the division and multiplication problems that could represent this scenario:

1. \( \frac{3}{4} \div \frac{13}{2} \) - This does not represent the scenario correctly. It suggests how many times the total amount can go into one batch.

2. \( \frac{13}{2} \div \frac{3}{4} \) - This is correct. It shows how many batches can be made from the total amount of potatoes.

3. \( \frac{3}{4} \times \frac{13}{2} \) - This does not represent the scenario; it suggests the total amount of potatoes used if you multiply the amount per batch by the total number of batches.

4. \( \frac{13}{2} \times \frac{3}{4} \) - This also does not represent the scenario for the same reason as above.

5. \( \frac{13}{2} \times \frac{4}{3} \) - This represents the multiplication of the total potatoes used by the reciprocal of the batch size. It is correct in the sense that it is used to solve for the number of batches (as derived earlier).

So the correct representations are:

• \( \frac{13}{2} \div \frac{3}{4} \)
• \( \frac{13}{2} \times \frac{4}{3} \) (as derived from the first operation)

The correct answers are:

• \( \frac{13}{2} \div \frac{3}{4} \)
• \( \frac{13}{2} \times \frac{4}{3} \)