A school club plans to package and sell dried fruit to raise money. The club purchased 12 containers of dried...

1. Translate the problem requirements: We need to find how many \(\frac{1}{4}\)-pound bags can be made from 12 containers, each containing \(16\frac{3}{4}\) pounds of dried fruit. This is asking for the total weight divided by the weight per bag.
2. Calculate total pounds of dried fruit available: Multiply the number of containers by the weight per container to find our total inventory.
3. Convert mixed numbers to improper fractions for easier division: Transform \(16\frac{3}{4}\) into an improper fraction to streamline calculations with the \(\frac{1}{4}\) pound bag size.
4. Determine maximum number of bags through division: Divide total pounds by pounds per bag to find how many complete bags can be made.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We have a school club that bought dried fruit in large containers, and they want to repackage it into smaller bags to sell.

Think of it like this: imagine you have several large jars of cookies, and you want to put them into small individual bags. To find out how many small bags you can make, you need to know:

• How much total cookie do you have?
• How much goes into each small bag?

In our case:

• We have 12 large containers of dried fruit
• Each container holds \(16\frac{3}{4}\) pounds
• We want to make bags that each contain \(\frac{1}{4}\) pound
• We need to find the maximum number of these small bags we can make

Process Skill: TRANSLATE

2. Calculate total pounds of dried fruit available

First, let's find out how much dried fruit the club has in total.

We have 12 containers, and each container has \(16\frac{3}{4}\) pounds.
So total pounds = \(12 \times 16\frac{3}{4}\)

Let's work with the mixed number \(16\frac{3}{4}\). We can think of this as:
\(16\frac{3}{4} = 16 + \frac{3}{4} = 16 + 0.75 = 16.75\) pounds per container

Total pounds = \(12 \times 16\frac{3}{4} = 12 \times (16 + \frac{3}{4})\)
= \(12 \times 16 + 12 \times \frac{3}{4}\)
= \(192 + 12 \times \frac{3}{4}\)
= \(192 + 9\)
= \(201\) pounds

So the club has 201 pounds of dried fruit in total.

3. Convert mixed numbers to improper fractions for easier division

Now we need to divide 201 pounds by \(\frac{1}{4}\) pound per bag. Let's think about this step by step.

When we divide by a fraction, it's the same as asking "How many quarter-pounds fit into 201 pounds?"

To make this easier, let's convert our mixed number calculation to fractions:
\(16\frac{3}{4} = \frac{16 \times 4 + 3}{4} = \frac{64 + 3}{4} = \frac{67}{4}\)

So our total pounds = \(12 \times \frac{67}{4} = \frac{12 \times 67}{4} = \frac{804}{4} = 201\) pounds

This confirms our earlier calculation.

4. Determine maximum number of bags through division

Now we need to find how many \(\frac{1}{4}\)-pound bags we can make from 201 pounds.

This is: \(201 \div \frac{1}{4}\)

When we divide by a fraction, we multiply by its reciprocal:
\(201 \div \frac{1}{4} = 201 \times \frac{4}{1} = 201 \times 4 = 804\)

Let's double-check this using our fraction form:
Total pounds = \(\frac{804}{4}\)
Number of bags = \(\frac{804}{4} \div \frac{1}{4} = \frac{804}{4} \times \frac{4}{1} = 804\)

This makes intuitive sense: if each bag contains \(\frac{1}{4}\) pound, then from 1 pound we can make 4 bags. From 201 pounds, we can make \(201 \times 4 = 804\) bags.

Final Answer

The club can make a maximum of 804 individual bags of dried fruit.

Looking at our answer choices:

1. 50
2. 64
3. 67
4. 768
5. 804

Our answer is 804, which corresponds to choice E.

Common Faltering Points

Errors while devising the approach

Students might think they need to account for packaging waste or assume they can't use all the fruit, when the problem is asking for a straightforward division to find how many complete \(\frac{1}{4}\)-pound bags can be made from the total available fruit.

Some students might try to work container by container (finding bags per container then multiplying) instead of recognizing this as a total-amount-divided-by-unit-size problem. This approach is more complex and prone to errors.

Errors while executing the approach

When calculating \(12 \times 16\frac{3}{4}\), students often make mistakes with mixed numbers. They might calculate \(12 \times 16 = 192\) and \(12 \times \frac{3}{4} = 9\), but then add incorrectly, or they might convert \(16\frac{3}{4}\) incorrectly (like writing it as 16.34 instead of 16.75).

When dividing 201 by \(\frac{1}{4}\), students frequently forget to multiply by the reciprocal. They might calculate \(201 \div 4 = 50.25\) instead of \(201 \times 4 = 804\), leading them to select answer choice A (50).

When converting \(16\frac{3}{4}\) to an improper fraction, students might incorrectly calculate \((16 \times 4 + 3)\) as 67, but then make errors in subsequent calculations, or they might get the conversion wrong entirely (like getting \(\frac{65}{4}\) instead of \(\frac{67}{4}\)).

Errors while selecting the answer

Students might select answer choice C (67) if they confused the improper fraction numerator (\(\frac{67}{4}\)) with the final answer, or answer choice B (64) if they only calculated \(12 \times 16\) and divided by \(\frac{1}{4}\) but forgot to account for the additional \(\frac{3}{4}\) pound per container.