Equal amounts of water were poured into two empty jars of different capacities, which made one jar 1/4 full and...

1. Translate the problem requirements: We need to understand that "equal amounts of water" means the same volume goes into both jars, but creates different fraction fills (\(\frac{1}{4}\) vs \(\frac{1}{3}\)) because the jars have different capacities. We're asked to find what fraction of the larger jar will be filled when all water is combined in it.
2. Establish capacity relationships using the equal water constraint: Since the same amount of water fills \(\frac{1}{4}\) of one jar and \(\frac{1}{3}\) of another, we can determine which jar is larger and find the relationship between their capacities.
3. Calculate the combined water effect: Determine what fraction of the larger jar will be occupied when we add the water from the smaller jar to the water already in the larger jar.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in plain English:

• We have two empty jars with different sizes (capacities)
• We pour the same amount of water into both jars
• This same amount of water fills \(\frac{1}{4}\) of one jar and \(\frac{1}{3}\) of the other jar
• We need to find what happens when we combine all the water in the larger jar

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

The key insight here is understanding what "equal amounts of water" means. If the same volume of water creates different fractions in different jars, it tells us something important about the relative sizes of the jars.

2. Establish capacity relationships using the equal water constraint

Now let's think about which jar is bigger. If I pour the same amount of water into two containers, and one becomes \(\frac{1}{4}\) full while the other becomes \(\frac{1}{3}\) full, which container is larger?

The container that becomes less full (\(\frac{1}{4}\)) must be the larger one! This makes intuitive sense - the same water spreads out more in a bigger container.

Let's call the amount of water we poured into each jar "W".

• Large jar capacity = L (water W fills \(\frac{1}{4}\) of it)
• Small jar capacity = S (water W fills \(\frac{1}{3}\) of it)

Since the same water W fills different fractions:

• \(\mathrm{W} = \frac{1}{4} \times \mathrm{L}\)
• \(\mathrm{W} = \frac{1}{3} \times \mathrm{S}\)

Since both expressions equal W, we can set them equal:

\(\frac{1}{4} \times \mathrm{L} = \frac{1}{3} \times \mathrm{S}\)

This tells us: \(\mathrm{L} = \frac{4}{3} \times \mathrm{S}\)

So the large jar is \(\frac{4}{3}\) times the capacity of the small jar.

3. Calculate the combined water effect

Now we need to figure out what fraction of the large jar will be filled when we pour all the water together.

Let's think about this step by step:

• The large jar already has water W in it (which is \(\frac{1}{4}\) of its capacity)
• We're adding water W from the small jar (the same amount)
• Total water = \(\mathrm{W} + \mathrm{W} = 2\mathrm{W}\)

Process Skill: SIMPLIFY - Breaking down the combination into manageable parts

What fraction of the large jar will 2W fill?

We know that \(\mathrm{W} = \frac{1}{4} \times \mathrm{L}\), so:

\(2\mathrm{W} = 2 \times \frac{1}{4} \times \mathrm{L} = \frac{2}{4} \times \mathrm{L} = \frac{1}{2} \times \mathrm{L}\)

This means 2W fills exactly \(\frac{1}{2}\) of the large jar!

4. Final Answer

When we pour all the water into the larger jar, it will be \(\frac{1}{2}\) full.

Let's verify this makes sense:

• Large jar was \(\frac{1}{4}\) full initially
• We added the same amount again (another \(\frac{1}{4}\))
• Total: \(\frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\)

The answer is C. \(\frac{1}{2}\)

Common Faltering Points

Errors while devising the approach

1. Misidentifying which jar is larger

Students often assume that since \(\frac{1}{3} > \frac{1}{4}\), the jar that becomes \(\frac{1}{3}\) full must be the larger jar. This is backwards thinking! The key insight is that equal amounts of water were poured into both jars. When the same volume fills a smaller fraction of a container, that container must be larger. The jar that becomes only \(\frac{1}{4}\) full is actually the larger jar because the same water spreads out more.

2. Confusing the water amounts vs. jar capacities

Students may get confused about what stays constant in this problem. The amount of water poured into each jar is the same, but the fractions filled and the jar capacities are different. Setting up equations like "\(\frac{1}{4}\) of large jar = \(\frac{1}{3}\) of small jar" incorrectly assumes the capacities are equal rather than the water amounts.

3. Overcomplicating with unnecessary variables

Students might assign specific numerical values to jar capacities (like "let large jar = 12 liters, small jar = 9 liters") instead of working with the fractional relationships directly. While this can work, it makes the problem unnecessarily complex and increases chances for arithmetic errors.

Errors while executing the approach

1. Incorrectly setting up the equal water equation

When establishing that \(\mathrm{W} = \frac{1}{4}\mathrm{L} = \frac{1}{3}\mathrm{S}\), students might write the relationship backwards or with wrong fractions. For example, writing "\(\mathrm{L} = \frac{1}{4}\mathrm{W}\)" instead of "\(\mathrm{W} = \frac{1}{4}\mathrm{L}\)", which completely changes the meaning of the relationship.

2. Arithmetic mistakes when finding total water

Students know they need to add the water from both jars but might make simple fraction arithmetic errors. For instance, they might incorrectly calculate \(\frac{1}{4} + \frac{1}{4} = \frac{1}{8}\) instead of \(\frac{2}{4} = \frac{1}{2}\), especially when working under time pressure.

3. Forgetting to express final answer in terms of the large jar

Students might correctly calculate that total water = 2W, but then express this as a fraction of the wrong jar capacity. They need to remember that the question asks for the fraction of the larger jar that will be filled.

Errors while selecting the answer

1. Selecting the fraction that would fill the small jar

Students might correctly calculate all the water relationships but then accidentally select what fraction the total water would fill in the small jar rather than the large jar. Since the small jar is \(\frac{3}{4}\) the size of the large jar, this would lead them to select \(\frac{2}{3}\) instead of \(\frac{1}{2}\).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient jar capacities

Since we need capacities where the same water amount fills \(\frac{1}{4}\) of one jar and \(\frac{1}{3}\) of another, let's choose values that make fraction calculations clean. The least common multiple of 4 and 3 is 12, so let's work with multiples of 12.

Let's set:

• Smaller jar capacity = 12 units
• Larger jar capacity = 16 units

We can verify this works: if 4 units of water fill \(\frac{1}{3}\) of the smaller jar (\(4 \div 12 = \frac{1}{3}\) ✓) and \(\frac{1}{4}\) of the larger jar (\(4 \div 16 = \frac{1}{4}\) ✓)

Step 2: Calculate water amounts

Amount of water in each jar = 4 units

Total water when combined = \(4 + 4 = 8\) units

Step 3: Find fraction of larger jar filled

When all 8 units of water are poured into the larger jar:

Fraction filled = \(\frac{\text{Total water}}{\text{Larger jar capacity}} = \frac{8}{16} = \frac{1}{2}\)

Answer: C. \(\frac{1}{2}\)

The smart numbers approach works excellently here because we can logically select jar capacities that satisfy the given fractional relationships, making all calculations concrete and straightforward.