At a certain reception, punch is served in glasses of two sizes, small and large. If a bowl of a...

1. Translate the problem requirements: One bowl can fill either 24 small glasses OR 15 large glasses (same volume of punch). We need to find how many large glasses can be filled by 10 bowls if 72 small glasses are also filled from those same 10 bowls.
2. Establish the volume relationship: Since one bowl fills either 24 small or 15 large glasses, we can determine the relative sizes of small vs large glasses.
3. Calculate remaining punch capacity: Determine how much punch capacity remains in the 10 bowls after filling 72 small glasses.
4. Convert remaining capacity to large glasses: Use the established relationship to find how many large glasses the remaining punch can fill.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

Think of it like this - imagine you have a big bowl of punch. If you use only small glasses, you can fill exactly 24 of them. But if you use only large glasses instead, that same bowl of punch fills exactly 15 large glasses.

Now we have 10 of these bowls. We're told that from these 10 bowls, we fill 72 small glasses and some number of large glasses. We need to find how many large glasses.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

What we know:
• 1 bowl = 24 small glasses worth of punch
• 1 bowl = 15 large glasses worth of punch
• 10 bowls are used to fill 72 small glasses + ? large glasses

2. Establish the volume relationship

Since one bowl can fill either 24 small glasses OR 15 large glasses, we can figure out how the glass sizes compare to each other.

Think about it this way: if the same amount of punch fills 24 small glasses or 15 large glasses, then the large glasses must hold more punch than the small glasses.

To find the exact relationship, let's use the fact that the total volume is the same:
• 24 small glasses = 15 large glasses (in terms of volume)

This means: 1 large glass = \(\frac{24}{15}\) small glasses = \(\frac{8}{5}\) small glasses = 1.6 small glasses

In other words, each large glass holds as much punch as 1.6 small glasses.

3. Calculate remaining punch capacity

Now let's figure out how much punch we've used and how much we have left.

Total punch available: 10 bowls
Since each bowl can fill 24 small glasses, 10 bowls can fill: \(10 \times 24 = 240\) small glasses worth of punch

Punch already used for small glasses: 72 small glasses worth

Remaining punch: \(240 - 72 = 168\) small glasses worth of punch

This remaining punch is what we'll use to fill the large glasses.

4. Convert remaining capacity to large glasses

We have 168 small glasses worth of punch remaining. We need to convert this to large glasses.

Since 1 large glass = 1.6 small glasses worth of punch:

Number of large glasses = \(168 \div 1.6 = 168 \div \frac{8}{5} = 168 \times \frac{5}{8} = \frac{168 \times 5}{8} = \frac{840}{8} = 105\)

Let's verify this makes sense:
• 72 small glasses used
• 105 large glasses, where each large glass = 1.6 small glasses
• So \(105 \times 1.6 = 168\) small glasses worth
• Total: \(72 + 168 = 240\) small glasses worth = 10 bowls ✓

Final Answer

The number of large glasses that can be filled is 105.

Looking at our answer choices: A) 90, B) 105, C) 120, D) 135, E) 150

Our answer matches choice B) 105.

Common Faltering Points

Errors while devising the approach

1. Misreading the constraint setup
Students often misinterpret what "10 such bowls, when filled with punch, will fill 72 small glasses and how many large glasses" means. They might think this is asking for the maximum capacity (10 bowls = 240 small glasses OR 150 large glasses) rather than understanding that 72 small glasses have already been filled from these 10 bowls, leaving remaining punch for large glasses.

2. Failing to establish the volume relationship between glass sizes
Students may try to work directly with the numbers without first determining how small and large glasses relate to each other in terms of volume. They might attempt to solve without realizing that 24 small glasses = 15 large glasses in volume, which is crucial for converting between the two glass types.

3. Approaching as separate problems rather than a distribution problem
Students might treat this as two independent calculations rather than understanding that the 10 bowls' total capacity is being distributed between small and large glasses, where some capacity is already used for small glasses.

Errors while executing the approach

1. Arithmetic errors in fraction conversion
When converting the relationship 24 small = 15 large glasses, students often make mistakes calculating \(\frac{24}{15} = \frac{8}{5} = 1.6\). Some might incorrectly simplify the fraction or make decimal conversion errors.

2. Incorrect division when converting remaining capacity
Students frequently struggle with dividing 168 by 1.6, especially when converting 1.6 to the fraction \(\frac{8}{5}\). They might incorrectly calculate \(168 \div \frac{8}{5}\) or make errors when multiplying by the reciprocal \(\frac{5}{8}\).

3. Calculation errors in determining remaining punch
When calculating total capacity \(10 \times 24 = 240\) and subtracting used capacity \(240 - 72 = 168\), students may make basic arithmetic mistakes, leading to an incorrect starting point for the final conversion.

Errors while selecting the answer

No likely faltering points - the calculation leads directly to 105, which clearly matches answer choice B.

Alternate Solutions

Smart Numbers Approach

We can solve this problem by choosing a convenient value for the bowl's punch capacity and working with concrete numbers throughout.

Step 1: Choose a smart number for bowl capacity
Since one bowl fills either 24 small glasses or 15 large glasses, let's choose the bowl capacity to be the LCM of 24 and 15.
\(\text{LCM}(24, 15) = 120\) units of punch per bowl

Step 2: Determine individual glass capacities
• Small glass capacity = \(120 \div 24 = 5\) units each
• Large glass capacity = \(120 \div 15 = 8\) units each

Step 3: Calculate total punch available
Total punch from 10 bowls = \(10 \times 120 = 1200\) units

Step 4: Calculate punch used for small glasses
Punch used for 72 small glasses = \(72 \times 5 = 360\) units

Step 5: Find remaining punch for large glasses
Remaining punch = \(1200 - 360 = 840\) units

Step 6: Calculate number of large glasses
Number of large glasses = \(840 \div 8 = 105\) large glasses

Why this smart number works: By choosing 120 (the LCM of 24 and 15) as our bowl capacity, we ensure that both small and large glass capacities are whole numbers, making all calculations clean and avoiding fractions throughout the solution.