How many fluid ounces are equivalent to 76 quarts? (8 fluid ounces = 1 cup and 4 cups = 1...

1. TRANSLATE the problem information

• Given conversions:
  • \(\mathrm{8\ fluid\ ounces = 1\ cup}\)
  • \(\mathrm{4\ cups = 1\ quart}\)
• Need to find: How many fluid ounces in 76 quarts

2. INFER the conversion strategy

• Since we're going from quarts to fluid ounces, but don't have a direct conversion, we need to convert in steps
• Path: 76 quarts → cups → fluid ounces
• Use conversion factors (ratios) where unwanted units cancel out

3. TRANSLATE the first conversion factor

• Convert quarts to cups: \(\mathrm{76\ quarts \times \frac{4\ cups}{1\ quart}}\)
• The "quart" units cancel, leaving us with cups

4. SIMPLIFY the first conversion

• \(\mathrm{76 \times 4 = 304\ cups}\)

5. TRANSLATE the second conversion factor

• Convert cups to fluid ounces: \(\mathrm{304\ cups \times \frac{8\ fluid\ ounces}{1\ cup}}\)
• The "cup" units cancel, leaving fluid ounces

6. SIMPLIFY the final calculation

• \(\mathrm{304 \times 8 = 2,432\ fluid\ ounces}\)

Answer: 2,432

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Setting up conversion factors backwards or incorrectly

Students might write: \(\mathrm{76\ quarts \times \frac{1\ quart}{4\ cups} = 19\ cups}\), then \(\mathrm{19\ cups \times \frac{1\ cup}{8\ fluid\ ounces} = 2.375\ fluid\ ounces}\). This happens when they flip the ratios and put the wrong units in numerator vs denominator. This leads to confusion and an unreasonably small answer that doesn't make sense.

Second Most Common Error:

Poor INFER reasoning: Not recognizing the need for multi-step conversion

Students may try to find a direct relationship between quarts and fluid ounces, getting confused about how to combine the given conversions. They might multiply or divide the conversion factors incorrectly: \(\mathrm{76 \times 8 \times 4}\) or \(\mathrm{76 \div (8 \times 4)}\), leading to answers like 2,432 or 2.375. This causes them to get stuck and guess among unreasonable answers.

The Bottom Line:

Unit conversion problems require systematic thinking about the conversion pathway and careful attention to which units go where in each ratio.