Question

1 gallon = 8 pints
1 quart = 2 pints
COLUMN A: The least number of half-pint bottles needed of to hold x quarts milk
COLUMN B: The least number of one-quart bottles needed to hold x gallons of milk

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

In unit conversion problems, it's crucial to be systematic. Convert all quantities to a common base unit (in this case, pints or quarts) before performing the final calculation. Writing out the conversion factors clearly helps prevent mistakes.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a unit conversion problem. We need to calculate the number of bottles required in two different scenarios and compare the results. The variable \(x\) should cancel out if the quantities are directly comparable.
Step 2: Key Formula or Approach:
For each column, we need to convert the total volume of milk to the units of the bottle size and then determine the number of bottles. Number of bottles = (Total Volume) / (Volume per bottle).
Step 3: Detailed Explanation:
Column A: Total volume of milk = \(x\) quarts.
Bottle size = half-pint = 0.5 pints.
We need to convert quarts to pints to have consistent units. From the given information, 1 quart = 2 pints. So, \(x\) quarts = \(x \times 2\) pints = \(2x\) pints.
Now, we can find the number of half-pint bottles needed: \[ \text{Number of bottles (A)} = \frac{\text{Total volume in pints}}{\text{Volume per bottle in pints}} = \frac{2x}{0.5} \] Dividing by 0.5 is the same as multiplying by 2. \[ \text{Number of bottles (A)} = 2x \times 2 = 4x \] Column B: Total volume of milk = \(x\) gallons.
Bottle size = 1 quart.
We need to convert gallons to quarts. First, let's find the relationship between gallons and quarts using pints as a bridge. 1 gallon = 8 pints. 1 quart = 2 pints, which means 1 pint = 0.5 quarts. So, 1 gallon = 8 pints = \(8 \times (0.5 \text{ quarts}) = 4\) quarts.
The total volume of milk is \(x\) gallons, which is \(x \times 4\) quarts = \(4x\) quarts.
The bottle size is 1 quart. \[ \text{Number of bottles (B)} = \frac{\text{Total volume in quarts}}{\text{Volume per bottle in quarts}} = \frac{4x}{1} = 4x \] Comparison: The quantity in Column A is \(4x\). The quantity in Column B is \(4x\). The two quantities are equal.
Step 4: Final Answer:
Both columns evaluate to \(4x\), so the quantities are equal.