Question

Quantity A: The product of the consecutive integers from 20 through 73, inclusive.
Quantity B: The product of the consecutive integers from 18 through 72, inclusive.

• A. if Quantity A is greater;
• B. if Quantity B is greater;
• C. if the two quantities are equal;
• D. if the relationship cannot be determined from the information given.
• E.

Hint:

When comparing large products or sums that share many common terms (like factorials or series), the most efficient strategy is to cancel the common parts and only compare the remaining, unique terms.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The problem asks to compare two large products of consecutive integers. Calculating the full products is impractical. The best approach is to identify and cancel out common factors.
Step 2: Key Formula or Approach:
We will write out the expressions for both quantities and identify the overlapping sequence of numbers. By "dividing out" or canceling these common terms, we can reduce the comparison to a much simpler calculation.
Step 3: Detailed Explanation:
Let's write out the products for Quantity A and Quantity B.
Quantity A = \(20 \times 21 \times 22 \times \dots \times 71 \times 72 \times 73\)
Quantity B = \(18 \times 19 \times 20 \times 21 \times \dots \times 71 \times 72\)
Both quantities share the product of the integers from 20 to 72. Let's call this common product \(C\). \[ C = 20 \times 21 \times \dots \times 72 \] Now, we can express Quantity A and Quantity B in terms of \(C\).
Quantity A = \(C \times 73\)
Quantity B = \(18 \times 19 \times C\)
Since \(C\) is a product of positive integers, it is a large positive number. To compare Quantity A and Quantity B, we only need to compare their unique factors: 73 and \(18 \times 19\).
Let's calculate the product \(18 \times 19\): \[ 18 \times 19 = 18 \times (20 - 1) = (18 \times 20) - (18 \times 1) = 360 - 18 = 342 \] Now we compare the unique factors: The unique factor for Quantity A is 73.
The unique factors for Quantity B multiply to 342.
Step 4: Final Answer:
We are comparing \(C \times 73\) with \(C \times 342\).
Since \(342 \textgreater 73\), and \(C\) is positive, it follows that \(C \times 342 \textgreater C \times 73\).
Therefore, Quantity B is greater than Quantity A.