For all positive integers j and k, the operation k ◇ j is defined to be the product of j...

1. Translate the problem requirements: Understand that \(\mathrm{k} ◇ \mathrm{j}\) means multiplying \(\mathrm{j}\) consecutive integers starting with \(\mathrm{k}\). We need to find \(\mathrm{a/b}\) where \(\mathrm{a} = 20 ◇ 20\) (20 consecutive integers starting from 20) and \(\mathrm{b} = 21 ◇ 20\) (20 consecutive integers starting from 21).
2. Express both products in expanded form: Write out what \(\mathrm{a}\) and \(\mathrm{b}\) actually represent in terms of the consecutive integer products to see their structure clearly.
3. Identify the relationship between the products: Compare the two products to find common factors and see how they relate to each other, focusing on what's different between them.
4. Simplify the ratio using cancellation: Cancel out the common terms in the numerator and denominator to arrive at the simplified fraction \(\mathrm{a/b}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this special operation ◇ actually means in everyday terms.

The operation \(\mathrm{k} ◇ \mathrm{j}\) means: "Start with the number \(\mathrm{k}\), then multiply it by the next \(\mathrm{j-1}\) numbers in sequence."

For our specific problem:

- \(\mathrm{a} = 20 ◇ 20\) means "start with 20, then multiply by the next 19 numbers"

- \(\mathrm{b} = 21 ◇ 20\) means "start with 21, then multiply by the next 19 numbers"

So we need to find the ratio \(\mathrm{a/b}\) where these are two different products of 20 consecutive integers each.

Process Skill: TRANSLATE - Converting the custom operation into clear mathematical understanding

2. Express both products in expanded form

Let's write out exactly what each product looks like:

\(\mathrm{a} = 20 ◇ 20 = 20 \times 21 \times 22 \times 23 \times \ldots \times 39\)

(This gives us 20 consecutive integers starting from 20)

\(\mathrm{b} = 21 ◇ 20 = 21 \times 22 \times 23 \times 24 \times \ldots \times 40\)

(This gives us 20 consecutive integers starting from 21)

Notice something interesting: both products have almost the same numbers! The only differences are:

- Product \(\mathrm{a}\) starts with 20 (which \(\mathrm{b}\) doesn't have)

- Product \(\mathrm{b}\) ends with 40 (which \(\mathrm{a}\) doesn't have)

- Everything from 21 to 39 appears in both products

3. Identify the relationship between the products

Let's see this relationship more clearly by identifying the common factors:

\(\mathrm{a} = 20 \times (21 \times 22 \times 23 \times \ldots \times 39)\)

\(\mathrm{b} = (21 \times 22 \times 23 \times \ldots \times 39) \times 40\)

The part in parentheses \((21 \times 22 \times 23 \times \ldots \times 39)\) is exactly the same in both products!

Let's call this common part \(\mathrm{C}\), so:

- \(\mathrm{a} = 20 \times \mathrm{C}\)

- \(\mathrm{b} = \mathrm{C} \times 40\)

Process Skill: VISUALIZE - Seeing the pattern in how the two products overlap

4. Simplify the ratio using cancellation

Now we can find \(\mathrm{a/b}\):

\(\mathrm{a/b} = \frac{20 \times \mathrm{C}}{\mathrm{C} \times 40}\)

Since \(\mathrm{C}\) appears in both numerator and denominator, we can cancel it out:

\(\mathrm{a/b} = \frac{20}{40}\)

Simplifying this fraction:

\(\mathrm{a/b} = \frac{20}{40} = \frac{1}{2}\)

4. Final Answer

The answer is \(\mathrm{a/b} = \frac{1}{2}\), which corresponds to choice (E).

To verify: We found that \(\mathrm{a} = 20 ◇ 20\) contains the factor 20 but not 40, while \(\mathrm{b} = 21 ◇ 20\) contains the factor 40 but not 20. All other factors (21 through 39) cancel out, leaving us with the simple ratio \(\frac{20}{40} = \frac{1}{2}\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the custom operation definition

Students often struggle with the notation \(\mathrm{k} ◇ \mathrm{j}\) and may confuse which number represents the starting point versus how many consecutive integers to multiply. For example, they might think \(20 ◇ 20\) means "20 multiplied by itself 20 times" (like exponentiation) rather than "20 consecutive integers starting from 20."

2. Miscounting the number of terms in the sequence

When working with "\(\mathrm{j}\) consecutive integers beginning with \(\mathrm{k}\)," students frequently miscount and include either \(\mathrm{j+1}\) terms or \(\mathrm{j-1}\) terms. For instance, with \(20 ◇ 20\), they might think it's 19 terms (20 through 38) or 21 terms (20 through 40) instead of exactly 20 terms (20 through 39).

3. Attempting to calculate the actual values instead of finding patterns

Seeing large products like \(\frac{20!}{19!}\), students may try to compute the enormous actual numerical values rather than recognizing that most terms will cancel out when forming the ratio \(\mathrm{a/b}\).

Errors while executing the approach

1. Incorrectly identifying the overlapping terms

When comparing \(\mathrm{a} = 20 \times 21 \times \ldots \times 39\) and \(\mathrm{b} = 21 \times 22 \times \ldots \times 40\), students might incorrectly identify which terms are common to both products. They may think 20 and 40 are both present in both products, or miss that terms 21 through 39 appear in both.

2. Errors in the cancellation process

Even when students correctly identify that most terms cancel, they may make algebraic errors when setting up the cancellation. For example, they might write \(\mathrm{a/b} = \frac{20 \times \text{common terms}}{\text{common terms} \times 40}\) but then incorrectly cancel to get \(\frac{40}{20}\) instead of \(\frac{20}{40}\).

Errors while selecting the answer

1. Forgetting to simplify the final fraction

Students might correctly arrive at \(\frac{20}{40}\) but forget to reduce it to its simplest form of \(\frac{1}{2}\), potentially looking for \(\frac{20}{40}\) among the answer choices and getting confused when it's not there.

2. Inverting the final ratio

After correctly finding that the unique factors are 20 in the numerator and 40 in the denominator, students might flip the fraction and select \(\frac{2}{1}\) or look for its decimal equivalent of 2, or select choice (A) \(\frac{21}{40}\) thinking they made a small arithmetic error.