In a certain corporation, the ratio of the number of vice presidents to the number of directors is equal to...

1. Translate the problem requirements: We need to understand that "ratio of VP to directors equals ratio of directors to senior managers" means \(\mathrm{VP:D = D:S}\), and we want to find \(\mathrm{VP:S}\) given \(\mathrm{VP=8}\) and \(\mathrm{D=12}\)
2. Set up the equal ratios relationship: Express the given condition as \(\mathrm{VP/D = D/S}\), which creates a proportion we can solve
3. Solve for the number of senior managers: Use cross-multiplication to find S, since we know \(\mathrm{VP=8}\) and \(\mathrm{D=12}\)
4. Calculate the final ratio: Find \(\mathrm{VP:S}\) using the values we now have for both VP and S, then simplify to match answer choices

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is telling us in plain English. We have a corporation with three levels of employees: vice presidents (VP), directors (D), and senior-level managers (S).

The key relationship is that "the ratio of vice presidents to directors equals the ratio of directors to senior-level managers." This means if we compare VP to D, we get the same numerical relationship as when we compare D to S.

We know there are 8 vice presidents and 12 directors. We want to find the ratio of vice presidents to senior-level managers.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up the equal ratios relationship

When two ratios are equal, we can write this as a proportion. In everyday terms, this means:

"VP compared to D" = "D compared to S"

Mathematically, we write this as:
\(\mathrm{VP/D = D/S}\)

Substituting our known values:
\(\mathrm{8/12 = 12/S}\)

This proportion tells us that whatever relationship exists between 8 and 12 is the same relationship that exists between 12 and S.

3. Solve for the number of senior managers

To solve this proportion, we use cross-multiplication. This means we multiply diagonally across the equals sign:

\(\mathrm{8 \times S = 12 \times 12}\)
\(\mathrm{8S = 144}\)
\(\mathrm{S = 144 ÷ 8 = 18}\)

So there are 18 senior-level managers in the corporation.

Let's verify this makes sense: The ratio \(\mathrm{8:12}\) simplifies to \(\mathrm{2:3}\), and the ratio \(\mathrm{12:18}\) also simplifies to \(\mathrm{2:3}\). Perfect!

4. Calculate the final ratio

Now we can find the ratio of vice presidents to senior-level managers:
\(\mathrm{VP:S = 8:18}\)

To simplify this ratio, we find the greatest common divisor of 8 and 18:

• Factors of 8: 1, 2, 4, 8
• Factors of 18: 1, 2, 3, 6, 9, 18
• Greatest common divisor: 2

Dividing both numbers by 2:
\(\mathrm{8:18 = 4:9}\)

Final Answer

The ratio of vice presidents to senior-level managers is \(\mathrm{4:9}\).

Looking at our answer choices, this matches option D: \(\mathrm{4:9}\).

We can double-check: \(\mathrm{VP:D = 8:12 = 2:3}\), and \(\mathrm{D:S = 12:18 = 2:3}\). These ratios are indeed equal, confirming our answer is correct.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the equal ratios relationship

Students often struggle to understand what "the ratio of VP to D equals the ratio of D to S" actually means. They might set up incorrect relationships like \(\mathrm{VP/S = D/D}\) or confuse the order of terms in the ratios, leading to wrong proportions from the start.

2. Setting up the proportion incorrectly

Even when students understand the relationship conceptually, they may write the proportion backwards or mix up the positions. For example, writing \(\mathrm{D/VP = D/S}\) instead of \(\mathrm{VP/D = D/S}\), which would lead to completely different calculations.

Errors while executing the approach

1. Cross-multiplication errors

When solving \(\mathrm{8/12 = 12/S}\), students frequently make mistakes during cross-multiplication. They might incorrectly multiply \(\mathrm{8 \times 12 = 12 \times S}\) instead of \(\mathrm{8 \times S = 12 \times 12}\), or make arithmetic errors when calculating \(\mathrm{12 \times 12 = 144}\).

2. Calculation mistakes when finding S

After getting \(\mathrm{8S = 144}\), students may make division errors. Common mistakes include \(\mathrm{144 ÷ 8 = 16}\) instead of 18, or forgetting to complete the division step entirely and leaving S in terms of the equation.

Errors while selecting the answer

1. Forgetting to simplify the final ratio

Students might correctly find that there are 18 senior managers and determine the ratio as \(\mathrm{8:18}\), but then select this as their final answer without simplifying it to \(\mathrm{4:9}\). Since \(\mathrm{8:18}\) doesn't appear in the answer choices, this would lead to confusion or an incorrect selection.

2. Providing an intermediate ratio instead of the final answer

Students may accidentally provide one of the intermediate ratios they calculated during the problem. For example, they might select \(\mathrm{2:3}\) (which is the simplified form of both \(\mathrm{VP:D}\) and \(\mathrm{D:S}\)) instead of \(\mathrm{4:9}\) (which is \(\mathrm{VP:S}\)), confusing the relationships they found along the way.