Four staff members at a certain company worked on a project. The amounts of time that the four staff members...

1. Translate the problem requirements: The four staff members worked in a ratio of \(2:3:5:6\) hours. One person worked exactly \(30\) hours. We need to find which total cannot occur when \(30\) hours corresponds to any of the four ratio parts.
2. Determine possible scenarios: Since any of the four staff members could be the one who worked \(30\) hours, we have four different scenarios to consider based on which ratio part equals \(30\).
3. Calculate total hours for each scenario: For each scenario, find the multiplier that makes one ratio part equal \(30\), then calculate the total hours for all four staff members.
4. Eliminate possible totals: Compare our calculated totals with the answer choices to identify which total cannot occur in any scenario.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're working with in everyday terms. We have four people working on a project, and the time they spent follows a specific pattern - a ratio of \(2:3:5:6\). Think of this like a recipe where the ingredients must be in these exact proportions.

We know that one person worked exactly \(30\) hours, but we don't know which of the four people this was. This is the key insight - any of the four ratio parts (\(2\), \(3\), \(5\), or \(6\)) could represent the \(30\) hours.

Our job is to figure out which total number of hours is impossible when we consider all four scenarios.

Process Skill: TRANSLATE - Converting the ratio language into concrete scenarios we can work with

2. Determine possible scenarios

Since any of the four staff members could be the one who worked \(30\) hours, let's identify our four scenarios:

• Scenario 1: The person with ratio part \(2\) worked \(30\) hours
• Scenario 2: The person with ratio part \(3\) worked \(30\) hours
• Scenario 3: The person with ratio part \(5\) worked \(30\) hours
• Scenario 4: The person with ratio part \(6\) worked \(30\) hours

For each scenario, we need to find what multiplier makes that ratio part equal to \(30\), then use that same multiplier for all four parts to get the total hours.

Process Skill: CONSIDER ALL CASES - We must check every possible scenario to ensure we don't miss any valid totals

3. Calculate total hours for each scenario

Let's work through each scenario step by step:

Scenario 1: Ratio part \(2 = 30\) hours
If \(2\) parts = \(30\) hours, then \(1\) part = \(15\) hours
Total hours = \((2 + 3 + 5 + 6) \times 15 = 16 \times 15 = 240\) hours

Scenario 2: Ratio part \(3 = 30\) hours
If \(3\) parts = \(30\) hours, then \(1\) part = \(10\) hours
Total hours = \((2 + 3 + 5 + 6) \times 10 = 16 \times 10 = 160\) hours

Scenario 3: Ratio part \(5 = 30\) hours
If \(5\) parts = \(30\) hours, then \(1\) part = \(6\) hours
Total hours = \((2 + 3 + 5 + 6) \times 6 = 16 \times 6 = 96\) hours

Scenario 4: Ratio part \(6 = 30\) hours
If \(6\) parts = \(30\) hours, then \(1\) part = \(5\) hours
Total hours = \((2 + 3 + 5 + 6) \times 5 = 16 \times 5 = 80\) hours

So our possible totals are: \(240\), \(160\), \(96\), and \(80\) hours.

4. Eliminate possible totals

Now let's check our calculated totals against the answer choices:

• (A) \(80\) hours ✓ (Scenario 4)
• (B) \(96\) hours ✓ (Scenario 3)
• (C) \(160\) hours ✓ (Scenario 2)
• (D) \(192\) hours ✗ (Not in our list)
• (E) \(240\) hours ✓ (Scenario 1)

The only total that cannot occur is \(192\) hours. No matter which staff member worked the \(30\) hours, we cannot get a total of \(192\) hours while maintaining the \(2:3:5:6\) ratio.

Final Answer

The answer is (D) \(192\). This total cannot be achieved because none of our four possible scenarios (where \(30\) hours corresponds to ratio parts \(2\), \(3\), \(5\), or \(6\)) results in a total of \(192\) hours. The possible totals are \(80\), \(96\), \(160\), and \(240\) hours only.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the constraint "one of the four staff members worked for \(30\) hours"
Students often assume that the \(30\) hours must correspond to a specific ratio part (like always being the largest or smallest). They fail to recognize that ANY of the four ratio parts (\(2\), \(3\), \(5\), or \(6\)) could equal \(30\) hours, leading them to consider only one scenario instead of all four possible cases.

2. Incorrectly setting up the ratio relationship
Some students get confused about how ratios work and might try to add \(30\) to each ratio part or use \(30\) as a multiplier directly, rather than understanding that \(30\) represents the actual hours for one specific ratio part that needs to be identified.

Errors while executing the approach

1. Arithmetic errors when calculating the multiplier
When determining how many hours each "part" represents, students might make division errors. For example, when \(5\) parts = \(30\) hours, they might incorrectly calculate \(1\) part = \(5\) hours instead of \(6\) hours, leading to wrong total calculations.

2. Forgetting to calculate the total for all scenarios
Students might correctly identify that multiple scenarios are possible but then calculate the total hours for only one or two scenarios, missing some of the possible totals. This incomplete execution leads them to incorrectly eliminate answer choices that are actually possible.

Errors while selecting the answer

1. Selecting a possible total instead of the impossible one
Since this is a "CANNOT be" question, students need to find the option that's NOT possible. Some students calculate all possible totals correctly but then mistakenly select one of the calculated totals (like \(240\) or \(160\)) instead of the one total (\(192\)) that doesn't appear in their list of possibilities.