{-10, -6, -5, -4, -2.5, -1, 0, 2.5, 4, 6, 7, 10} A number is to be selected at random...

1. Translate the problem requirements: We need to find how many numbers from the given set satisfy the equation \((\mathrm{x} - 5)(\mathrm{x} + 10)(2\mathrm{x} - 5) = 0\), then calculate what fraction this represents of the total 12 numbers
2. Find all solutions to the equation: Set each factor equal to zero to find the values of x that make the entire product equal to zero
3. Check which solutions exist in the given set: Compare the equation solutions with the numbers available in our set {-10, -6, -5, -4, -2.5, -1, 0, 2.5, 4, 6, 7, 10}
4. Calculate the probability: Determine the ratio of favorable outcomes (numbers that are solutions) to total possible outcomes (total numbers in the set)

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us to do in plain English. We have a set of 12 numbers, and we're going to pick one at random. We want to know: what's the chance that the number we pick will make the equation \((\mathrm{x} - 5)(\mathrm{x} + 10)(2\mathrm{x} - 5) = 0\) true?

To find this probability, we need two things:

• How many numbers from our set actually work in the equation (favorable outcomes)
• How many total numbers we could pick from (total outcomes = 12)

The probability will be: (numbers that work) ÷ (total numbers) = (favorable outcomes) ÷ 12

Process Skill: TRANSLATE - Converting the probability question into a clear counting problem

2. Find all solutions to the equation

When we have an equation written as a product of factors equal to zero, like \((\mathrm{x} - 5)(\mathrm{x} + 10)(2\mathrm{x} - 5) = 0\), we can use a simple principle: if you multiply several things together and get zero, then at least one of those things must be zero.

So our equation is true when any of these is true:

• First factor: \((\mathrm{x} - 5) = 0\), which means \(\mathrm{x} = 5\)
• Second factor: \((\mathrm{x} + 10) = 0\), which means \(\mathrm{x} = -10\)
• Third factor: \((2\mathrm{x} - 5) = 0\), which means \(2\mathrm{x} = 5\), so \(\mathrm{x} = \frac{5}{2} = 2.5\)

Therefore, the complete set of solutions to our equation is: \(\mathrm{x} = 5\), \(\mathrm{x} = -10\), and \(\mathrm{x} = 2.5\)

3. Check which solutions exist in the given set

Now let's check if each of our solutions appears in the given set {-10, -6, -5, -4, -2.5, -1, 0, 2.5, 4, 6, 7, 10}:

• \(\mathrm{x} = 5\): Looking through our set... No, 5 is not in the set
• \(\mathrm{x} = -10\): Looking through our set... Yes, -10 is the first number in our set
• \(\mathrm{x} = 2.5\): Looking through our set... Yes, 2.5 appears in our set

So out of the 3 mathematical solutions to our equation, exactly 2 of them (-10 and 2.5) are actually available in our set of numbers to choose from.

Process Skill: APPLY CONSTRAINTS - Recognizing that only solutions within the given set count as favorable outcomes

4. Calculate the probability

Now we can calculate our probability:

• Favorable outcomes (numbers that satisfy the equation): 2 numbers (-10 and 2.5)
• Total possible outcomes: 12 numbers in the set
• Probability = \(\frac{2}{12} = \frac{1}{6}\)

Let's verify this matches our answer choices:

1. \(\frac{1}{12}\) ❌
2. \(\frac{1}{6}\) ✓
3. \(\frac{1}{4}\) ❌
4. \(\frac{1}{3}\) ❌
5. \(\frac{1}{2}\) ❌

Final Answer

The probability that a randomly selected number from the given set will be a solution to the equation \((\mathrm{x} - 5)(\mathrm{x} + 10)(2\mathrm{x} - 5) = 0\) is \(\frac{1}{6}\).

Answer: B

Common Faltering Points

Errors while devising the approach

Students may think they need to find values that make the entire expression equal to some specific number, rather than understanding that they need values that make the equation equal to zero. This fundamental misunderstanding would lead them down completely the wrong path.

Students might solve the equation correctly but forget that they can only select numbers from the given set {-10, -6, -5, -4, -2.5, -1, 0, 2.5, 4, 6, 7, 10}. They may calculate probability using all mathematical solutions rather than only those present in the set.

Students may try to solve the equation by expanding \((\mathrm{x} - 5)(\mathrm{x} + 10)(2\mathrm{x} - 5) = 0\) into a cubic equation rather than recognizing they can use the zero product property to set each factor equal to zero separately.

Errors while executing the approach

When solving \(2\mathrm{x} - 5 = 0\), students commonly make the error of getting \(\mathrm{x} = -\frac{5}{2} = -2.5\) instead of the correct \(\mathrm{x} = \frac{5}{2} = 2.5\). This sign error or fraction manipulation mistake would lead them to look for -2.5 instead of 2.5 in the given set.

Students may correctly find that the mathematical solutions are \(\mathrm{x} = 5\), \(\mathrm{x} = -10\), and \(\mathrm{x} = 2.5\), but then carelessly assume all three are in the given set without actually verifying each one. Missing that 5 is not in the set would lead to counting 3 favorable outcomes instead of 2.

Errors while selecting the answer

Even with the correct count of 2 favorable outcomes out of 12 total, students may make arithmetic errors when simplifying \(\frac{2}{12}\). They might get \(\frac{2}{12} = \frac{1}{6}\) but then mistakenly select \(\frac{1}{12}\) or fail to reduce the fraction properly.

If students incorrectly counted 3 solutions (including \(\mathrm{x} = 5\)), they would calculate \(\frac{3}{12} = \frac{1}{4}\) and select answer choice C instead of the correct answer B.