A group of 20 people are drinking coffee. The total number taking cream in their coffee is 7 less than...

1. Translate the problem requirements: Convert the word problem into mathematical relationships using variables for each group (cream takers, sugar takers, both, neither) and establish the given constraints clearly.
2. Set up the constraint equations: Express the three given conditions as mathematical equations using the principle that total people equals the sum of all non-overlapping categories.
3. Solve the system using substitution: Use the constraint that people taking both equals people taking neither to simplify the equations and solve for the number of people taking cream.
4. Verify the solution: Check that our answer satisfies all original constraints and makes logical sense within the context of the problem.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have in plain English. We have 20 people drinking coffee, and we need to organize them into different groups based on what they add to their coffee.

Think of it like this: some people take cream only, some take sugar only, some take both cream and sugar, and some take neither. These four groups don't overlap and together they make up all 20 people.

Let's define our variables for each group:

• Let \(\mathrm{C}\) = total number of people taking cream
• Let \(\mathrm{S}\) = total number of people taking sugar
• Let \(\mathrm{B}\) = number of people taking both cream and sugar
• Let \(\mathrm{N}\) = number of people taking neither

Now let's translate the given information:

• Total people = 20
• "The total number taking cream is 7 less than twice the total number taking sugar" means: \(\mathrm{C} = 2\mathrm{S} - 7\)
• "The number taking both cream and sugar is the same as the number taking neither" means: \(\mathrm{B} = \mathrm{N}\)

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

2. Set up the constraint equations

Now we need to connect all our variables using the fundamental principle that everyone must be counted exactly once.

Think about it this way: if we count people taking cream only, plus people taking sugar only, plus people taking both, plus people taking neither, we should get exactly 20 people.

• People taking cream only = \(\mathrm{C} - \mathrm{B}\) (total cream takers minus those taking both)
• People taking sugar only = \(\mathrm{S} - \mathrm{B}\) (total sugar takers minus those taking both)
• People taking both = \(\mathrm{B}\)
• People taking neither = \(\mathrm{N}\)

So our fundamental equation is:

\((\mathrm{C} - \mathrm{B}) + (\mathrm{S} - \mathrm{B}) + \mathrm{B} + \mathrm{N} = 20\)

Simplifying: \(\mathrm{C} + \mathrm{S} - \mathrm{B} + \mathrm{N} = 20\)

Since we know \(\mathrm{B} = \mathrm{N}\), we can substitute:

\(\mathrm{C} + \mathrm{S} - \mathrm{B} + \mathrm{B} = 20\)

Therefore: \(\mathrm{C} + \mathrm{S} = 20\)

3. Solve the system using substitution

Now we have a simple system with our key relationships:

• \(\mathrm{C} = 2\mathrm{S} - 7\) (from the problem statement)
• \(\mathrm{C} + \mathrm{S} = 20\) (from our constraint equation)

Let's substitute the first equation into the second:

\((2\mathrm{S} - 7) + \mathrm{S} = 20\)

\(3\mathrm{S} - 7 = 20\)

\(3\mathrm{S} = 27\)

\(\mathrm{S} = 9\)

Now we can find C:

\(\mathrm{C} = 2\mathrm{S} - 7 = 2(9) - 7 = 18 - 7 = 11\)

So 11 people take cream.

Process Skill: SIMPLIFY - Using the constraint B = N to eliminate variables and create a solvable system

4. Verify the solution

Let's check our answer makes sense:

• People taking cream: \(\mathrm{C} = 11\)
• People taking sugar: \(\mathrm{S} = 9\)
• Total: \(\mathrm{C} + \mathrm{S} = 11 + 9 = 20\) ✓
• Is \(\mathrm{C} = 2\mathrm{S} - 7\)? Is \(11 = 2(9) - 7 = 18 - 7 = 11\)? ✓

We can also verify the individual groups work out:

• Since \(\mathrm{C} + \mathrm{S} = 20\) and \(\mathrm{B} = \mathrm{N}\), and we know \((\mathrm{C} - \mathrm{B}) + (\mathrm{S} - \mathrm{B}) + \mathrm{B} + \mathrm{N} = 20\)
• This gives us \(\mathrm{C} + \mathrm{S} - \mathrm{B} + \mathrm{N} = 20\), and since \(\mathrm{B} = \mathrm{N}\): \(\mathrm{C} + \mathrm{S} = 20\) ✓

Our answer of 11 people taking cream corresponds to choice C.

Final Answer

The number of people in the group who take cream is 11.

Answer: C

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the constraint "taking both cream and sugar is the same as taking neither"

Students often struggle with this constraint because it's not immediately obvious how \(\mathrm{B} = \mathrm{N}\) connects to the total count. They might try to set up separate equations for each individual group (cream only, sugar only, both, neither) without recognizing that this constraint significantly simplifies the problem by allowing them to eliminate variables.

2. Incorrectly defining what "total number taking cream" means

The phrase "total number taking cream" includes both people who take cream only AND people who take both cream and sugar. Students might mistakenly think this refers only to people taking cream exclusively, leading them to set up wrong equations. The total taking cream (C) must include everyone who has cream in their coffee, regardless of whether they also add sugar.

3. Missing the fundamental counting principle

Students often fail to recognize that the four groups (cream only, sugar only, both, neither) must add up to exactly 20 people. Without establishing this core relationship \((\mathrm{C} - \mathrm{B}) + (\mathrm{S} - \mathrm{B}) + \mathrm{B} + \mathrm{N} = 20\), they cannot create the essential constraint equation \(\mathrm{C} + \mathrm{S} = 20\) that makes the problem solvable.

Errors while executing the approach

1. Algebraic mistakes when substituting and simplifying

When substituting \(\mathrm{C} = 2\mathrm{S} - 7\) into \(\mathrm{C} + \mathrm{S} = 20\), students commonly make errors like: forgetting to distribute properly, getting \((2\mathrm{S} - 7) + \mathrm{S} = 3\mathrm{S} - 7\) wrong, or making arithmetic errors when solving \(3\mathrm{S} - 7 = 20\). These mechanical errors lead to incorrect values for S and subsequently C.

2. Incorrectly applying the constraint B = N

Even when students recognize that \(\mathrm{B} = \mathrm{N}\), they might not properly use this to simplify their constraint equation. They may try to solve for B and N individually instead of recognizing that this constraint allows the equation \((\mathrm{C} - \mathrm{B}) + (\mathrm{S} - \mathrm{B}) + \mathrm{B} + \mathrm{N} = 20\) to simplify directly to \(\mathrm{C} + \mathrm{S} = 20\).

Errors while selecting the answer

No likely faltering points - once students correctly solve for \(\mathrm{C} = 11\), the answer selection is straightforward as it directly matches choice C.