Data set A: 72, 73, 73, 76, 76 Data set B: 61, 64, 74, 85, x Data set A and...

1. TRANSLATE the problem information

• Given information:
  • Data set A: 72, 73, 73, 76, 76 (5 numbers)
  • Data set B: 61, 64, 74, 85, x (5 numbers)
  • The mean of data set A equals the mean of data set B
• What this tells us: We need to find x such that both data sets have the same average

2. INFER the approach

• First calculate the mean of data set A since all values are known
• Then set up an equation where mean of A equals mean of B
• Solve the equation for x

3. Find the mean of data set A

• Mean = \((72 + 73 + 73 + 76 + 76) \div 5\)
• Sum = 370, so mean = \(370 \div 5 = 74\)

4. TRANSLATE the equal means condition into an equation

• Mean of data set B = \((61 + 64 + 74 + 85 + \mathrm{x}) \div 5\)
• Sum of known values: \(61 + 64 + 74 + 85 = 284\)
• So mean of B = \((284 + \mathrm{x}) \div 5\)
• Since means are equal: \(74 = (284 + \mathrm{x}) \div 5\)

5. SIMPLIFY to solve for x

• Multiply both sides by 5: \(74 \times 5 = 284 + \mathrm{x}\)
• This gives us: \(370 = 284 + \mathrm{x}\)
• Subtract 284 from both sides: \(\mathrm{x} = 370 - 284 = 86\)

Answer: C. 86

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Arithmetic errors when adding the numbers in data set A or when solving the equation

Students might incorrectly calculate \(72 + 73 + 73 + 76 + 76\), getting a sum other than 370, which leads to a wrong mean for data set A. Alternatively, they might make errors in the equation-solving steps, such as incorrectly multiplying \(74 \times 5\) or making subtraction errors.

This may lead them to select Choice A (77) or Choice D (95) depending on the specific arithmetic mistake.

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing which values belong in which data set or misunderstanding what "equal means" requires

Some students might think they need to make x equal to one of the existing values, or they might set up the equation incorrectly by not properly expressing the mean of data set B.

This may lead them to select Choice B (85) since 85 is already a value in data set B and might seem like a reasonable guess.

The Bottom Line:

This problem tests whether students can systematically work with the mean formula and set up equations from word problems. Success requires careful arithmetic and methodical algebraic manipulation.