Each student in Mrs. Halstead’s class was assigned a random group of 10 seventh graders. They asked everyone in each group the following question: “Do you prefer vanilla or chocolate ice cream?” The results of some of the students' findings are shown below.

Question

Marcy’s group: 4 vanilla and 6 chocolate
Owen’s group: 3 vanilla and 7 chocolate
Spencer’s group: 1 vanilla and 9 chocolate
Vivian’s group: 2 vanilla and 8 chocolate

When the class combined all of the results, they found that 110 seventh graders preferred vanilla and 190 preferred chocolate. Which student’s results were closest to the class results?

A. Marcy

B. Owen

C. Spencer

D. Vivian

Answer 1

To determine which student's results were closest to the class results, we start by calculating the proportion of vanilla and chocolate preferences for each student and then comparing these proportions to the total results of the class.

The total number of students surveyed is:

• Total preferred vanilla = 110
• Total preferred chocolate = 190
• Total students surveyed = 110 + 190 = 300

Now, we can calculate the overall proportions:

• Proportion of vanilla = 110 / 300 = 11 / 30
• Proportion of chocolate = 190 / 300 = 19 / 30

Next, we calculate the proportions for each student:

Marcy's group:

• 4 vanilla and 6 chocolate
• Total = 10
• Proportion of vanilla = 4 / 10 = 2 / 5
• Proportion of chocolate = 6 / 10 = 3 / 5

Owen's group:

• 3 vanilla and 7 chocolate
• Total = 10
• Proportion of vanilla = 3 / 10
• Proportion of chocolate = 7 / 10

Spencer's group:

• 1 vanilla and 9 chocolate
• Total = 10
• Proportion of vanilla = 1 / 10
• Proportion of chocolate = 9 / 10

Vivian's group:

• 2 vanilla and 8 chocolate
• Total = 10
• Proportion of vanilla = 2 / 10 = 1 / 5
• Proportion of chocolate = 8 / 10 = 4 / 5

Now we find the absolute differences between the class proportions and each student’s proportions.

1. Marcy's group:

   • Difference for vanilla: |11/30 - 2/5| = |11/30 - 12/30| = 1/30
   • Difference for chocolate: |19/30 - 3/5| = |19/30 - 18/30| = 1/30
   • Total difference = (1/30) + (1/30) = 2/30 = 1/15

2. Owen's group:

   • Difference for vanilla: |11/30 - 3/10| = |11/30 - 9/30| = 2/30
   • Difference for chocolate: |19/30 - 7/10| = |19/30 - 21/30| = 2/30
   • Total difference = (2/30) + (2/30) = 4/30 = 2/15

3. Spencer's group:

   • Difference for vanilla: |11/30 - 1/10| = |11/30 - 3/30| = 8/30
   • Difference for chocolate: |19/30 - 9/10| = |19/30 - 27/30| = 8/30
   • Total difference = (8/30) + (8/30) = 16/30 = 8/15

4. Vivian's group:

   • Difference for vanilla: |11/30 - 1/5| = |11/30 - 6/30| = 5/30
   • Difference for chocolate: |19/30 - 4/5| = |19/30 - 24/30| = 5/30
   • Total difference = (5/30) + (5/30) = 10/30 = 1/3

Now we can summarize the total differences:

• Marcy: 1/15
• Owen: 2/15
• Spencer: 8/15
• Vivian: 1/3 (which is approximately 5/15)

The smallest total difference is Marcy's with 1/15.

Therefore, the student whose results were closest to the class results is:

A. Marcy