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 need to analyze the preferences of the entire class in relation to each student's findings.

The combined class results show that:

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

Now, we will calculate the percentage of students from the class who preferred each flavor:

• Percentage preferring vanilla = \( \frac{110}{300} \times 100 = 36.67%\)
• Percentage preferring chocolate = \( \frac{190}{300} \times 100 = 63.33%\)

Next, we calculate the preferences for each student and their corresponding percentages:

1. Marcy's group:

   • Vanilla: 4
   • Chocolate: 6
   • Total: 10
   • Percent preferring vanilla = \( \frac{4}{10} \times 100 = 40%\)
   • Percent preferring chocolate = \( \frac{6}{10} \times 100 = 60%\)

2. Owen's group:

   • Vanilla: 3
   • Chocolate: 7
   • Total: 10
   • Percent preferring vanilla = \( \frac{3}{10} \times 100 = 30%\)
   • Percent preferring chocolate = \( \frac{7}{10} \times 100 = 70%\)

3. Spencer's group:

   • Vanilla: 1
   • Chocolate: 9
   • Total: 10
   • Percent preferring vanilla = \( \frac{1}{10} \times 100 = 10%\)
   • Percent preferring chocolate = \( \frac{9}{10} \times 100 = 90%\)

4. Vivian's group:

   • Vanilla: 2
   • Chocolate: 8
   • Total: 10
   • Percent preferring vanilla = \( \frac{2}{10} \times 100 = 20%\)
   • Percent preferring chocolate = \( \frac{8}{10} \times 100 = 80%\)

Now let's compare each student's percentages to the class percentages (36.67% for vanilla and 63.33% for chocolate):

• Marcy: |40 - 36.67| = 3.33% (for vanilla), |60 - 63.33| = 3.33% (for chocolate)
  Total difference: \(3.33 + 3.33 = 6.66%\)

• Owen: |30 - 36.67| = 6.67% (for vanilla), |70 - 63.33| = 6.67% (for chocolate)
  Total difference: \(6.67 + 6.67 = 13.34%\)

• Spencer: |10 - 36.67| = 26.67% (for vanilla), |90 - 63.33| = 26.67% (for chocolate)
  Total difference: \(26.67 + 26.67 = 53.34%\)

• Vivian: |20 - 36.67| = 16.67% (for vanilla), |80 - 63.33| = 16.67% (for chocolate)
  Total difference: \(16.67 + 16.67 = 33.34%\)

Based on the total differences calculated, Marcy had the smallest total difference (6.66%).

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

A. Marcy

Answer 2

The table lists the numbers of rolls of paper towels produced each day this week by two companies.

Paper Giant Wipe It Clean
653 756
647 582
651 734
658 542
655 784
Which inference can be drawn from the data?

A.
The workers at Wipe It Clean are more consistent than the workers at Paper Giant.

B.
Wipe It Clean shows a wider variation in the numbers of paper towels produced than Paper Giant.

C.
On average, Paper Giant produces more paper towels than Wipe It Clean.

D.
The data for Paper Giant has a higher median than the data for Wipe It Clean.

Answer 3

To analyze which inference can be drawn from the data, we need to examine the production figures for both companies and calculate some relevant statistics, such as the averages and medians.

Data:

• Paper Giant: 653, 647, 651, 658, 655
• Wipe It Clean: 756, 582, 734, 542, 784

Let’s calculate the averages for both companies:

Average for Paper Giant: \[ \text{Average} = \frac{653 + 647 + 651 + 658 + 655}{5} = \frac{3264}{5} = 652.8 \]

Average for Wipe It Clean: \[ \text{Average} = \frac{756 + 582 + 734 + 542 + 784}{5} = \frac{3898}{5} = 779.6 \]

Next, let's find the medians:

Paper Giant: Ordering the numbers: 647, 651, 653, 655, 658

• Median is the middle value, which is 653.

Wipe It Clean: Ordering the numbers: 542, 582, 734, 756, 784

• Median is the middle value, which is 734.

Now let's analyze the four inferences based on these calculations:

A. The workers at Wipe It Clean are more consistent than the workers at Paper Giant.

• This might be true if we calculate the range or variance, but it cannot be immediately confirmed based on the averages and medians alone.

B. Wipe It Clean shows a wider variation in the numbers of paper towels produced than Paper Giant.

• We can see that Wipe It Clean has values that are considerably lower and higher than its average, suggesting a wider variation.

C. On average, Paper Giant produces more paper towels than Wipe It Clean.

• This is false based on our calculated average: 652.8 (Paper Giant) vs. 779.6 (Wipe It Clean).

D. The data for Paper Giant has a higher median than the data for Wipe It Clean.

• This statement is also false, as 653 (Paper Giant) is less than 734 (Wipe It Clean).

With the above analysis, the strongest inference that can be drawn is:

B. Wipe It Clean shows a wider variation in the numbers of paper towels produced than Paper Giant.

Answer 4

Greg's soccer team is selling sets of 3 raffle tickets for $5.00. What is the constant of proportionality in dollars per ticket, to the nearest cent?
A.
$0.60

B.
$1.67

C.
$5.00

D.
$15.00

Answer 5

To find the constant of proportionality in dollars per ticket, we need to calculate the cost of each individual ticket when they are sold in sets of 3 for $5.00.

1. Cost of the set of 3 tickets: $5.00
2. Number of tickets in the set: 3

To find the cost per ticket, we divide the total cost by the number of tickets:

\[ \text{Cost per ticket} = \frac{\text{Total cost}}{\text{Number of tickets}} = \frac{5.00}{3} \]

Now, we calculate \( \frac{5.00}{3} \):

\[ \frac{5.00}{3} \approx 1.6667 \]

Rounding to the nearest cent gives us $1.67.

Thus, the constant of proportionality in dollars per ticket is:

B. $1.67

Answer 6

The table below shows the total number of brake pads produced by machines throughout a 12-hour work day.

The table shows the number of brake pads produced in 3 hours is 48, 6 hours is 96, 9 hours is 144, and 12 hours is 192.

What is the unit rate for brake pads produced per hour?

A.
4 brake pads per hour

B.
3 brake pads per hour

C.
12 brake pads per hour

D.
16 brake pads per hour

Answer 7

To find the unit rate for brake pads produced per hour, we can use the total number of brake pads produced over the total number of hours worked.

From the table:

• After 12 hours, a total of 192 brake pads were produced.

Now, we can calculate the unit rate (brake pads produced per hour) using the formula:

\[ \text{Unit Rate} = \frac{\text{Total Brake Pads Produced}}{\text{Total Hours}} \]

Substituting the values:

\[ \text{Unit Rate} = \frac{192 \text{ brake pads}}{12 \text{ hours}} = 16 \text{ brake pads per hour} \]

Thus, the unit rate for brake pads produced per hour is:

D. 16 brake pads per hour