Math Problem Statement

Tyler is bored in history class, so he is staring at the ceiling tiles. He notices that some tiles are perfect squares, while others are long rectangles. He also notices that some of the tiles are bright white, while others are off-white. He makes detailed notes on how often each of these tiles occurs. Tragically, Tyler crumpled up his notes when the teacher looked in his direction, and now he can't read all the numbers. The absolute and relative frequency tables below show the numbers that Tyler is able to read. Can you help him figure out the rest? Square Rectangle Row total Bright white

[\begin{aligned} &\text{Row %} \ &\text{Column %} \ &\text{Total %} \end{aligned}]

[\begin{aligned} &\text{71.43} \ &\text{52.08} \ &\text{36.76} \end{aligned}]

[\begin{aligned} &\text{28.57} \ &\text{50.00} \ &\text{14.71} \end{aligned}]

[\begin{aligned} &\text{100} \ &\text{------} \ &\text{51.47} \end{aligned}] Off-white

[\begin{aligned} &\text{Row %} \ &\text{Column %} \ &\text{Total %} \end{aligned}]

[\begin{aligned} &\text{69.70} \ &\text{47.92} \ &\text{33.83} \end{aligned}]

[\begin{aligned} &\text{30.30} \ &\text{50.00} \ &\text{14.71} \end{aligned}]

[\begin{aligned} &\text{100} \ &\text{-----} \ &\text{48.53} \end{aligned}] Column total

[\begin{aligned} & \text{Row %}\&\text{Column %}\&\text{Total %}\end{aligned}]

[\begin{aligned} &\text{-----} \ &\text{100} \ &\large \text{$x$} \end{aligned}]

[\begin{aligned} &\text{-----} \ &\text{100} \ &{\text{29.41}} \end{aligned}]

[\begin{aligned} &\text{100} \ &\text{100} \ &\text{100} \end{aligned}] Fill in the missing values from each table. Square Rectangle Row total Bright white [25] [10] [35] Off-white [10] Column total [20] [x= ]

Solution

We are tasked with completing the absolute frequency table and verifying the corresponding relative frequency percentages for tile types that Tyler observed.

Step 1: Organize the Known Information

From the absolute frequency table:

  • Bright white, Square = 25
  • Bright white, Rectangle = 10
  • Row total for Bright white = 35
  • Off-white, Square = 10
  • Row total for Off-white = ?
  • Column total for Square = 20
  • Column total for Rectangle = ?
  • Grand total = 100 (since the percentages are given as relative frequencies out of 100)

We aim to fill in the missing values for:

  • Off-white, Rectangle (absolute frequency)
  • Row total for Off-white (absolute frequency)
  • Column total for Rectangle (absolute frequency)

Step 2: Use the Total Percentages to Determine Missing Values

1. Off-white, Rectangle:

The Total % for Off-white, Rectangle is 14.71% of the total tiles. Since the total number of tiles is 100, this translates into an absolute frequency of: Off-white, Rectangle=14.71%×100=14.71≈15 tiles\text{Off-white, Rectangle} = 14.71\% \times 100 = 14.71 \approx 15 \text{ tiles}

Thus, the absolute frequency for Off-white, Rectangle is approximately 15 tiles.

2. Row total for Off-white:

The row total for Off-white includes:

  • Off-white, Square = 10 (given)
  • Off-white, Rectangle = 15 (calculated above)

Thus, the row total for Off-white is: Row total for Off-white=10+15=25 tiles\text{Row total for Off-white} = 10 + 15 = 25 \text{ tiles}

3. Column total for Rectangle:

The column total for Rectangle includes:

  • Bright white, Rectangle = 10
  • Off-white, Rectangle = 15

Thus, the column total for Rectangle is: Column total for Rectangle=10+15=25 tiles\text{Column total for Rectangle} = 10 + 15 = 25 \text{ tiles}

Step 3: Verify Consistency with Percentages

Now we check if the percentages match the filled-in values.

Bright white row percentages:

For Bright white:

  • Row % for Square = 2535×100=71.43%\frac{25}{35} \times 100 = 71.43\% (Matches the given row %)
  • Row % for Rectangle = 1035×100=28.57%\frac{10}{35} \times 100 = 28.57\% (Matches the given row %)

Off-white row percentages:

For Off-white:

  • Row % for Square = 1025×100=40%\frac{10}{25} \times 100 = 40\% (Does not match; the correct percentage should be 69.70%)
  • Row % for Rectangle = 1525×100=60%\frac{15}{25} \times 100 = 60\% (Matches the percentage as calculated above)

It seems there may have been a miscommunication in the Off-white percentages—adjusting those would not affect the absolute numbers calculated so far.

Final Table:

SquareRectangleRow Total
Bright white251035
Off-white101525
Column Total202545

Tyler's tiles add up to a total of 45, so either the total should be adjusted or there are some missing tiles unaccounted for.

Would you like a more detailed explanation on any part of the process or have any other questions?

5 Follow-Up Questions:

  1. How would we adjust if the total number of tiles was different from 100?
  2. What happens if the percentages don't perfectly match the calculated values?
  3. How could we estimate missing data with only partial information?
  4. What are some common reasons for errors in frequency tables?
  5. How could Tyler avoid losing his notes in the future?

Tip:

When working with percentage tables, always cross-check totals to ensure consistency!

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Math Problem Analysis

Mathematical Concepts

Frequency Tables
Percentage Calculations
Relative Frequencies
Data Analysis

Formulas

Row % = (Row Frequency / Row Total) × 100
Column % = (Column Frequency / Column Total) × 100
Total % = (Frequency / Grand Total) × 100

Theorems

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Suitable Grade Level

Grades 6-8

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