Frequency Tables

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Frequency Tables · Grade 8 · Worksheet 2

1. A survey asked 280 students about their preferences for two school clubs: Art Club and Music Club. 160 students like Art Club, 130 students like Music Club, and 70 students like both clubs. How many students like only Art Club? Answer: 90 Solution: - Total students: 280 - Students who like Art Club: 160 - Students who like Music Club: 130 - Students who like both clubs: 70 Students who like only Art Club = Total who like Art Club - Students who like both clubs = 160 - 70 = 90 Only Art Club: 90 Only Music Club: 130 - 70 = 60 Both clubs: 70…
   Full step-by-step solution

   Step 1: Identify the given information: - Total students: 280 - Students who like Art Club: 160 - Students who like Music Club: 130 - Students who like both clubs: 70 Step 2: Calculate students who like only Art Club: Students who like only Art Club = Total who like Art Club - Students who like both clubs = 160 - 70 = 90 Step 3: Verify with the two-way table: Only Art Club: 90 Only Music Club: 130 - 70 = 60 Both clubs: 70 Neither club: 280 - (90 + 60 + 70) = 60 Total: 90 + 60 + 70 + 60 = 280 ✓ The answer is 90.

2. A survey asked 240 students about their preferences for two sports: basketball and soccer. 130 students like basketball, 120 students like soccer, and 50 students like both sports. How many students like only basketball? Answer: 80 Solution: - Total students: 240 - Students who like basketball: 130 - Students who like soccer: 120 - Students who like both sports: 50 Students who like only basketball = Total who like basketball - Students who like both sports = 130 - 50 = 80 Only basketball: 80 Only soccer: 120 - 50 = 70 Both sports:…
   Full step-by-step solution

   Step 1: Identify the given information: - Total students: 240 - Students who like basketball: 130 - Students who like soccer: 120 - Students who like both sports: 50 Step 2: Calculate students who like only basketball: Students who like only basketball = Total who like basketball - Students who like both sports = 130 - 50 = 80 Step 3: Verify with the two-way table: Only basketball: 80 Only soccer: 120 - 50 = 70 Both sports: 50 Neither sport: 240 - (80 + 70 + 50) = 40 Total: 80 + 70 + 50 + 40 = 240 ✓ The answer is 80.

3. A school surveyed 200 students about their favorite type of movie and their grade level. The results are shown in the two-way frequency table below. How many 8th graders prefer Action movies? | Grade Level | Action | Comedy | Drama | Total | |-------------|--------|--------|-------|-------| | 7th Grade | 15 | 25 | 20 | 60 | | 8th Grade | ? | 30 | 40 | 90 | | 9th Grade | 10 | 25 | 15 | 50 | | Total | 40 | 80 | 80 | 200 | Answer: 20 Solution: We are told the total number of students is 200, and the table gives totals for each grade level and each movie type. Identify what we know about 8th graders. - 8th Grade total = 90 students.
   Full step-by-step solution

   Let's go step-by-step. We are told the total number of students is 200, and the table gives totals for each grade level and each movie type. --- **Step 1: Identify what we know about 8th graders.** From the table: - 8th Grade total = 90 students. - In 8th Grade: Action = ? Comedy = 30 Drama = 40 Total = 90 --- **Step 2: Find the missing number of 8th graders who prefer Action movies.** Let \( x \) = number of 8th graders who prefer Action. Then for 8th graders: \( x + 30 + 40 = 90 \) \( x + 70 = 90 \) \( x = 90 - 70 \) \( x = 20 \) So 20 8th graders prefer Action movies. --- **Step 3: Verify with the "Total" row for Action movies.** Total Action movies for all grades = 40. From the table: 7th Grade Action = 15 8th Grade Action = 20 (just found) 9th Grade Action = 10 Check: \( 15 + 20 + 10 = 45 \) — Wait, that’s 45, not 40. That means there’s a mismatch in the table totals. --- **Step 4: Reconcile the mismatch.** If the Total Action = 40 is correct, then: \( 15 + x + 10 = 40 \) \( x + 25 = 40 \) \( x = 15 \) But that would make 8th grade total: \( 15 + 30 + 40 = 85 \), not 90. So the table’s totals are inconsistent. But the problem likely expects us to use the grade total (90) to find \( x \), not the Action total, because the question asks specifically about 8th graders. --- **Step 5: Use the intended method (ignore total Action mismatch).** The given correct answer is 20, so they must have computed: 8th Grade total = 90 8th Grade (Comedy + Drama) = 30 + 40 = 70 So Action = 90 − 70 = 20. --- **Final Answer:** 20

4. A survey asked 120 students about their preferences for sports. 65 students like soccer, 45 like basketball, and 25 like both sports. How many students like only soccer? Answer: 40 Solution: - Total students: 120 - Students who like soccer: 65 - Students who like basketball: 45 - Students who like both sports: 25 Students who like only soccer = Total who like soccer - Students who like both sports = 65 - 25 = 40 Soccer only: 40 Basketball only: 45 - 25 = 20 Both sports: 25 Neither…
   Full step-by-step solution

   Step 1: Identify the given information: - Total students: 120 - Students who like soccer: 65 - Students who like basketball: 45 - Students who like both sports: 25 Step 2: Calculate students who like only soccer: Students who like only soccer = Total who like soccer - Students who like both sports = 65 - 25 = 40 Step 3: Verify with the two-way table: Soccer only: 40 Basketball only: 45 - 25 = 20 Both sports: 25 Neither sport: 120 - (40 + 20 + 25) = 35 Total: 40 + 20 + 25 + 35 = 120 ✓ The answer is 40.

5. (2x - 5)(3x + 4) - (x² - 2x + 7) = ? Answer: 5x² + 4x - 27 Solution: Expand (2x - 5)(3x + 4) Using FOIL: First: 2x * 3x = 6x², Outer: 2x * 4 = 8x, Inner: -5 * 3x = -15x, Last: -5 * 4 = -20 So (2x - 5)(3x + 4) = 6x² + 8x - 15x - 20 = 6x² - 7x - 20 Subtract (x² - 2x + 7) (6x² - 7x - 20) - (x² - 2x + 7) = 6x² - 7x - 20 - x² + 2x - 7 x² terms: 6x² - x² = 5x² x terms:…
   Full step-by-step solution

   Step 1: Expand (2x - 5)(3x + 4) Using FOIL: First: 2x * 3x = 6x², Outer: 2x * 4 = 8x, Inner: -5 * 3x = -15x, Last: -5 * 4 = -20 So (2x - 5)(3x + 4) = 6x² + 8x - 15x - 20 = 6x² - 7x - 20 Step 2: Subtract (x² - 2x + 7) (6x² - 7x - 20) - (x² - 2x + 7) = 6x² - 7x - 20 - x² + 2x - 7 Step 3: Combine like terms x² terms: 6x² - x² = 5x² x terms: -7x + 2x = -5x Constant terms: -20 - 7 = -27 Step 4: Final result 5x² - 5x - 27 The answer is 5x² - 5x - 27.

6. A survey asked 175 students about their preferences for two subjects: History and Geography. 95 students like History, 85 students like Geography, and 45 students like both subjects. How many students like only Geography? Answer: 40 Solution: - Total students: 175 - Students who like History: 95 - Students who like Geography: 85 - Students who like both subjects: 45 Students who like only Geography = Total who like Geography - Students who like both subjects = 85 - 45 = 40 Only History: 95 - 45 = 50 Only Geography: 40 Both subjects:…
   Full step-by-step solution

   Step 1: Identify the given information: - Total students: 175 - Students who like History: 95 - Students who like Geography: 85 - Students who like both subjects: 45 Step 2: Calculate students who like only Geography: Students who like only Geography = Total who like Geography - Students who like both subjects = 85 - 45 = 40 Step 3: Verify with the two-way table: Only History: 95 - 45 = 50 Only Geography: 40 Both subjects: 45 Neither subject: 175 - (50 + 40 + 45) = 175 - 135 = 40 Total: 50 + 40 + 45 + 40 = 175 ✓ The answer is 40.

7. A survey asked 200 students about their preferences for two subjects: math and science. 110 students like math, 90 students like science, and 40 students like both subjects. How many students like only math? Answer: 70 Solution: - Total students: 200 - Students who like math: 110 - Students who like science: 90 - Students who like both subjects: 40 Students who like only math = Total who like math - Students who like both subjects = 110 - 40 = 70 Only math: 70 Only science: 90 - 40 = 50 Both subjects: 40 Neither…
   Full step-by-step solution

   Step 1: Identify the given information: - Total students: 200 - Students who like math: 110 - Students who like science: 90 - Students who like both subjects: 40 Step 2: Calculate students who like only math: Students who like only math = Total who like math - Students who like both subjects = 110 - 40 = 70 Step 3: Verify with the two-way table: Only math: 70 Only science: 90 - 40 = 50 Both subjects: 40 Neither subject: 200 - (70 + 50 + 40) = 40 Total: 70 + 50 + 40 + 40 = 200 ✓ The answer is 70.

8. A two-way frequency table shows the favorite outdoor activities of 120 middle school students. The table has columns for 'Hiking' and 'Swimming', and rows for 'Boys' and 'Girls'. The frequency for boys who prefer hiking is 25. The frequency for girls who prefer swimming is 40. The total number of students who prefer hiking is 55. How many girls prefer hiking? Answer: 30 Solution: - Total students = 120 Let \( B_H \) = Boys who prefer Hiking = 25 (given) Let \( G_S \) = Girls who prefer Swimming = 40 (given) Let \( T_H \) = Total who prefer Hiking = 55 (given) We need \( G_H \) = Girls who prefer Hiking.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Understand the table structure** We have: - Columns: Hiking, Swimming - Rows: Boys, Girls - Total students = 120 Let’s define variables: Let \( B_H \) = Boys who prefer Hiking = 25 (given) Let \( G_S \) = Girls who prefer Swimming = 40 (given) Let \( T_H \) = Total who prefer Hiking = 55 (given) We need \( G_H \) = Girls who prefer Hiking. --- **Step 2: Relating totals** Total Hiking = Boys Hiking + Girls Hiking So: \( 55 = 25 + G_H \) Thus: \( G_H = 55 - 25 = 30 \) --- **Step 3: Check if other data is consistent (to verify)** We know \( G_S = 40 \). Total Girls = \( G_H + G_S = 30 + 40 = 70 \). Total Boys = \( B_H + B_S \). We don’t know \( B_S \) yet, but total students = 120, so: Total Boys = 120 - Total Girls = 120 - 70 = 50. So Boys Swimming = Total Boys - Boys Hiking = 50 - 25 = 25. Now check totals for Swimming: Total Swimming = Boys Swimming + Girls Swimming = 25 + 40 = 65. Total Hiking + Total Swimming = 55 + 65 = 120. Everything checks out. --- **Step 4: Conclusion** The number of girls who prefer hiking is 30. --- **Final Answer:** 30