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Relative Frequencies

Grade 8 · Statistics · Worksheet 1

  1. (2.4 × 10^8) ÷ (6 × 10^3) = ? Answer: ______________
  2. A coordinate plane shows triangle ABC with vertices at A(1, 2), B(4, 5), and C(7, 2). Triangle A'B'C' is created by applying the transformation (x, y) → (2x, 3y) to all vertices of triangle ABC. What is the ratio of the area of triangle A'B'C' to the area of triangle ABC? Answer: ______________
  3. A wildlife biologist is tracking the population growth of two different species of birds in a nature reserve. Species A started with 120 birds and increases by 15 birds each month. Species B started with 80 birds and increases by 20 birds each month. After how many months will both bird populations be equal? Answer: ______________
  4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (0,12). A line is drawn from the vertex at (0,12) to the point (5,0), creating the hypotenuse. What is the length of this hypotenuse? Answer: ______________
  5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A second triangle is created by applying the transformation (x, y) → (2x, 2y) to all vertices of the first triangle. What is the ratio of the area of the second triangle to the area of the first triangle? Answer: ______________
  6. A survey of 240 students asked about their preferred sport (Soccer or Basketball) and whether they exercise daily. The results are shown in the two-way table below: | | Soccer | Basketball | Total | |--------------|--------|------------|-------| | Daily | 72 | 48 | 120 | | Not Daily | 48 | 72 | 120 | | Total | 120 | 120 | 240 | Calculate the relative frequency of students who prefer Soccer given that they exercise daily. Then, determine if the relative frequencies suggest an association between preferring Soccer and exercising daily. Answer: ______________
  7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (0,12). A line is drawn from the vertex at (0,12) perpendicular to the hypotenuse, meeting it at point P. What is the length of this perpendicular segment from (0,12) to point P? Answer: ______________
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Answer Key & Explanations

Relative Frequencies · Grade 8 · Worksheet 1

  1. (2.4 × 10^8) ÷ (6 × 10^3) = ? Answer: 40000 Solution: Write the expression as (2.4 ÷ 6) × (10^8 ÷ 10^3). Calculate 2.4 ÷ 6 = 0.4. Calculate 10^8 ÷ 10^3 = 10^(8-3) = 10^5.
    Full step-by-step solution

    Step 1: Write the expression as (2.4 ÷ 6) × (10^8 ÷ 10^3). Step 2: Calculate 2.4 ÷ 6 = 0.4. Step 3: Calculate 10^8 ÷ 10^3 = 10^(8-3) = 10^5. Step 4: Multiply the results: 0.4 × 10^5 = 4 × 10^4. Step 5: Convert to standard form: 4 × 10^4 = 40,000. The answer is 40000.

  2. A coordinate plane shows triangle ABC with vertices at A(1, 2), B(4, 5), and C(7, 2). Triangle A'B'C' is created by applying the transformation (x, y) → (2x, 3y) to all vertices of triangle ABC. What is the ratio of the area of triangle A'B'C' to the area of triangle ABC? Answer: 6 Solution: Find the area of original triangle ABC with vertices A(1,2), B(4,5), C(7,2) Area = 1/2 × |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| Area = 1/2 × |1(5-2) + 4(2-2) + 7(2-5)| Area = 1/2 × |1(3) + 4(0) + 7(-3)| Area = 1/2 × |3 + 0 - 21| Area = 1/2 × |-18| Area = 1/2 × 18 = 9 square units Apply the…
    Full step-by-step solution

    Step 1: Find the area of original triangle ABC with vertices A(1,2), B(4,5), C(7,2) Using the shoelace formula: Area = 1/2 × |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| Area = 1/2 × |1(5-2) + 4(2-2) + 7(2-5)| Area = 1/2 × |1(3) + 4(0) + 7(-3)| Area = 1/2 × |3 + 0 - 21| Area = 1/2 × |-18| Area = 1/2 × 18 = 9 square units Step 2: Apply the transformation (x, y) → (2x, 3y) to each vertex A'(2×1, 3×2) = A'(2, 6) B'(2×4, 3×5) = B'(8, 15) C'(2×7, 3×2) = C'(14, 6) Step 3: Find the area of transformed triangle A'B'C' Using the shoelace formula: Area = 1/2 × |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| Area = 1/2 × |2(15-6) + 8(6-6) + 14(6-15)| Area = 1/2 × |2(9) + 8(0) + 14(-9)| Area = 1/2 × |18 + 0 - 126| Area = 1/2 × |-108| Area = 1/2 × 108 = 54 square units Step 4: Find the ratio of the areas Ratio = Area of A'B'C' / Area of ABC = 54 / 9 = 6 The answer is 6.

  3. A wildlife biologist is tracking the population growth of two different species of birds in a nature reserve. Species A started with 120 birds and increases by 15 birds each month. Species B started with 80 birds and increases by 20 birds each month. After how many months will both bird populations be equal? Answer: 8 Solution: Write the population equations for each species. Species A starts with 120 birds and increases by 15 birds per month. Population of A = \( 120 + 15m \) Species B starts with 80 birds and increases by 20 birds per month.
    Full step-by-step solution

    Let's solve this step by step. --- **Step 1: Write the population equations for each species.** Species A starts with 120 birds and increases by 15 birds per month. After \( m \) months: Population of A = \( 120 + 15m \) Species B starts with 80 birds and increases by 20 birds per month. After \( m \) months: Population of B = \( 80 + 20m \) --- **Step 2: Set the populations equal to find when they are the same.** We want: \( 120 + 15m = 80 + 20m \) --- **Step 3: Solve for \( m \).** Subtract \( 15m \) from both sides: \( 120 = 80 + 5m \) Subtract 80 from both sides: \( 40 = 5m \) Divide both sides by 5: \( m = 8 \) --- **Step 4: Interpret the result.** After 8 months, both populations will be equal. --- **Step 5: Check the populations at 8 months.** Species A: \( 120 + 15 \times 8 = 120 + 120 = 240 \) Species B: \( 80 + 20 \times 8 = 80 + 160 = 240 \) Yes, both are 240. --- **Final answer:** 8 months

  4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (0,12). A line is drawn from the vertex at (0,12) to the point (5,0), creating the hypotenuse. What is the length of this hypotenuse? Answer: 13 Solution: Identify the legs of the right triangle. The horizontal leg extends from (0,0) to (5,0), so its length is 5 units. The vertical leg extends from (0,0) to (0,12), so its length is 12 units.
    Full step-by-step solution

    Step 1: Identify the legs of the right triangle. The horizontal leg extends from (0,0) to (5,0), so its length is 5 units. The vertical leg extends from (0,0) to (0,12), so its length is 12 units. Step 2: Apply the Pythagorean Theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse. Step 3: Substitute the values: 5² + 12² = c² Step 4: Calculate: 25 + 144 = c² Step 5: Simplify: 169 = c² Step 6: Take the square root: c = sqrt(169) = 13 The answer is 13.

  5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A second triangle is created by applying the transformation (x, y) → (2x, 2y) to all vertices of the first triangle. What is the ratio of the area of the second triangle to the area of the first triangle? Answer: 4 Solution: A = (0, 0) B = (6, 0) C = (6, 8) This is a right triangle with legs along the x-axis and a vertical line at x = 6.
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Identify the first triangle's vertices and area** Vertices of the first triangle: A = (0, 0) B = (6, 0) C = (6, 8) This is a right triangle with legs along the x-axis and a vertical line at x = 6. Length of AB = 6 (horizontal leg) Length of BC = 8 (vertical leg) Area of first triangle = (1/2) × base × height = (1/2) × 6 × 8 = (1/2) × 48 = 24 --- **Step 2: Apply the transformation (x, y) → (2x, 2y)** A' = (0, 0) B' = (12, 0) C' = (12, 16) --- **Step 3: Find the area of the second triangle** Second triangle has vertices: A' = (0, 0) B' = (12, 0) C' = (12, 16) Horizontal leg A'B' = 12 Vertical leg B'C' = 16 Area = (1/2) × 12 × 16 = (1/2) × 192 = 96 --- **Step 4: Ratio of areas** Area of second triangle / Area of first triangle = 96 / 24 = 4 --- **Step 5: Reasoning check** The transformation (x, y) → (2x, 2y) is a dilation by factor 2 from the origin. In a dilation with scale factor k, area scales by k². Here k = 2, so area ratio = 2² = 4. This matches our computed ratio. --- **Final answer:** 4

  6. A survey of 240 students asked about their preferred sport (Soccer or Basketball) and whether they exercise daily. The results are shown in the two-way table below: | | Soccer | Basketball | Total | |--------------|--------|------------|-------| | Daily | 72 | 48 | 120 | | Not Daily | 48 | 72 | 120 | | Total | 120 | 120 | 240 | Calculate the relative frequency of students who prefer Soccer given that they exercise daily. Then, determine if the relative frequencies suggest an association between preferring Soccer and exercising daily. Answer: 0.6; No association Solution: Identify the condition: students who exercise daily. There are 120 daily exercisers total. Among daily exercisers, 72 prefer Soccer.
    Full step-by-step solution

    Step 1: Identify the condition: students who exercise daily. There are 120 daily exercisers total. Step 2: Among daily exercisers, 72 prefer Soccer. Step 3: Conditional relative frequency = 72 / 120 = 0.6. Step 4: Overall relative frequency of Soccer preference = 120 / 240 = 0.5. Step 5: Since 0.6 is close to 0.5 (difference of 0.1), there is no strong association between preferring Soccer and exercising daily. The answer is 0.6; No association.

  7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (0,12). A line is drawn from the vertex at (0,12) perpendicular to the hypotenuse, meeting it at point P. What is the length of this perpendicular segment from (0,12) to point P? Answer: 4.615 Solution: Calculate the area of the triangle using the legs as base and height. Area = (1/2) × base × height = (1/2) × 5 × 12 = 30 square units Find the length of the hypotenuse using the Pythagorean theorem.
    Full step-by-step solution

    Step 1: Calculate the area of the triangle using the legs as base and height. Area = (1/2) × base × height = (1/2) × 5 × 12 = 30 square units Step 2: Find the length of the hypotenuse using the Pythagorean theorem. Hypotenuse = sqrt(5² + 12²) = sqrt(25 + 144) = sqrt(169) = 13 units Step 3: The perpendicular from (0,12) to the hypotenuse creates a right triangle where this perpendicular is the height when the hypotenuse is used as the base. Area = (1/2) × hypotenuse × perpendicular = (1/2) × 13 × perpendicular Step 4: Set the two area expressions equal to each other. 30 = (1/2) × 13 × perpendicular Step 5: Solve for the perpendicular length. 30 = 6.5 × perpendicular perpendicular = 30 ÷ 6.5 = 4.615 units (rounded to 3 decimal places) The answer is 4.615.