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Rational Multiplication/Division

Grade 7 · Ratios · Worksheet 1

  1. Hana is tracking the temperature changes during a cold snap. At midnight, the temperature was -8.4°C. By sunrise, the temperature had dropped by 3/5 of its midnight value. What was the temperature at sunrise? Answer: ______________
  2. (-3/4) × (8/9) ÷ (-1/6) = ? Answer: ______________
  3. Mere is designing a rectangular garden on a coordinate grid. The vertices of the garden are at (0, 0), (0, 60), (84, 60), and (84, 0). She wants to create a triangular flower bed with vertices at (0, 0), (84, 0), and (0, 60). The remaining area of the garden (outside the triangle) will be divided into 14 equal rectangular plots for vegetables. What is the area of each rectangular vegetable plot? Answer: ______________
  4. (-3/4) × (5/6) ÷ (-2/3) = ? Answer: ______________
  5. A large rectangular banner is displayed on a coordinate grid with vertices at (0, 0), (0, 48), (72, 48), and (72, 0). A triangular section of the banner is painted with a cultural design, with vertices at (0, 0), (0, 48), and (72, 0). The unpainted part of the banner is then divided into 12 equal rectangular sections for community messages. What is the area of each rectangular section? Answer: ______________
  6. Liam is designing a rectangular garden for his school project. The garden's length is 3.5 meters and its width is 2.8 meters. He needs to calculate the area to determine how much soil to buy. What is the area of Liam's garden in square meters?
    Answer: ______________
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Answer Key & Explanations

Rational Multiplication/Division · Grade 7 · Worksheet 1

  1. Hana is tracking the temperature changes during a cold snap. At midnight, the temperature was -8.4°C. By sunrise, the temperature had dropped by 3/5 of its midnight value. What was the temperature at sunrise? Answer: -13.44°C Solution: The midnight temperature is -8.4°C. The temperature drops by 3/5 of its midnight value. This means we need to find 3/5 of -8.4 and then subtract that from -8.4 (or add the negative change).
    Full step-by-step solution

    Step 1: Understand the problem. The midnight temperature is -8.4°C. The temperature drops by 3/5 of its midnight value. This means we need to find 3/5 of -8.4 and then subtract that from -8.4 (or add the negative change). Step 2: Calculate the amount of temperature drop. Drop = 3/5 × (-8.4) Drop = (3 × -8.4) / 5 Drop = -25.2 / 5 Drop = -5.04°C Step 3: Calculate the sunrise temperature. Sunrise temperature = midnight temperature + drop Sunrise temperature = -8.4 + (-5.04) Sunrise temperature = -13.44°C The answer is -13.44°C.

  2. (-3/4) × (8/9) ÷ (-1/6) = ? Answer: 4 Solution: We have: (-3/4) × (8/9) ÷ (-1/6) Dividing by (-1/6) is the same as multiplying by its reciprocal. Reciprocal of (-1/6) is -6/1 = -6.
    Full step-by-step solution

    Let's solve step-by-step. We have: (-3/4) × (8/9) ÷ (-1/6) --- **Step 1: Understand the division** Dividing by (-1/6) is the same as multiplying by its reciprocal. Reciprocal of (-1/6) is -6/1 = -6. So the expression becomes: (-3/4) × (8/9) × (-6) --- **Step 2: Multiply the first two fractions** (-3/4) × (8/9) = (-3 × 8) / (4 × 9) = (-24) / (36) Simplify: divide numerator and denominator by 12: -24 ÷ 12 = -2 36 ÷ 12 = 3 So (-24/36) = -2/3 Now we have: (-2/3) × (-6) --- **Step 3: Multiply (-2/3) × (-6)** First, (-2/3) × (-6/1) Multiply numerators: (-2) × (-6) = 12 Multiply denominators: 3 × 1 = 3 So result = 12/3 = 4 --- **Step 4: Final answer** The product is positive because multiplying two negatives gives a positive. Answer: 4

  3. Mere is designing a rectangular garden on a coordinate grid. The vertices of the garden are at (0, 0), (0, 60), (84, 60), and (84, 0). She wants to create a triangular flower bed with vertices at (0, 0), (84, 0), and (0, 60). The remaining area of the garden (outside the triangle) will be divided into 14 equal rectangular plots for vegetables. What is the area of each rectangular vegetable plot? Answer: 180 Solution: Find the area of the entire rectangular garden. Length = 84, Width = 60. Area = 84 × 60 = 5040 square units.
    Full step-by-step solution

    Step 1: Find the area of the entire rectangular garden. Length = 84, Width = 60. Area = 84 × 60 = 5040 square units. Step 2: Find the area of the triangular flower bed. The triangle has base = 84 (along the bottom from (0,0) to (84,0)) and height = 60 (along the left side from (0,0) to (0,60)). Area of triangle = 1/2 × base × height = 1/2 × 84 × 60 = 1/2 × 5040 = 2520 square units. Step 3: Find the remaining area (outside the triangle). Remaining area = Total area - Triangle area = 5040 - 2520 = 2520 square units. Step 4: Divide the remaining area into 14 equal rectangular plots. Area of each plot = 2520 ÷ 14 = 180 square units. The answer is 180.

  4. (-3/4) × (5/6) ÷ (-2/3) = ? Answer: 15/16 Solution: Start with (-3/4) × (5/6) ÷ (-2/3) Division by a fraction is multiplication by its reciprocal: (-3/4) × (5/6) × (-3/2) Multiply the numerators: (-3) × 5 × (-3) = 45 Multiply the denominators: 4 × 6 × 2 = 48 The result is 45/48 Simplify by dividing numerator and denominator by 3: 45 ÷ 3 = 15, 48…
    Full step-by-step solution

    Step 1: Start with (-3/4) × (5/6) ÷ (-2/3) Step 2: Division by a fraction is multiplication by its reciprocal: (-3/4) × (5/6) × (-3/2) Step 3: Multiply the numerators: (-3) × 5 × (-3) = 45 Step 4: Multiply the denominators: 4 × 6 × 2 = 48 Step 5: The result is 45/48 Step 6: Simplify by dividing numerator and denominator by 3: 45 ÷ 3 = 15, 48 ÷ 3 = 16 Step 7: The simplified fraction is 15/16 The answer is 15/16.

  5. A large rectangular banner is displayed on a coordinate grid with vertices at (0, 0), (0, 48), (72, 48), and (72, 0). A triangular section of the banner is painted with a cultural design, with vertices at (0, 0), (0, 48), and (72, 0). The unpainted part of the banner is then divided into 12 equal rectangular sections for community messages. What is the area of each rectangular section? Answer: 144 Solution: Find the area of the entire rectangular banner. Length = 72, Width = 48. Area = 72 × 48 = 3456 square units.
    Full step-by-step solution

    Step 1: Find the area of the entire rectangular banner. Length = 72, Width = 48. Area = 72 × 48 = 3456 square units. Step 2: Find the area of the painted triangular section. The triangle has base = 72 (along the bottom edge from (0,0) to (72,0)) and height = 48 (vertical from (0,0) to (0,48)). Area of triangle = (1/2) × base × height = (1/2) × 72 × 48 = (1/2) × 3456 = 1728 square units. Step 3: Find the unpainted area. Unpainted area = Total area - Painted area = 3456 - 1728 = 1728 square units. Step 4: Divide the unpainted area into 12 equal rectangular sections. Area of each section = 1728 ÷ 12 = 144 square units. The answer is 144.

  6. Liam is designing a rectangular garden for his school project. The garden's length is 3.5 meters and its width is 2.8 meters. He needs to calculate the area to determine how much soil to buy. What is the area of Liam's garden in square meters? Answer: 9.8 Solution: To find the area of a rectangle, we use the formula: Area = length × width. Write down the given measurements. Length = 3.5 meters Width = 2.8 meters Multiply the length by the width.
    Full step-by-step solution

    To find the area of a rectangle, we use the formula: Area = length × width. Step 1: Write down the given measurements. Length = 3.5 meters Width = 2.8 meters Step 2: Multiply the length by the width. 3.5 × 2.8 Step 3: Break the multiplication into simpler parts. First, multiply 3.5 by 2: 3.5 × 2 = 7.0 Step 4: Now multiply 3.5 by 0.8: 3.5 × 0.8 = 2.8 Step 5: Add the two results together: 7.0 + 2.8 = 9.8 Step 6: Include the units. Since both length and width are in meters, the area is in square meters. Final answer: The area of Liam's garden is 9.8 square meters.