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Proportional Equations

Grade 7 · Algebra · Worksheet 3

  1. Olivia is designing a rectangular mural on a coordinate grid. The mural's corners are at (0, 0), (35, 0), (35, 20), and (0, 20). She draws a straight diagonal line from (0, 0) to (35, 20), dividing the mural into two triangular sections. The amount of red paint needed for the larger triangle is proportional to its area. If 15 square units of area require 6 liters of paint, write an equation in the form y = kx that represents the relationship between the area of the larger triangle (x) and the liters of red paint needed (y). Answer: ______________
  2. Aroha is drawing a design for a rectangular mural on a coordinate grid. The mural's corners are at (1, 3), (43, 3), (43, 19), and (1, 19). She plans to place a square photo in the center of the mural with a side length equal to one-fifth of the mural's shorter side length. The photo is placed so its sides are parallel to the mural's sides, and its center coincides with the mural's center. The remaining area of the mural will be painted with a proportional pattern where the amount of blue paint used (y, in milliliters) is proportional to the area painted (x, in square units). If 117 milliliters of blue paint are needed for every 9 square units of area, write an equation that represents the proportional relationship between the blue paint needed (y) and the area painted (x). Answer: ______________
  3. Mere is designing a mosaic on a coordinate grid. The mosaic is in the shape of a rectangle with vertices at (0, 0), (24, 0), (24, 18), and (0, 18). She wants to place a triangular pattern inside the mosaic with vertices at (0, 0), (24, 0), and (12, 18). The area of the triangular pattern is proportional to the area of the entire rectangular mosaic. Write an equation in the form y = kx that represents the proportional relationship where y is the area of the triangular pattern and x is the area of the rectangular mosaic. Then find the value of k as a simplified fraction. Answer: ______________
  4. A construction company needs to mix concrete using cement, sand, and gravel in the ratio 2:3:5. For a large building project, they need to prepare 15,000 kilograms of concrete mixture. How many kilograms of sand should they use in this batch? Answer: ______________
  5. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). If the garden's length and width are in a 3:2 ratio, and you need to create a scale drawing where 1 unit on the coordinate plane represents 4 feet in real life, what is the actual area of the garden in square feet? Answer: ______________
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Answer Key & Explanations

Proportional Equations · Grade 7 · Worksheet 3

  1. Olivia is designing a rectangular mural on a coordinate grid. The mural's corners are at (0, 0), (35, 0), (35, 20), and (0, 20). She draws a straight diagonal line from (0, 0) to (35, 20), dividing the mural into two triangular sections. The amount of red paint needed for the larger triangle is proportional to its area. If 15 square units of area require 6 liters of paint, write an equation in the form y = kx that represents the relationship between the area of the larger triangle (x) and the liters of red paint needed (y). Answer: y = 0.4x Solution: Identify the proportional relationship. The amount of red paint (y) is proportional to the area of the larger triangle (x). This means y = kx, where k is the constant of proportionality.
    Full step-by-step solution

    Step 1: Identify the proportional relationship. The amount of red paint (y) is proportional to the area of the larger triangle (x). This means y = kx, where k is the constant of proportionality. Step 2: Find the constant of proportionality (k). The problem states that 15 square units of area require 6 liters of paint. So, when x = 15, y = 6. Step 3: Substitute these values into the equation y = kx to solve for k. 6 = k * 15 Step 4: Divide both sides by 15 to isolate k. k = 6 / 15 Step 5: Simplify the fraction. k = 2 / 5 = 0.4 Step 6: Write the final equation. y = 0.4x The answer is y = 0.4x.

  2. Aroha is drawing a design for a rectangular mural on a coordinate grid. The mural's corners are at (1, 3), (43, 3), (43, 19), and (1, 19). She plans to place a square photo in the center of the mural with a side length equal to one-fifth of the mural's shorter side length. The photo is placed so its sides are parallel to the mural's sides, and its center coincides with the mural's center. The remaining area of the mural will be painted with a proportional pattern where the amount of blue paint used (y, in milliliters) is proportional to the area painted (x, in square units). If 117 milliliters of blue paint are needed for every 9 square units of area, write an equation that represents the proportional relationship between the blue paint needed (y) and the area painted (x). Answer: y = 13x Solution: Find the constant of proportionality k. The problem states that 117 milliliters of blue paint are needed for 9 square units of area. k = y / x = 117 / 9 = 13 Write the proportional relationship equation.
    Full step-by-step solution

    Step 1: Find the constant of proportionality k. The problem states that 117 milliliters of blue paint are needed for 9 square units of area. k = y / x = 117 / 9 = 13 Step 2: Write the proportional relationship equation. Since y is proportional to x with constant k = 13, the equation is y = 13x. The answer is y = 13x.

  3. Mere is designing a mosaic on a coordinate grid. The mosaic is in the shape of a rectangle with vertices at (0, 0), (24, 0), (24, 18), and (0, 18). She wants to place a triangular pattern inside the mosaic with vertices at (0, 0), (24, 0), and (12, 18). The area of the triangular pattern is proportional to the area of the entire rectangular mosaic. Write an equation in the form y = kx that represents the proportional relationship where y is the area of the triangular pattern and x is the area of the rectangular mosaic. Then find the value of k as a simplified fraction. Answer: y = 1/2 x Solution: Find the area of the rectangle. Length = 24 units, Width = 18 units. Area of rectangle = 24 × 18 = 432 square units.
    Full step-by-step solution

    Step 1: Find the area of the rectangle. Length = 24 units, Width = 18 units. Area of rectangle = 24 × 18 = 432 square units. Step 2: Find the area of the triangle. Base = 24 units (from (0,0) to (24,0)), Height = 18 units (vertical distance from base to the third vertex at (12,18)). Area of triangle = 1/2 × base × height = 1/2 × 24 × 18 = 1/2 × 432 = 216 square units. Step 3: The relationship is proportional: y = kx, where y is the triangle's area and x is the rectangle's area. Substitute the known values: 216 = k × 432. Step 4: Solve for k: k = 216 / 432 = 1/2. Step 5: Write the equation: y = 1/2 x. The answer is y = 1/2 x.

  4. A construction company needs to mix concrete using cement, sand, and gravel in the ratio 2:3:5. For a large building project, they need to prepare 15,000 kilograms of concrete mixture. How many kilograms of sand should they use in this batch? Answer: 4500 Solution: Step 1: Identify the ratio parts: cement = 2 parts, sand = 3 parts, gravel = 5 parts Step 2: Calculate total parts: 2 + 3 + 5 = 10 parts Step 3: Determine what fraction of the total mixture is sand: 3 parts sand out of 10 total parts = 3/10 Step 4: Calculate kilograms of sand needed: 3/10 ×…
    Full step-by-step solution

    Step 1: Identify the ratio parts: cement = 2 parts, sand = 3 parts, gravel = 5 parts Step 2: Calculate total parts: 2 + 3 + 5 = 10 parts Step 3: Determine what fraction of the total mixture is sand: 3 parts sand out of 10 total parts = 3/10 Step 4: Calculate kilograms of sand needed: 3/10 × 15,000 = 4,500 Step 5: Verify: 4,500 kg of sand represents 3 parts, so each part = 4,500 ÷ 3 = 1,500 kg Cement: 2 × 1,500 = 3,000 kg Gravel: 5 × 1,500 = 7,500 kg Total: 3,000 + 4,500 + 7,500 = 15,000 kg ✓ The answer is 4500 kilograms of sand.

  5. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). If the garden's length and width are in a 3:2 ratio, and you need to create a scale drawing where 1 unit on the coordinate plane represents 4 feet in real life, what is the actual area of the garden in square feet? Answer: 384 Solution: Identify the coordinates and find the side lengths on the coordinate plane. Vertices: (2, 1), (8, 1), (8, 5), (2, 5). (2, 1) to (8, 1) → horizontal side length = 8 - 2 = 6 units.
    Full step-by-step solution

    Let's go step by step. --- **Step 1: Identify the coordinates and find the side lengths on the coordinate plane.** Vertices: (2, 1), (8, 1), (8, 5), (2, 5). Plotting order: (2, 1) to (8, 1) → horizontal side length = 8 - 2 = 6 units. (8, 1) to (8, 5) → vertical side length = 5 - 1 = 4 units. So on the coordinate plane, the rectangle has length = 6 units, width = 4 units. --- **Step 2: Check the given ratio of length to width.** Ratio given: 3 : 2. Our length : width = 6 : 4 = 3 : 2. ✓ Matches. --- **Step 3: Apply the scale to find actual dimensions.** Scale: 1 unit on coordinate plane = 4 feet in real life. Actual length = 6 units × 4 ft/unit = 24 feet. Actual width = 4 units × 4 ft/unit = 16 feet. --- **Step 4: Calculate actual area in square feet.** Area = length × width = 24 ft × 16 ft. 24 × 16 = 384. --- **Step 5: Final answer.** Actual area = 384 square feet. --- **Answer:** 384