Proportional Graphs

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Answer Key & Explanations

Proportional Graphs · Grade 7 · Worksheet 3

1. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (14, 1), (14, 7), and (2, 7). A path runs diagonally from the vertex at (2, 1) to the vertex at (14, 7), dividing the garden into two triangular sections. What is the ratio of the area of the smaller triangular section to the area of the entire rectangular garden? Answer: 1:2 Solution: Vertices: (2, 1), (14, 1), (14, 7), (2, 7). Length along x-axis: from x = 2 to x = 14 → length = 14 − 2 = 12 units. Height along y-axis: from y = 1 to y = 7 → height = 7 − 1 = 6 units.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the rectangle dimensions** Vertices: (2, 1), (14, 1), (14, 7), (2, 7). Length along x-axis: from x = 2 to x = 14 → length = 14 − 2 = 12 units. Height along y-axis: from y = 1 to y = 7 → height = 7 − 1 = 6 units. Area of rectangle = length × height = 12 × 6 = 72 square units. --- **Step 2: Identify the diagonal** Diagonal from (2, 1) to (14, 7) splits the rectangle into two triangles. Triangle 1: vertices (2, 1), (14, 7), (2, 7) Triangle 2: vertices (2, 1), (14, 7), (14, 1) We need the smaller triangular section. --- **Step 3: Determine which triangle is smaller** Draw mentally: - Triangle 1: vertices (2, 1), (14, 7), (2, 7) Base along left side from (2, 1) to (2, 7) has length 6. But easier: area by coordinates. Triangle 1 vertices: A = (2, 1), B = (14, 7), C = (2, 7). Area formula for triangle given coordinates: Area = 1/2 × | x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2) | = 1/2 × | 2(7 − 7) + 14(7 − 1) + 2(1 − 7) | = 1/2 × | 0 + 14(6) + 2(−6) | = 1/2 × | 84 − 12 | = 1/2 × 72 = 36. Triangle 2 vertices: A = (2, 1), B = (14, 7), C = (14, 1). Area = 1/2 × | 2(7 − 1) + 14(1 − 1) + 14(1 − 7) | = 1/2 × | 2(6) + 0 + 14(−6) | = 1/2 × | 12 − 84 | = 1/2 × 72 = 36. Both triangles have equal area: 36 square units. --- **Step 4: Identify the "smaller triangular section"** Since both triangles have the same area, the "smaller" is just either one, both are equal. --- **Step 5: Ratio of smaller triangle area to entire rectangle** Smaller triangle area = 36. Rectangle area = 72. Ratio = 36 : 72 = 1 : 2. --- **Final Answer:** 1:2

2. A proportional relationship is graphed on a coordinate plane, showing the cost of printing photos at a photo lab. The line passes through points (0, 0) and (8, 12), where x represents the number of photos printed and y represents the total cost in dollars. What is the cost per photo? Answer: 1.5 Solution: In a proportional relationship, the ratio y/x is constant and represents the unit rate. Using the point (8, 12), calculate the ratio: 12/8 Simplify the fraction: 12 ÷ 8 = 1.5 This means for every photo printed, the cost increases by $1.50 The cost per photo is $1.50 The answer is 1.5.
   Full step-by-step solution

   Step 1: In a proportional relationship, the ratio y/x is constant and represents the unit rate. Step 2: Using the point (8, 12), calculate the ratio: 12/8 Step 3: Simplify the fraction: 12 ÷ 8 = 1.5 Step 4: This means for every photo printed, the cost increases by $1.50 Step 5: The cost per photo is $1.50 The answer is 1.5.

3. Hana is filling a large tank with water. The graph of the proportional relationship between time (in minutes) and the volume of water (in liters) is a straight line through the origin. The line passes through the point (15, 180). If the tank holds 540 liters, how many minutes will it take to fill the tank completely? Answer: 45 Solution: Find the constant of proportionality (rate of flow). The point (15, 180) means 180 liters in 15 minutes. Rate = volume / time = 180 / 15 = 12 liters per minute.
   Full step-by-step solution

   Step 1: Find the constant of proportionality (rate of flow). The point (15, 180) means 180 liters in 15 minutes. Rate = volume / time = 180 / 15 = 12 liters per minute. Step 2: Write the equation of the proportional relationship: y = 12x, where y is volume in liters and x is time in minutes. Step 3: The tank holds 540 liters, so substitute y = 540 into the equation: 540 = 12x. Step 4: Solve for x: x = 540 / 12 = 45. The tank will take 45 minutes to fill completely.

4. A proportional relationship is graphed on a coordinate plane, showing the cost of printing photos at a photo lab. The line passes through point (8, 12) representing 8 photos costing $12. The line extends through the origin (0,0). What is the cost per photo, and how much would 15 photos cost? Answer: 22.5 Solution: Identify that this is a proportional relationship since the line passes through the origin. Find the cost per photo using the given point (8, 12).
   Full step-by-step solution

   Step 1: Identify that this is a proportional relationship since the line passes through the origin. Step 2: Find the cost per photo using the given point (8, 12). Cost per photo = total cost ÷ number of photos = 12 ÷ 8 = 1.5 Step 3: Each photo costs $1.50. Step 4: Calculate the cost for 15 photos: 15 × 1.5 = 22.5 Step 5: The cost for 15 photos is $22.50.

5. If y = 2.5x and x = 12, then y = ? Answer: 30 Solution: The equation is y = 2.5x. Substitute x = 12 into the equation: y = 2.5 * 12. Multiply 2.5 by 12: 2.5 * 12 = 30.
   Full step-by-step solution

   Step 1: The equation is y = 2.5x. Step 2: Substitute x = 12 into the equation: y = 2.5 * 12. Step 3: Multiply 2.5 by 12: 2.5 * 12 = 30. The answer is 30.

6. y = 12x, x = 15, y = ? Answer: 180 Solution: The equation is y = 12x, which is a proportional relationship. Substitute x = 15 into the equation: y = 12 × 15. Multiply: 12 × 15 = 180.
   Full step-by-step solution

   Step 1: The equation is y = 12x, which is a proportional relationship. Step 2: Substitute x = 15 into the equation: y = 12 × 15. Step 3: Multiply: 12 × 15 = 180. Step 4: Therefore, y = 180. The answer is 180.

7. If y = 2.75x + 8 and x = 24, then y = ? Answer: 74 Solution: Start with the equation y = 2.75x + 8 Substitute x = 24 into the equation: y = 2.75 * 24 + 8 Multiply first: 2.75 * 24 = 66 Then add: 66 + 8 = 74 Therefore, y = 74 The answer is 74.
   Full step-by-step solution

   Step 1: Start with the equation y = 2.75x + 8 Step 2: Substitute x = 24 into the equation: y = 2.75 * 24 + 8 Step 3: Multiply first: 2.75 * 24 = 66 Step 4: Then add: 66 + 8 = 74 Step 5: Therefore, y = 74 The answer is 74.

8. A city is planning a new bike path that needs to be painted with special reflective paint. The paint company provides a graph showing that 4 gallons of paint can cover 1,200 feet of path. If the city needs to paint 15,000 feet of bike path, how many gallons of paint should they order? Answer: 50 Solution: Identify the proportional relationship from the graph: 4 gallons covers 1,200 feet. Find the coverage rate per gallon: 1,200 feet ÷ 4 gallons = 300 feet per gallon.
   Full step-by-step solution

   Step 1: Identify the proportional relationship from the graph: 4 gallons covers 1,200 feet. Step 2: Find the coverage rate per gallon: 1,200 feet ÷ 4 gallons = 300 feet per gallon. Step 3: Calculate gallons needed for 15,000 feet: 15,000 feet ÷ 300 feet/gallon = 50 gallons. The answer is 50.