Proportional Graphs

lessonbunny.com

Answer Key & Explanations

Proportional Graphs · Grade 7 · Worksheet 1

1. 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 the x-axis represents number of photos and the y-axis represents total cost in dollars. If a customer wants to print 18 photos, what would be the total cost? Answer: 27 Solution: Find the constant of proportionality (unit rate) using the given points (0, 0) and (8, 12). The constant of proportionality = y/x = 12/8 = 1.5 dollars per photo.
   Full step-by-step solution

   Step 1: Find the constant of proportionality (unit rate) using the given points (0, 0) and (8, 12). Step 2: The constant of proportionality = y/x = 12/8 = 1.5 dollars per photo. Step 3: For 18 photos, multiply the number of photos by the unit rate: 18 × 1.5 = 27. Step 4: The total cost for 18 photos is $27.

2. Tane is a conservation ranger who tracks the growth of native trees in a reforestation project. He discovers that the height of a particular kauri tree is directly proportional to the number of years since it was planted. After 12 years, the tree is 9 meters tall. If the tree continues to grow at the same constant rate, how tall will it be after 28 years? Answer: 21 Solution: Identify that height and time are directly proportional, so height = k × time, where k is the constant of proportionality. Use the given pair (12 years, 9 meters) to find k: 9 = k × 12, so k = 9 ÷ 12 = 3/4 or 0.75.
   Full step-by-step solution

   Step 1: Identify that height and time are directly proportional, so height = k × time, where k is the constant of proportionality. Step 2: Use the given pair (12 years, 9 meters) to find k: 9 = k × 12, so k = 9 ÷ 12 = 3/4 or 0.75. Step 3: Write the proportional equation: height = 0.75 × time. Step 4: Substitute time = 28 years: height = 0.75 × 28 = 21. Step 5: The tree will be 21 meters tall after 28 years. The answer is 21.

3. 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

4. If y = 3x + 2, what is y when x = 5? Answer: 17 Solution: We are given the equation: y = 3x + 2 We are told x = 5. Substitute x = 5 into the equation. y = 3 * 5 + 2 Perform the multiplication first (order of operations).
   Full step-by-step solution

   We are given the equation: y = 3x + 2 We are told x = 5. Step 1: Substitute x = 5 into the equation. y = 3 * 5 + 2 Step 2: Perform the multiplication first (order of operations). 3 * 5 = 15 So now: y = 15 + 2 Step 3: Perform the addition. 15 + 2 = 17 Step 4: Write the final answer. y = 17 So when x = 5, y equals 17.

5. Isabella is filling orders for her online craft store. She knows that to make 12 beaded bracelets, she needs 84 inches of elastic cord. If she needs to fill an order for 27 bracelets, and she wants to graph the proportional relationship between the number of bracelets and the length of cord needed, what is the constant of proportionality (the slope) that she should use for her graph? Give your answer as a decimal. Answer: 7 Solution: Identify the two quantities: number of bracelets (x) and length of cord in inches (y). The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality.
   Full step-by-step solution

   Step 1: Identify the two quantities: number of bracelets (x) and length of cord in inches (y). Step 2: The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality. Step 3: We know that when x = 12 bracelets, y = 84 inches of cord. Step 4: Substitute into the equation: 84 = k * 12. Step 5: Solve for k by dividing both sides by 12: k = 84 / 12. Step 6: Calculate: 84 / 12 = 7. Step 7: The constant of proportionality is 7, meaning each bracelet requires 7 inches of cord. The answer is 7.

6. Emma is training for a charity run. She runs at a constant speed, and the distance she covers is proportional to the time she runs. After 45 minutes, she has run 6,750 meters. If she continues at the same rate, how many meters will she run in 75 minutes? Answer: 11250 Solution: Identify the proportional relationship: distance = k × time, where k is the constant speed. Use the given values: after 45 minutes, distance = 6,750 meters.
   Full step-by-step solution

   Step 1: Identify the proportional relationship: distance = k × time, where k is the constant speed. Step 2: Use the given values: after 45 minutes, distance = 6,750 meters. Step 3: Find the constant of proportionality k: 6,750 = k × 45 → k = 6,750 ÷ 45 = 150 meters per minute. Step 4: Use the constant to find the distance for 75 minutes: distance = 150 × 75 = 11,250 meters. The answer is 11,250.

7. y = 9x, x = 13, y = ? Answer: 117 Solution: Start with the equation y = 9x. Substitute x = 13 into the equation: y = 9 * 13. Multiply: 9 * 13 = 117.
   Full step-by-step solution

   Step 1: Start with the equation y = 9x. Step 2: Substitute x = 13 into the equation: y = 9 * 13. Step 3: Multiply: 9 * 13 = 117. Step 4: The final answer is 117.