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Graph Proportional

Grade 8 · Ratios · Worksheet 2

  1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A line is drawn from the origin (0,0) to the point (6,8), which represents the hypotenuse. What is the slope of this hypotenuse? Answer: ______________
  2. A line passes through points (3, 7) and (9, 19). Find the slope of this line.
    • A. 1.5
    • B. 2
    • C. 2.5
    • D. 3
  3. Aroha is filling her bird feeder with seeds. She notices that the number of seeds eaten, y, is proportional to the number of days, x. After 3 days, 21 seeds have been eaten. If she graphs this relationship with days on the x-axis and seeds eaten on the y-axis, what is the slope of the line, and what does it represent? Answer: ______________
  4. Kaia is filling her bird feeder at a constant rate. She uses a scoop that holds 9 ounces of birdseed, and it takes her 12 seconds to pour one full scoop into the feeder. The relationship between the number of scoops (x) and the total birdseed in ounces (y) is proportional. Write the equation that represents this proportional relationship, identify the slope, and explain what the slope means in this context. Answer: ______________
  5. Noah is filling jars with homemade lemonade for a school fundraiser. Each jar holds the same amount of lemonade. He notices that after filling 6 jars, he has used 4.2 liters of lemonade. If the relationship between the number of jars and the liters of lemonade used is proportional, what is the constant of proportionality (slope) that represents the liters of lemonade per jar? Express your answer as a decimal. Answer: ______________
  6. Mere is filling her bird feeder at a constant rate. She notices that after 4 seconds, 12 ounces of birdseed have been poured. After 6 seconds, 18 ounces have been poured. If the relationship between time (in seconds) and the amount of birdseed (in ounces) is proportional, what is the slope of the line representing this relationship, and what does it represent in this context? Answer: ______________
  7. A wheelchair ramp is being designed to meet accessibility standards. The ramp forms a right triangle on a coordinate plane with its base along the x-axis from (0,0) to (15,0) and its vertical rise along the y-axis from (0,0) to (0,1.5). What is the slope of the ramp's inclined surface? Answer: ______________
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Answer Key & Explanations

Graph Proportional · Grade 8 · Worksheet 2

  1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A line is drawn from the origin (0,0) to the point (6,8), which represents the hypotenuse. What is the slope of this hypotenuse? Answer: 4/3 Solution: Identify the two points on the line. The line goes from (0,0) to (6,8). So Point 1: (x1, y1) = (0, 0) Point 2: (x2, y2) = (6, 8) Recall the slope formula.
    Full step-by-step solution

    Step 1: Identify the two points on the line. The line goes from (0,0) to (6,8). So Point 1: (x1, y1) = (0, 0) Point 2: (x2, y2) = (6, 8) Step 2: Recall the slope formula. Slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Step 3: Substitute the coordinates into the formula. Change in y = 8 - 0 = 8 Change in x = 6 - 0 = 6 So Slope = 8 / 6 Step 4: Simplify the fraction. 8/6 can be simplified by dividing numerator and denominator by 2: 8 ÷ 2 = 4 6 ÷ 2 = 3 So Slope = 4/3 Step 5: Interpret the result. The slope of the hypotenuse from (0,0) to (6,8) is 4/3. Final Answer: 4/3

  2. A line passes through points (3, 7) and (9, 19). Find the slope of this line. Answer: B. 2 Solution: Identify the coordinates: (3, 7) and (9, 19) Use the slope formula: slope = (y2 - y1) ÷ (x2 - x1) Substitute the values: slope = (19 - 7) ÷ (9 - 3) Calculate the numerator: 19 - 7 = 12 Calculate the denominator: 9 - 3 = 6 Divide: 12 ÷ 6 = 2 The correct answer is 2.
    Full step-by-step solution

    Step 1: Identify the coordinates: (3, 7) and (9, 19) Step 2: Use the slope formula: slope = (y2 - y1) ÷ (x2 - x1) Step 3: Substitute the values: slope = (19 - 7) ÷ (9 - 3) Step 4: Calculate the numerator: 19 - 7 = 12 Step 5: Calculate the denominator: 9 - 3 = 6 Step 6: Divide: 12 ÷ 6 = 2 The correct answer is 2.

  3. Aroha is filling her bird feeder with seeds. She notices that the number of seeds eaten, y, is proportional to the number of days, x. After 3 days, 21 seeds have been eaten. If she graphs this relationship with days on the x-axis and seeds eaten on the y-axis, what is the slope of the line, and what does it represent? Answer: 7 Solution: Identify the given information. The relationship is proportional, so the equation is y = kx, where k is the slope. After 3 days (x = 3), 21 seeds have been eaten (y = 21).
    Full step-by-step solution

    Step 1: Identify the given information. The relationship is proportional, so the equation is y = kx, where k is the slope. After 3 days (x = 3), 21 seeds have been eaten (y = 21). Step 2: Substitute the values into the equation y = kx: 21 = k * 3. Step 3: Solve for k by dividing both sides by 3: k = 21 / 3 = 7. Step 4: Interpret the slope. The slope k = 7 means that 7 seeds are eaten per day. This is the unit rate. The slope is 7, representing 7 seeds eaten each day.

  4. Kaia is filling her bird feeder at a constant rate. She uses a scoop that holds 9 ounces of birdseed, and it takes her 12 seconds to pour one full scoop into the feeder. The relationship between the number of scoops (x) and the total birdseed in ounces (y) is proportional. Write the equation that represents this proportional relationship, identify the slope, and explain what the slope means in this context. Answer: y = 9x; slope = 9; the slope represents the unit rate of 9 ounces per scoop. Solution: Identify the proportional relationship. The total birdseed (y) is proportional to the number of scoops (x). The equation has the form y = kx, where k is the constant of proportionality (slope).
    Full step-by-step solution

    Step 1: Identify the proportional relationship. The total birdseed (y) is proportional to the number of scoops (x). The equation has the form y = kx, where k is the constant of proportionality (slope). Step 2: Find the unit rate. One scoop holds 9 ounces. So when x = 1, y = 9. This means k = 9/1 = 9. Step 3: Write the equation. Substitute k = 9 into y = kx to get y = 9x. Step 4: Identify the slope. The slope of the line is the same as k, which is 9. Step 5: Interpret the slope. The slope of 9 means that for every 1 scoop of birdseed, the feeder gets 9 ounces. The slope is the unit rate: 9 ounces per scoop. The answer is: y = 9x; slope = 9; the slope represents the unit rate of 9 ounces per scoop.

  5. Noah is filling jars with homemade lemonade for a school fundraiser. Each jar holds the same amount of lemonade. He notices that after filling 6 jars, he has used 4.2 liters of lemonade. If the relationship between the number of jars and the liters of lemonade used is proportional, what is the constant of proportionality (slope) that represents the liters of lemonade per jar? Express your answer as a decimal. Answer: 0.7 Solution: Identify the two quantities in the proportional relationship. The number of jars is the independent variable (x). The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality (slope).
    Full step-by-step solution

    Step 1: Identify the two quantities in the proportional relationship. The number of jars is the independent variable (x). The liters of lemonade is the dependent variable (y). Step 2: The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality (slope). Step 3: Use the given point (6 jars, 4.2 liters) to find k. Substitute x = 6 and y = 4.2 into y = kx: 4.2 = k * 6 Step 4: Solve for k by dividing both sides by 6: k = 4.2 / 6 k = 0.7 Step 5: Interpret the result. The constant of proportionality is 0.7, which means Noah uses 0.7 liters of lemonade per jar. The answer is 0.7.

  6. Mere is filling her bird feeder at a constant rate. She notices that after 4 seconds, 12 ounces of birdseed have been poured. After 6 seconds, 18 ounces have been poured. If the relationship between time (in seconds) and the amount of birdseed (in ounces) is proportional, what is the slope of the line representing this relationship, and what does it represent in this context? Answer: 3 ounces per second Solution: Identify the two points: (4, 12) and (6, 18). Slope = change in y / change in x = (18 - 12) / (6 - 4) = 6 / 2 = 3.
    Full step-by-step solution

    Step 1: Identify the two points: (4, 12) and (6, 18). Step 2: Slope = change in y / change in x = (18 - 12) / (6 - 4) = 6 / 2 = 3. Step 3: Since the relationship is proportional and passes through the origin, the slope equals the constant of proportionality k. Step 4: The slope of 3 means that for every 1 second, 3 ounces of birdseed are poured. The slope is 3 ounces per second.

  7. A wheelchair ramp is being designed to meet accessibility standards. The ramp forms a right triangle on a coordinate plane with its base along the x-axis from (0,0) to (15,0) and its vertical rise along the y-axis from (0,0) to (0,1.5). What is the slope of the ramp's inclined surface? Answer: 0.1 Solution: Identify the two points that define the ramp's inclined surface: (0,0) and (15,1.5) Recall that slope = (change in y) / (change in x) Calculate the change in y: 1.5 - 0 = 1.5 Calculate the change in x: 15 - 0 = 15 Divide the change in y by the change in x: 1.5 ÷ 15 = 0.1 The slope of the ramp is…
    Full step-by-step solution

    Step 1: Identify the two points that define the ramp's inclined surface: (0,0) and (15,1.5) Step 2: Recall that slope = (change in y) / (change in x) Step 3: Calculate the change in y: 1.5 - 0 = 1.5 Step 4: Calculate the change in x: 15 - 0 = 15 Step 5: Divide the change in y by the change in x: 1.5 ÷ 15 = 0.1 Step 6: The slope of the ramp is 0.1, which means it rises 0.1 units vertically for every 1 unit horizontally.