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Graph Proportional

Grade 8 · Ratios · Worksheet 2

  1. On a coordinate plane, a line is drawn through the origin (0,0) and the point (7, 42). This line represents the proportional relationship between the number of hours Mason works (x-axis) and his total earnings in dollars (y-axis). What is the slope of this line, and what does it represent as the unit rate? Answer: ______________
  2. Tane walks at a constant speed. The distance he walks, y meters, is proportional to the time, x minutes. After 7 minutes, he has walked 91 meters. What is the slope of the graph of this proportional relationship, and what does it represent? Answer: ______________
  3. Aroha is making traditional woven flax kete (baskets). She finds that the amount of flax ribbon she needs is proportional to the number of kete she makes. For 3 kete, she needs 21 meters of ribbon. If she graphs the relationship with the number of kete on the x-axis and the meters of ribbon on the y-axis, what is the slope of the line, and what does it represent? Answer: ______________
  4. Mason earns money at a constant rate. After 9 hours of work, he has earned $126. If the relationship between hours worked and money earned is proportional, what is the slope of the graph representing this relationship? Answer: ______________
  5. Mason earns money at a constant rate. After working 7 hours, he earns $42. What is the slope of the line representing his earnings (y) as a function of hours worked (x)? Answer: ______________
  6. Liam is tracking his savings for a new video game. He starts with $40 and saves $15 each week. Create a graph showing the relationship between weeks (x-axis) and total savings (y-axis). What is the slope of this line and what does it represent in this situation? Answer: ______________
  7. 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: ______________
  8. (3² × 4) ÷ (6 - 2) = ? Answer: ______________
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Answer Key & Explanations

Graph Proportional · Grade 8 · Worksheet 2

  1. On a coordinate plane, a line is drawn through the origin (0,0) and the point (7, 42). This line represents the proportional relationship between the number of hours Mason works (x-axis) and his total earnings in dollars (y-axis). What is the slope of this line, and what does it represent as the unit rate? Answer: 6 Solution: Identify the two points on the line: (0,0) and (7,42). The slope formula is slope = (change in y) / (change in x). Calculate the change in y: 42 - 0 = 42.
    Full step-by-step solution

    Step 1: Identify the two points on the line: (0,0) and (7,42). Step 2: The slope formula is slope = (change in y) / (change in x). Step 3: Calculate the change in y: 42 - 0 = 42. Step 4: Calculate the change in x: 7 - 0 = 7. Step 5: Divide: 42 / 7 = 6. Step 6: The slope is 6. In this context, the slope represents the unit rate, meaning Mason earns $6 for every hour he works. Final Answer: 6

  2. Tane walks at a constant speed. The distance he walks, y meters, is proportional to the time, x minutes. After 7 minutes, he has walked 91 meters. What is the slope of the graph of this proportional relationship, and what does it represent? Answer: 13 Solution: The relationship is proportional, so it can be written as y = kx, where k is the slope (unit rate). We are given that when x = 7, y = 91.
    Full step-by-step solution

    Step 1: The relationship is proportional, so it can be written as y = kx, where k is the slope (unit rate). Step 2: We are given that when x = 7, y = 91. Step 3: Substitute into the equation: 91 = k * 7 Step 4: Solve for k: k = 91 / 7 = 13 Step 5: The slope is 13, which represents the unit rate: Tane walks 13 meters per minute. The answer is 13.

  3. Aroha is making traditional woven flax kete (baskets). She finds that the amount of flax ribbon she needs is proportional to the number of kete she makes. For 3 kete, she needs 21 meters of ribbon. If she graphs the relationship with the number of kete on the x-axis and the meters of ribbon on the y-axis, what is the slope of the line, and what does it represent? Answer: 7 Solution: Identify the point from the problem. When 3 kete are made, 21 meters of ribbon are used. This gives the point (3, 21) on the graph.
    Full step-by-step solution

    Step 1: Identify the point from the problem. When 3 kete are made, 21 meters of ribbon are used. This gives the point (3, 21) on the graph. Since the relationship is proportional, the line passes through (0, 0) as well. Step 2: Calculate the slope using the two points (0, 0) and (3, 21). Slope = (change in y) / (change in x) = (21 - 0) / (3 - 0) = 21 / 3. Step 3: Simplify 21 / 3 = 7. Step 4: Interpret the slope. The slope of 7 means that for every 1 kete Aroha makes, she needs 7 meters of ribbon. The slope is the unit rate (meters of ribbon per kete). The answer is 7.

  4. Mason earns money at a constant rate. After 9 hours of work, he has earned $126. If the relationship between hours worked and money earned is proportional, what is the slope of the graph representing this relationship? Answer: 14 Solution: In a proportional relationship, the slope is the unit rate. Here, the unit rate is the amount earned per hour. Mason earned $126 in 9 hours.
    Full step-by-step solution

    Step 1: In a proportional relationship, the slope is the unit rate. Here, the unit rate is the amount earned per hour. Step 2: Mason earned $126 in 9 hours. Step 3: Divide the total earnings by the number of hours to find the unit rate: 126 / 9 = 14. Step 4: The slope of the graph is 14, meaning Mason earns $14 per hour. The answer is 14.

  5. Mason earns money at a constant rate. After working 7 hours, he earns $42. What is the slope of the line representing his earnings (y) as a function of hours worked (x)? Answer: 6 Solution: The relationship is proportional, so it can be written as y = kx, where k is the slope (unit rate). We know that when x = 7 hours, y = $42.
    Full step-by-step solution

    Step 1: The relationship is proportional, so it can be written as y = kx, where k is the slope (unit rate). Step 2: We know that when x = 7 hours, y = $42. Step 3: Substitute into the equation: 42 = k * 7 Step 4: Solve for k: k = 42 / 7 = 6 Step 5: The slope is 6, meaning Mason earns $6 per hour. The answer is 6.

  6. Liam is tracking his savings for a new video game. He starts with $40 and saves $15 each week. Create a graph showing the relationship between weeks (x-axis) and total savings (y-axis). What is the slope of this line and what does it represent in this situation? Answer: 15, which represents the amount saved per week Solution: Liam starts with $40 and saves $15 each week. - \( x \) = number of weeks - \( y \) = total savings in dollars Initial savings = $40 (when \( x = 0 \)) Each week adds $15.
    Full step-by-step solution

    Let's go step by step. --- **Step 1: Understand the problem** Liam starts with $40 and saves $15 each week. Let: - \( x \) = number of weeks - \( y \) = total savings in dollars The relationship is: Initial savings = $40 (when \( x = 0 \)) Each week adds $15. --- **Step 2: Write the equation** Total savings = starting amount + (amount saved per week) × (number of weeks) So: \( y = 40 + 15x \) --- **Step 3: Identify the slope** The equation \( y = 40 + 15x \) is in slope-intercept form \( y = mx + b \), where: - \( m \) = slope - \( b \) = y-intercept Here, \( m = 15 \), \( b = 40 \). --- **Step 4: Interpret the slope** The slope is the change in \( y \) per unit change in \( x \). Since \( x \) is weeks and \( y \) is total savings, slope = change in savings / change in weeks = 15 dollars per week. So the slope represents the **amount saved per week**. --- **Step 5: Final answer** Slope = 15, which represents the amount saved per week.

  7. 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

  8. (3² × 4) ÷ (6 - 2) = ? Answer: 9 Solution: Solve inside the parentheses: (6 - 2) = 4 Calculate the exponent: 3² = 9 Multiply: 9 × 4 = 36 Divide: 36 ÷ 4 = 9 The answer is 9.
    Full step-by-step solution

    Step 1: Solve inside the parentheses: (6 - 2) = 4 Step 2: Calculate the exponent: 3² = 9 Step 3: Multiply: 9 × 4 = 36 Step 4: Divide: 36 ÷ 4 = 9 The answer is 9.