Graph Proportional

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Answer Key & Explanations

Graph Proportional · Grade 8 · Worksheet 1

1. Tane earns $12 for every hour he works. If the relationship between hours worked (x) and total earnings (y) is proportional, what is the slope of the graph representing this relationship? Answer: 12 Solution: The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality (the slope). Tane earns $12 per hour, so the unit rate is 12 dollars per hour.
   Full step-by-step solution

   Step 1: The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality (the slope). Step 2: Tane earns $12 per hour, so the unit rate is 12 dollars per hour. Step 3: In the equation y = kx, k is the slope. Here, k = 12. Step 4: Therefore, the slope of the graph is 12. The answer is 12.

2. Emma earns money at a constant rate. She earns $45 for working 3 hours. If the relationship is proportional, what is the slope of the graph showing her earnings (y) versus time in hours (x)? Answer: 15 Solution: Identify the given information: Emma earns $45 for 3 hours. This gives the point (3, 45) on the graph. Since the relationship is proportional, the graph passes through the origin (0, 0).
   Full step-by-step solution

   Step 1: Identify the given information: Emma earns $45 for 3 hours. This gives the point (3, 45) on the graph. Step 2: Since the relationship is proportional, the graph passes through the origin (0, 0). Step 3: The slope is the unit rate, which is the earnings per hour. Calculate: slope = (change in y) / (change in x) = (45 - 0) / (3 - 0) = 45 / 3 = 15. Step 4: The slope is 15, meaning Emma earns $15 per hour. The answer is 15.

3. Sophia is filling orders for her small business making handmade candles. She uses 6 ounces of wax for each candle. If she makes 11 candles, the total wax used is 66 ounces. What is the constant of proportionality (slope) that represents the number of ounces of wax per candle? Graph the proportional relationship between the number of candles (x) and total ounces of wax used (y). Answer: 6 Solution: Identify the proportional relationship. The total ounces of wax (y) is proportional to the number of candles (x), so y = kx where k is the constant of proportionality (slope).
   Full step-by-step solution

   Step 1: Identify the proportional relationship. The total ounces of wax (y) is proportional to the number of candles (x), so y = kx where k is the constant of proportionality (slope). Step 2: Use the given point: when x = 11 candles, y = 66 ounces. Step 3: Substitute into the equation: 66 = k * 11. Step 4: Solve for k: k = 66 / 11 = 6. Step 5: The slope is 6, which means Sophia uses 6 ounces of wax per candle. Step 6: To graph, plot the origin (0,0) and the point (11, 66). Draw a straight line through them. The slope is the unit rate, 6. The answer is 6.

4. Liam is designing a wheelchair ramp for his school's new accessibility project. The ramp needs to rise 3 feet over a horizontal distance of 36 feet. What is the slope of the ramp? Express your answer as a simplified fraction. Answer: 1/12 Solution: Slope is defined as the ratio of the vertical change (rise) to the horizontal change (run). So, slope = rise / run. Identify the rise and run from the problem.
   Full step-by-step solution

   Step 1: Understand the slope formula. Slope is defined as the ratio of the vertical change (rise) to the horizontal change (run). So, slope = rise / run. Step 2: Identify the rise and run from the problem. The ramp rises 3 feet, so rise = 3. The horizontal distance is 36 feet, so run = 36. Step 3: Write the slope as a fraction. Slope = 3 / 36. Step 4: Simplify the fraction. Both 3 and 36 can be divided by 3. 3 ÷ 3 = 1 36 ÷ 3 = 12 So, 3/36 simplifies to 1/12. Step 5: Interpret the result. The slope of the ramp is 1/12. This means for every 12 feet of horizontal distance, the ramp rises 1 foot. Final answer: 1/12

5. Emma is designing a rainwater collection system for her school's garden. The gutter needs to slope downward to direct water toward the collection barrel. For every 8 feet the gutter extends horizontally, it drops 1.5 feet vertically. What is the slope of Emma's gutter? Express your answer as a simplified fraction. Answer: 3/16 Solution: Identify the vertical change (rise) and horizontal change (run) Vertical change = 1.5 feet Horizontal change = 8 feet Slope = 1.5 / 8 Convert 1.5 to a fraction 1.5 = 3/2 Slope = (3/2) / 8 = 3/2 × 1/8 3/2 × 1/8 = 3/16 3/16 is already in simplest form since 3 and 16 share no common factors The…
   Full step-by-step solution

   Step 1: Identify the vertical change (rise) and horizontal change (run) Vertical change = 1.5 feet Horizontal change = 8 feet Step 2: Write slope as rise over run Slope = 1.5 / 8 Step 3: Convert 1.5 to a fraction 1.5 = 3/2 Step 4: Write the slope as a fraction Slope = (3/2) / 8 = 3/2 × 1/8 Step 5: Multiply the fractions 3/2 × 1/8 = 3/16 Step 6: Check if the fraction can be simplified 3/16 is already in simplest form since 3 and 16 share no common factors The slope of Emma's gutter is 3/16.

6. Noah is driving from his home to a national park. The distance he travels is proportional to the time he drives. After 2 hours, he has traveled 140 miles. After 7 hours, he has traveled 490 miles. What is the slope of the line representing this proportional relationship, and what does the slope represent in the context of Noah's trip? Answer: 70 miles per hour Solution: Identify two points from the problem. The points are (2, 140) and (7, 490), where x is time in hours and y is distance in miles.
   Full step-by-step solution

   Step 1: Identify two points from the problem. The points are (2, 140) and (7, 490), where x is time in hours and y is distance in miles. Step 2: Calculate the slope using the formula: slope = (change in y) / (change in x) = (490 - 140) / (7 - 2). Step 3: Subtract: 490 - 140 = 350, and 7 - 2 = 5. Step 4: Divide: 350 / 5 = 70. Step 5: The slope is 70. Since the relationship is proportional, the slope represents the unit rate, which is the distance traveled per hour. So the slope of 70 means Noah drives at a constant speed of 70 miles per hour. The answer is 70 miles per hour.

7. A proportional relationship is represented by the equation y = kx. The graph of this relationship passes through the point (6, 36). What is the slope of the line? Answer: 6 Solution: The equation for a proportional relationship is y = kx, where k is the slope and unit rate. The graph passes through the origin (0,0) and the point (6, 36).
   Full step-by-step solution

   Step 1: The equation for a proportional relationship is y = kx, where k is the slope and unit rate. Step 2: The graph passes through the origin (0,0) and the point (6, 36). Step 3: Substitute the point (6, 36) into y = kx: 36 = k × 6 Step 4: Solve for k: k = 36 ÷ 6 = 6 Step 5: Therefore, the slope of the line is 6. The answer is 6.

8. Tane earns money at a constant rate. The graph of his earnings (y dollars) versus time (x hours) is a straight line through the origin. After 7 hours, he has earned $91. What is the slope of the line? Answer: 13 Solution: Identify the given point on the line: (7, 91). Since the line goes through the origin (0,0), use the slope formula: slope = (y2 - y1) / (x2 - x1). Substitute the points: slope = (91 - 0) / (7 - 0).
   Full step-by-step solution

   Step 1: Identify the given point on the line: (7, 91). Step 2: Since the line goes through the origin (0,0), use the slope formula: slope = (y2 - y1) / (x2 - x1). Step 3: Substitute the points: slope = (91 - 0) / (7 - 0). Step 4: Simplify: slope = 91 / 7. Step 5: Calculate: 91 ÷ 7 = 13. The slope is 13, which means Tane earns $13 per hour.