Proportional Equations

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Answer Key & Explanations

Proportional Equations · Grade 7 · Worksheet 2

1. Charlotte earns money at a constant rate. She earns $221 for working 13 hours. Write the equation y = kx that represents her total earnings y for x hours worked. Answer: y = 17x Solution: Identify the given values. Total earnings y = $221, hours worked x = 13. The relationship is proportional, so y = kx.
   Full step-by-step solution

   Step 1: Identify the given values. Total earnings y = $221, hours worked x = 13. Step 2: The relationship is proportional, so y = kx. Substitute the known values: 221 = k * 13. Step 3: Solve for k by dividing both sides by 13: k = 221 / 13 = 17. Step 4: Write the equation: y = 17x. The answer is y = 17x.

2. If y = 3x + 5, what is y when x = 7? Answer: 26 Solution: We are given the equation: y = 3x + 5 We are told x = 7. Substitute x = 7 into the equation. y = 3 * 7 + 5 Perform the multiplication first (order of operations).
   Full step-by-step solution

   We are given the equation: y = 3x + 5 We are told x = 7. Step 1: Substitute x = 7 into the equation. y = 3 * 7 + 5 Step 2: Perform the multiplication first (order of operations). 3 * 7 = 21 So now: y = 21 + 5 Step 3: Perform the addition. 21 + 5 = 26 Step 4: Conclusion. Therefore, when x = 7, y = 26.

3. Mason is planning a road trip. The car uses 1 gallon of gas for every 15 miles driven. If Mason drives 105 miles, how many gallons of gas will be used? Write an equation in the form y = kx to represent this relationship, where y is gallons and x is miles. Answer: 7 Solution: Identify the constant of proportionality k. The car uses 1 gallon for 15 miles, so k = 1/15 gallons per mile. The proportional relationship is y = kx, where y is gallons and x is miles.
   Full step-by-step solution

   Step 1: Identify the constant of proportionality k. The car uses 1 gallon for 15 miles, so k = 1/15 gallons per mile. Step 2: The proportional relationship is y = kx, where y is gallons and x is miles. Step 3: Substitute k = 1/15: y = (1/15)x Step 4: For 105 miles, y = (1/15) × 105 = 105/15 = 7 gallons. The equation is y = (1/15)x, and for 105 miles, 7 gallons are used.

4. Aisha is planning a road trip and uses a map app that shows the distance between cities. The app indicates that 3.5 centimeters on the map represents 210 kilometers in real life. If the distance between two cities measures 8.2 centimeters on the map, what is the actual distance between them in kilometers? Answer: 492 Solution: Identify the proportional relationship. The map scale is 3.5 cm : 210 km. Find the unit rate, which is the real distance for 1 cm on the map.
   Full step-by-step solution

   Step 1: Identify the proportional relationship. The map scale is 3.5 cm : 210 km. Step 2: Find the unit rate, which is the real distance for 1 cm on the map. Divide 210 km by 3.5 cm. 210 / 3.5 = 60. So, 1 cm on the map equals 60 km in real life. Step 3: Multiply the unit rate by the new map distance to find the actual distance. 8.2 cm * 60 km/cm = 492 km. The actual distance between the two cities is 492 kilometers.

5. Noah earns money at a constant rate. He earns $221 for working 13 hours. Write the equation y = kx that represents his total earnings y for x hours worked. Answer: y = 17x Solution: Identify the given values. Total earnings y = $221, hours worked x = 13. The relationship is proportional, so y = kx.
   Full step-by-step solution

   Step 1: Identify the given values. Total earnings y = $221, hours worked x = 13. Step 2: The relationship is proportional, so y = kx. Substitute the known values: 221 = k * 13. Step 3: Solve for k by dividing both sides by 13: k = 221 / 13 = 17. Step 4: Write the equation: y = 17x. The answer is y = 17x.

6. If y = 3x and y = 15, then x = ? Answer: 5 Solution: 1) y = 3x 2) y = 15 Since both equations are equal to y, we can set them equal to each other. 3x = 15 To solve for x, we need to isolate x. Since x is multiplied by 3, we divide both sides of the equation by 3.
   Full step-by-step solution

   We are given two equations: 1) y = 3x 2) y = 15 Step 1: Since both equations are equal to y, we can set them equal to each other. That means: 3x = 15 Step 2: To solve for x, we need to isolate x. Since x is multiplied by 3, we divide both sides of the equation by 3. So: x = 15 / 3 Step 3: Perform the division. 15 divided by 3 is 5. Step 4: Conclusion. Therefore, x = 5.

7. A school is planning a field trip and needs to calculate transportation costs. The bus company charges a fixed fee of $120 plus $15 per student. If the school has budgeted $600 for transportation, how many students can go on the field trip? Answer: 32 Solution: Write the equation for the total cost: Total cost = Fixed fee + (Cost per student × Number of students) Plug in the known values: 600 = 120 + (15 × n) Subtract the fixed fee from both sides: 600 - 120 = 15 × n Calculate: 480 = 15 × n Divide both sides by 15: n = 480 ÷ 15 Calculate: n = 32…
   Full step-by-step solution

   Step 1: Write the equation for the total cost: Total cost = Fixed fee + (Cost per student × Number of students) Step 2: Plug in the known values: 600 = 120 + (15 × n) Step 3: Subtract the fixed fee from both sides: 600 - 120 = 15 × n Step 4: Calculate: 480 = 15 × n Step 5: Divide both sides by 15: n = 480 ÷ 15 Step 6: Calculate: n = 32 Therefore, 32 students can go on the field trip.

8. A rectangular swimming pool is drawn on a coordinate plane with vertices at (0, 0), (20, 0), (20, 12), and (0, 12). A diagonal safety rope is installed from the bottom-left corner to the top-right corner. The pool owner wants to paint the larger triangular section of the pool floor a different color. What is the area of this larger triangular section in square units? Answer: 120 Solution: Identify the rectangle's dimensions from the coordinates. The rectangle has length from (0,0) to (20,0), so length = 20 units. The rectangle has width from (0,0) to (0,12), so width = 12 units.
   Full step-by-step solution

   Step 1: Identify the rectangle's dimensions from the coordinates. The rectangle has length from (0,0) to (20,0), so length = 20 units. The rectangle has width from (0,0) to (0,12), so width = 12 units. Step 2: Calculate the area of the entire rectangle. Area of rectangle = length × width = 20 × 12 = 240 square units. Step 3: Understand how the diagonal divides the rectangle. A diagonal splits a rectangle into two congruent right triangles of equal area. Step 4: Calculate the area of one triangular section. Area of one triangle = (Area of rectangle) ÷ 2 = 240 ÷ 2 = 120 square units. Step 5: Since both triangles are equal, the larger triangular section has area 120 square units. The answer is 120.