Proportional Equations

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Proportional Equations · Grade 7 · Worksheet 1

1. A science museum is creating a scale model of the solar system where 1 centimeter represents 12,000 kilometers. If the actual distance from Earth to Mars is approximately 78 million kilometers, what should be the distance between Earth and Mars in the scale model, measured in centimeters? Answer: 6500 Solution: Identify the scale ratio - 1 cm represents 12,000 km Write this as a proportion: 1 cm / 12,000 km The actual distance is 78,000,000 km Set up the equation: model distance / 78,000,000 km = 1 cm / 12,000 km Solve for model distance: model distance = (78,000,000 km × 1 cm) / 12,000 km Calculate:…
   Full step-by-step solution

   Step 1: Identify the scale ratio - 1 cm represents 12,000 km Step 2: Write this as a proportion: 1 cm / 12,000 km Step 3: The actual distance is 78,000,000 km Step 4: Set up the equation: model distance / 78,000,000 km = 1 cm / 12,000 km Step 5: Solve for model distance: model distance = (78,000,000 km × 1 cm) / 12,000 km Step 6: Calculate: 78,000,000 ÷ 12,000 = 6,500 Step 7: The model distance should be 6,500 centimeters

2. A construction company is building a scale model of a new bridge. The actual bridge will be 420 meters long, and the scale model uses a ratio of 1:35. If the model bridge is 12 meters long, how many centimeters wide should the model bridge be if the actual bridge is 28 meters wide? Answer: 80 Solution: Identify the scale ratio. The model uses 1:35, meaning 1 unit on the model represents 35 units in real life. The actual bridge width is 28 meters.
   Full step-by-step solution

   Step 1: Identify the scale ratio. The model uses 1:35, meaning 1 unit on the model represents 35 units in real life. Step 2: The actual bridge width is 28 meters. To find the model width, divide the actual width by the scale ratio: 28 ÷ 35 = 0.8 meters. Step 3: Convert meters to centimeters. Since 1 meter = 100 centimeters, multiply 0.8 × 100 = 80 centimeters. Step 4: The model bridge should be 80 centimeters wide to maintain the correct proportion with the actual bridge.

3. 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.

4. Hana earns money at a constant rate. She earns $168 for working 14 hours. Write the equation y = kx that represents her total earnings y for x hours worked. Answer: y = 12x Solution: Identify the given values. Total earnings y = $168, hours worked x = 14. The relationship is proportional, so y = kx.
   Full step-by-step solution

   Step 1: Identify the given values. Total earnings y = $168, hours worked x = 14. Step 2: The relationship is proportional, so y = kx. Substitute the known values: 168 = k * 14. Step 3: Solve for k by dividing both sides by 14: k = 168 / 14 = 12. Step 4: Write the equation: y = 12x. The answer is y = 12x.

5. A map of a national park uses a scale where 1 inch represents 2.5 miles. If the hiking trail from the visitor center to the waterfall measures 3.75 inches on the map, how many miles is the actual hiking trail? Answer: 9.375 Solution: The map scale is 1 inch = 2.5 miles The trail measures 3.75 inches on the map Multiply the map measurement by the scale factor: 3.75 × 2.5 Calculate 3.75 × 2.5 = 9.375 The actual hiking trail is 9.375 miles long
   Full step-by-step solution

   Step 1: The map scale is 1 inch = 2.5 miles Step 2: The trail measures 3.75 inches on the map Step 3: Multiply the map measurement by the scale factor: 3.75 × 2.5 Step 4: Calculate 3.75 × 2.5 = 9.375 Step 5: The actual hiking trail is 9.375 miles long

6. Charlotte earns money at a constant rate. She earns $255 for working 15 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 = $255, hours worked x = 15. The relationship is proportional, so y = kx.
   Full step-by-step solution

   Step 1: Identify the given values. Total earnings y = $255, hours worked x = 15. Step 2: The relationship is proportional, so y = kx. Substitute the known values: 255 = k * 15. Step 3: Solve for k by dividing both sides by 15: k = 255 / 15 = 17. Step 4: Write the equation: y = 17x. The answer is y = 17x.

7. Mia is saving money to buy a new video game. For every hour Mia helps with chores, Mia earns $11. If Mia works for 10 hours, write an equation that shows the relationship between the total money earned, y, and the number of hours worked, x. Then use your equation to find how much money Mia will earn. Answer: 110 Solution: The relationship is proportional, so the equation is y = kx, where k is the constant of proportionality. The constant of proportionality is the amount earned per hour, which is $11. So k = 11.
   Full step-by-step solution

   Step 1: The relationship is proportional, so the equation is y = kx, where k is the constant of proportionality. Step 2: The constant of proportionality is the amount earned per hour, which is $11. So k = 11. Step 3: The equation is y = 11x. Step 4: To find the total money earned for 10 hours, substitute x = 10 into the equation: y = 11 × 10 = 110. Step 5: Mia will earn $110.

8. If y = 4.2x and y = 63, then x = ? Answer: 15 Solution: Start with the given equation: y = 4.2x Substitute the known value of y: 63 = 4.2x To solve for x, divide both sides of the equation by 4.2: x = 63 ÷ 4.2 Calculate the division: 63 ÷ 4.2 = 15 Therefore, x = 15 The answer is 15.
   Full step-by-step solution

   Step 1: Start with the given equation: y = 4.2x Step 2: Substitute the known value of y: 63 = 4.2x Step 3: To solve for x, divide both sides of the equation by 4.2: x = 63 ÷ 4.2 Step 4: Calculate the division: 63 ÷ 4.2 = 15 Step 5: Therefore, x = 15 The answer is 15.