Proportional Equations

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Proportional Equations · Grade 7 · Worksheet 1

1. A rectangular garden is drawn on a coordinate plane with vertices at (1, 2), (13, 2), (13, 10), and (1, 10). A circular fountain is placed in the center of the garden with a diameter equal to half the garden's width. What is the area of the garden that is NOT covered by the fountain? (Use π = 3.14) Answer: 85.76 Solution: Length = difference in x-coordinates = 13 - 1 = 12 units Width = difference in y-coordinates = 10 - 2 = 8 units Area_rectangle = length × width = 12 × 8 = 96 square units Diameter = half the garden's width = 8 ÷ 2 = 4 units Radius = diameter ÷ 2 = 4 ÷ 2 = 2 units Area_fountain = π × radius² =…
   Full step-by-step solution

   Step 1: Find the dimensions of the rectangular garden Length = difference in x-coordinates = 13 - 1 = 12 units Width = difference in y-coordinates = 10 - 2 = 8 units Step 2: Calculate the area of the rectangular garden Area_rectangle = length × width = 12 × 8 = 96 square units Step 3: Find the diameter of the circular fountain Diameter = half the garden's width = 8 ÷ 2 = 4 units Step 4: Calculate the radius of the fountain Radius = diameter ÷ 2 = 4 ÷ 2 = 2 units Step 5: Calculate the area of the circular fountain Area_fountain = π × radius² = 3.14 × 2² = 3.14 × 4 = 12.56 square units Step 6: Calculate the area NOT covered by the fountain Area_not_covered = Area_rectangle - Area_fountain = 96 - 12.56 = 85.76 square units The answer is 85.76.

2. Liam is baking cookies for a school fundraiser. His recipe uses 2.5 cups of flour to make 15 cookies. If he wants to make 75 cookies for the event, how many cups of flour will he need? Answer: 12.5 Solution: 2.5 cups of flour make 15 cookies. We want to find how many cups (call it x) are needed for 75 cookies.
   Full step-by-step solution

   First, identify the ratio from the recipe: 2.5 cups of flour make 15 cookies. So the ratio of cups to cookies is 2.5 / 15. We want to find how many cups (call it x) are needed for 75 cookies. Set up a proportion: 2.5 / 15 = x / 75 To solve for x, multiply both sides by 75: x = (2.5 / 15) * 75 Simplify step-by-step: First, 2.5 / 15 = 25 / 150 = 5 / 30 = 1 / 6. So 2.5 / 15 = 1/6. Now multiply by 75: x = (1/6) * 75 = 75 / 6 75 / 6 = 12.5 So Liam needs 12.5 cups of flour for 75 cookies.

3. A construction company is building a scale model of a new bridge. The actual bridge will be 480 meters long, and the model uses a scale of 1:60. If the model bridge requires 3.2 liters of special coating paint to cover its entire surface, how many liters of the same paint would be needed to coat the actual bridge? Answer: 11520 Solution: The scale is 1:60, meaning 1 unit on the model represents 60 units on the actual bridge. For surface area, the scale factor is squared because area is two-dimensional.
   Full step-by-step solution

   Step 1: The scale is 1:60, meaning 1 unit on the model represents 60 units on the actual bridge. Step 2: For surface area, the scale factor is squared because area is two-dimensional. So the area scale factor is 60 × 60 = 3600. Step 3: The model requires 3.2 liters of paint for its surface area. Step 4: The actual bridge will require 3.2 × 3600 = 11520 liters of paint. Step 5: 3.2 × 3600 = 3.2 × (3000 + 600) = 9600 + 1920 = 11520 The answer is 11520 liters.

4. Liam is mixing paint for an art project. He needs to create a specific shade of purple by mixing red and blue paint in a 3:5 ratio. If Liam uses 2.4 liters of blue paint, how many liters of red paint should he use to maintain the correct proportion? Answer: 1.44 Solution: We are told the ratio of red to blue paint is 3:5. That means for every 3 parts red, there are 5 parts blue. Write the ratio as a fraction.
   Full step-by-step solution

   Let's solve this step by step. We are told the ratio of red to blue paint is 3:5. That means for every 3 parts red, there are 5 parts blue. Step 1: Write the ratio as a fraction. Red / Blue = 3 / 5 Step 2: We know Liam uses 2.4 liters of blue paint. Let R = liters of red paint needed. So: R / 2.4 = 3 / 5 Step 3: Solve for R. Multiply both sides by 2.4: R = (3 / 5) * 2.4 Step 4: Calculate. First, 3 / 5 = 0.6 Then, 0.6 * 2.4 = 1.44 Step 5: Conclusion. Liam needs 1.44 liters of red paint to maintain the 3:5 ratio with 2.4 liters of blue paint. Final answer: 1.44

5. 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.

6. Mere 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. Mere 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.

8. Noah is planning a road trip and needs to calculate his fuel costs. His car can travel 285 miles on 15 gallons of gas. If gas costs $3.80 per gallon and Noah plans to drive 1,140 miles, how much will he spend on gas for the entire trip? Answer: $228 Solution: Find the car's miles per gallon (fuel efficiency). 285 miles ÷ 15 gallons = 19 miles per gallon Calculate how many gallons are needed for 1,140 miles.
   Full step-by-step solution

   Step 1: Find the car's miles per gallon (fuel efficiency). 285 miles ÷ 15 gallons = 19 miles per gallon Step 2: Calculate how many gallons are needed for 1,140 miles. 1,140 miles ÷ 19 miles per gallon = 60 gallons Step 3: Calculate the total cost of gas. 60 gallons × $3.80 per gallon = $228 The answer is $228.