Proportional Equations

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Proportional Equations · Grade 7 · Worksheet 2

1. 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

2. Kai is filling a swimming pool with a hose. The pool gains 12 gallons of water every minute. If the pool already has 0 gallons at the start, write an equation showing the relationship between the total gallons of water, y, and the number of minutes, x. After 19 minutes, how many gallons of water are in the pool? Answer: 228 Solution: The amount of water is proportional to time because it starts at 0 and gains a constant amount each minute. The equation is y = kx, where k is the constant rate. Here, k = 12 gallons per minute.
   Full step-by-step solution

   Step 1: The amount of water is proportional to time because it starts at 0 and gains a constant amount each minute. Step 2: The equation is y = kx, where k is the constant rate. Here, k = 12 gallons per minute. Step 3: The equation is y = 12x. Step 4: After 19 minutes, substitute x = 19: y = 12 × 19 = 228. Step 5: There are 228 gallons in the pool after 19 minutes.

3. 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.

4. Emma is creating a pattern on a coordinate grid. She plots points that form a straight line through the origin (0, 0). The line passes through the point (15, 45). If the relationship between x and y is proportional, what is the equation of the line in the form y = kx? Answer: y = 3x Solution: The relationship is proportional, so the equation is y = kx. The point (15, 45) lies on the line, so we substitute x = 15 and y = 45 into the equation: 45 = k * 15.
   Full step-by-step solution

   Step 1: The relationship is proportional, so the equation is y = kx. Step 2: The point (15, 45) lies on the line, so we substitute x = 15 and y = 45 into the equation: 45 = k * 15. Step 3: Solve for k by dividing both sides by 15: k = 45 / 15 = 3. Step 4: Write the equation: y = 3x. The answer is y = 3x.

5. 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.

6. Isabella 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.

7. A construction company needs to mix concrete using cement, sand, and gravel in the ratio 2:3:5. If they want to make 1500 kilograms of concrete, how many kilograms of sand should they use? Answer: 450 Solution: Add all the ratio parts together: 2 + 3 + 5 = 10 total parts The sand represents 3 parts out of the total 10 parts Calculate the fraction of the mixture that is sand: 3/10 Multiply the total concrete amount by the sand fraction: 1500 × 3/10 Simplify: 1500 × 3 = 4500, then 4500 ÷ 10 = 450…
   Full step-by-step solution

   Step 1: Add all the ratio parts together: 2 + 3 + 5 = 10 total parts Step 2: The sand represents 3 parts out of the total 10 parts Step 3: Calculate the fraction of the mixture that is sand: 3/10 Step 4: Multiply the total concrete amount by the sand fraction: 1500 × 3/10 Step 5: Simplify: 1500 × 3 = 4500, then 4500 ÷ 10 = 450 Step 6: Therefore, they need 450 kilograms of sand Final answer: 450

8. A construction company is building a scale model of a new bridge. The actual bridge will be 1,200 meters long, and the scale model uses a ratio of 1:150. If the model bridge is 8 meters long, how many centimeters longer does the model need to be to match the correct scale proportion? Answer: 0 Solution: Calculate what the model length should be using the scale ratio. The scale is 1:150, meaning 1 unit on the model represents 150 units in reality.
   Full step-by-step solution

   Step 1: Calculate what the model length should be using the scale ratio. The scale is 1:150, meaning 1 unit on the model represents 150 units in reality. Actual bridge length: 1,200 meters Model length should be: 1,200 ÷ 150 = 8 meters Step 2: Compare the calculated model length to the given model length. Calculated model length: 8 meters Given model length: 8 meters Step 3: Determine how much longer the model needs to be. 8 meters - 8 meters = 0 meters 0 meters = 0 centimeters The model is already the correct length, so it doesn't need to be any longer.