Proportional Equations

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Answer Key & Explanations

Proportional Equations · Grade 7 · Worksheet 3

1. Charlotte is paid a constant hourly wage. 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. Hana is paid a constant hourly wage. She earns $195.50 for working 17 hours. Write the equation y = kx that represents her total earnings y for x hours worked. Answer: y = 11.5x Solution: Identify the given values. Total earnings y = $195.50, hours worked x = 17. The relationship is proportional, so y = kx.
   Full step-by-step solution

   Step 1: Identify the given values. Total earnings y = $195.50, hours worked x = 17. Step 2: The relationship is proportional, so y = kx. Substitute the known values: 195.50 = k * 17. Step 3: Solve for k by dividing both sides by 17: k = 195.50 / 17. Step 4: Calculate 195.50 ÷ 17 = 11.5. Step 5: Write the equation: y = 11.5x. The answer is y = 11.5x.

3. Charlotte earns money at a constant rate. She earns $195 for working 15 hours. Write the equation y = kx that represents her total earnings y for x hours worked. Answer: y = 13x Solution: Identify the given values. Total earnings y = $195, 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 = $195, hours worked x = 15. Step 2: The relationship is proportional, so y = kx. Substitute the known values: 195 = k * 15. Step 3: Solve for k by dividing both sides by 15: k = 195 / 15 = 13. Step 4: Write the equation: y = 13x. The answer is y = 13x.

4. Mason 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.

5. Kaia earns money at a constant rate. She earns $216 for working 18 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 = $216, hours worked x = 18. The relationship is proportional, so y = kx.
   Full step-by-step solution

   Step 1: Identify the given values. Total earnings y = $216, hours worked x = 18. Step 2: The relationship is proportional, so y = kx. Substitute the known values: 216 = k * 18. Step 3: Solve for k by dividing both sides by 18: k = 216 / 18 = 12. Step 4: Write the equation: y = 12x. The answer is y = 12x.

6. A construction company needs to mix concrete using cement, sand, and gravel in the ratio 2:3:5. If they want to make 15,000 kilograms of concrete, how many kilograms of sand should they use? Answer: 4500 Solution: The ratio of cement:sand:gravel is 2:3:5 Add the ratio parts: 2 + 3 + 5 = 10 total parts Sand represents 3 parts out of 10 total parts, so the fraction is 3/10 Multiply the total concrete amount by the sand fraction: 15,000 × 3/10 Calculate: 15,000 ÷ 10 = 1,500, then 1,500 × 3 = 4,500 The…
   Full step-by-step solution

   Step 1: The ratio of cement:sand:gravel is 2:3:5 Step 2: Add the ratio parts: 2 + 3 + 5 = 10 total parts Step 3: Sand represents 3 parts out of 10 total parts, so the fraction is 3/10 Step 4: Multiply the total concrete amount by the sand fraction: 15,000 × 3/10 Step 5: Calculate: 15,000 ÷ 10 = 1,500, then 1,500 × 3 = 4,500 Step 6: The company needs 4,500 kilograms of sand Final answer: 4500

7. A rectangular garden is drawn on a coordinate plane with vertices at (0,0), (12,0), (12,8), and (0,8). A diagonal path is drawn from (0,0) to (12,8), dividing the garden into two triangular sections. If the gardener plants flowers in the larger triangular section, what is the area of the flower garden in square units? Answer: 48 Solution: (0,0), (12,0), (12,8), (0,8) The diagonal from (0,0) to (12,8) splits the rectangle into two triangles. We need the area of the larger triangular section.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangle with vertices: (0,0), (12,0), (12,8), (0,8) The diagonal from (0,0) to (12,8) splits the rectangle into two triangles. We need the area of the larger triangular section. --- **Step 2: Visualize the triangles** The diagonal from (0,0) to (12,8) divides the rectangle into: - Triangle A: vertices (0,0), (12,0), (12,8) - Triangle B: vertices (0,0), (0,8), (12,8) --- **Step 3: Check if the triangles are equal** Actually, let's check: Triangle A: vertices (0,0), (12,0), (12,8) Base from (0,0) to (12,0) is length 12, height is 8 (up to (12,8)). Area = (1/2) × base × height = (1/2) × 12 × 8 = 48. Triangle B: vertices (0,0), (0,8), (12,8) Base from (0,0) to (0,8) is length 8, height is 12 (over to (12,8)). Area = (1/2) × base × height = (1/2) × 8 × 12 = 48. So both triangles have the same area. --- **Step 4: Which is larger?** They are equal in area. So the "larger triangular section" is just either one, area 48. --- **Step 5: Conclusion** The area of the flower garden (the larger triangular section) is 48 square units. --- **Final answer:** 48

8. If y = 12.5x and y = 187.5, then x = ? Answer: 15 Solution: The equation is y = 12.5x and we know y = 187.5. Substitute the known value into the equation: 187.5 = 12.5x. To solve for x, divide both sides of the equation by 12.5: x = 187.5 ÷ 12.5.
   Full step-by-step solution

   Step 1: The equation is y = 12.5x and we know y = 187.5. Step 2: Substitute the known value into the equation: 187.5 = 12.5x. Step 3: To solve for x, divide both sides of the equation by 12.5: x = 187.5 ÷ 12.5. Step 4: Perform the division: 187.5 ÷ 12.5 = 15. Step 5: Check the answer: 12.5 × 15 = 187.5, which matches the given y value. The answer is 15.

9. A construction company needs to mix concrete using cement, sand, and gravel in the ratio 2:3:5. For a large building project, they need to prepare 15,000 kilograms of concrete mixture. How many kilograms of sand should they use in this batch? Answer: 4500 Solution: Step 1: Identify the ratio parts: cement = 2 parts, sand = 3 parts, gravel = 5 parts Step 2: Calculate total parts: 2 + 3 + 5 = 10 parts Step 3: Determine what fraction of the total mixture is sand: 3 parts sand out of 10 total parts = 3/10 Step 4: Calculate kilograms of sand needed: 3/10 ×…
   Full step-by-step solution

   Step 1: Identify the ratio parts: cement = 2 parts, sand = 3 parts, gravel = 5 parts Step 2: Calculate total parts: 2 + 3 + 5 = 10 parts Step 3: Determine what fraction of the total mixture is sand: 3 parts sand out of 10 total parts = 3/10 Step 4: Calculate kilograms of sand needed: 3/10 × 15,000 = 4,500 Step 5: Verify: 4,500 kg of sand represents 3 parts, so each part = 4,500 ÷ 3 = 1,500 kg Cement: 2 × 1,500 = 3,000 kg Gravel: 5 × 1,500 = 7,500 kg Total: 3,000 + 4,500 + 7,500 = 15,000 kg ✓ The answer is 4500 kilograms of sand.