Proportion Applications

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Proportion Applications · Grade 7 · Worksheet 3

1. Isabella is a beekeeper who manages several beehives. From her 17 beehives, she harvested a total of 357 kilograms of honey last season. She wants to expand her apiary and estimates that if she adds 7 more beehives with the same average yield, how many total kilograms of honey can she expect to harvest from all 24 beehives next season? Answer: 504 Solution: Find the average honey yield per beehive. 357 kilograms ÷ 17 beehives = 21 kilograms per beehive. Find the total number of beehives after expansion.
   Full step-by-step solution

   Step 1: Find the average honey yield per beehive. 357 kilograms ÷ 17 beehives = 21 kilograms per beehive. Step 2: Find the total number of beehives after expansion. 17 + 7 = 24 beehives. Step 3: Multiply the yield per beehive by the total number of beehives. 21 kilograms × 24 beehives = 504 kilograms. The answer is 504 kilograms.

2. Liam is designing a scale model of a new building. The actual building will be 180 meters tall, and his model uses a scale where 3 centimeters represents 15 meters. How many centimeters tall should Liam make his model? Answer: 36 Solution: The scale says 3 centimeters in the model represents 15 meters in real life. So, 3 cm : 15 m. Find how many centimeters represent 1 meter in real life.
   Full step-by-step solution

   Step 1: Understand the scale. The scale says 3 centimeters in the model represents 15 meters in real life. So, 3 cm : 15 m. Step 2: Find how many centimeters represent 1 meter in real life. Divide both sides of the scale by 15: 3 cm / 15 m = 1/5 cm per meter. So, 1 meter in real life = 1/5 cm in the model. Step 3: The actual building height is 180 meters. Multiply the actual height by the scale factor (1/5 cm per meter): 180 m × (1/5 cm/m) = 180/5 cm = 36 cm. Step 4: Conclusion. Liam should make his model 36 centimeters tall.

3. Emma is planning a community garden and needs to create a soil mixture using compost, topsoil, and sand in a ratio of 2:5:3. If she wants to make 2500 kilograms of soil mixture for the garden beds, how many kilograms of topsoil should she use? Answer: 1250 Solution: Step 1: Add the ratio parts together: 2 + 5 + 3 = 10 total parts Step 2: Determine what fraction of the mixture is topsoil: 5 parts out of 10 total parts = 5/10 = 1/2 Step 3: Calculate the amount of topsoil needed: 1/2 × 2500 kg = 1250 kg Step 4: Verify the answer makes sense: 1250 kg is exactly…
   Full step-by-step solution

   Step 1: Add the ratio parts together: 2 + 5 + 3 = 10 total parts Step 2: Determine what fraction of the mixture is topsoil: 5 parts out of 10 total parts = 5/10 = 1/2 Step 3: Calculate the amount of topsoil needed: 1/2 × 2500 kg = 1250 kg Step 4: Verify the answer makes sense: 1250 kg is exactly half of 2500 kg The answer is 1250 kilograms of topsoil.

4. Noah is building a bookshelf for his room. He has a board that is 216 centimeters long. The design requires that the shelves be cut in a ratio of 3:4:5 for the bottom, middle, and top shelves respectively. How long, in centimeters, will the middle shelf be? Answer: 72 Solution: The ratio of the shelves is 3:4:5. Add the parts: 3 + 4 + 5 = 12 total parts. The total length is 216 cm.
   Full step-by-step solution

   Step 1: The ratio of the shelves is 3:4:5. Add the parts: 3 + 4 + 5 = 12 total parts. Step 2: The total length is 216 cm. Find the length of one part: 216 ÷ 12 = 18 cm per part. Step 3: The middle shelf has 4 parts: 4 × 18 = 72 cm. The middle shelf is 72 cm long.

5. Noah is a landscape designer who is creating a custom soil blend for a botanical garden. The blend requires sand, peat moss, and loam in a ratio of 6:11:1. If Noah needs to prepare 3,600 kilograms of this soil blend for the garden beds, how many kilograms of peat moss should he use? Answer: 2200 Solution: The ratio of sand:peat moss:loam is 6:11:1. The total number of parts is 6 + 11 + 1 = 18 parts. Peat moss represents 11 parts out of the total 18 parts, or the fraction 11/18.
   Full step-by-step solution

   Step 1: The ratio of sand:peat moss:loam is 6:11:1. The total number of parts is 6 + 11 + 1 = 18 parts. Step 2: Peat moss represents 11 parts out of the total 18 parts, or the fraction 11/18. Step 3: The total mixture is 3,600 kilograms. To find the kilograms of peat moss, multiply the total by the fraction: (11/18) * 3600. Step 4: Calculate: 3600 / 18 = 200, then 200 * 11 = 2200. Therefore, Noah needs 2,200 kilograms of peat moss.

6. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A circular fountain with a radius of 1 unit is placed at the center of the garden. What is the area of the garden that is not covered by the fountain? (Use π = 3.14) Answer: 20.86 Solution: Find the area of the rectangular garden. The vertices are (2,1), (8,1), (8,5), (2,5). Length along x-axis: from x=2 to x=8 → length = 8 - 2 = 6 units.
   Full step-by-step solution

   Step 1: Find the area of the rectangular garden. The vertices are (2,1), (8,1), (8,5), (2,5). Length along x-axis: from x=2 to x=8 → length = 8 - 2 = 6 units. Width along y-axis: from y=1 to y=5 → width = 5 - 1 = 4 units. Area of rectangle = length × width = 6 × 4 = 24 square units. Step 2: Find the center of the garden. Center of rectangle = midpoint of (2,1) and (8,5) or average of x's and y's. x-coordinate of center = (2 + 8)/2 = 10/2 = 5. y-coordinate of center = (1 + 5)/2 = 6/2 = 3. Center is at (5, 3). Step 3: Find the area of the circular fountain. Radius r = 1 unit. Area of circle = π × r² = 3.14 × (1)² = 3.14 square units. Step 4: Find the area not covered by the fountain. Area not covered = Area of rectangle − Area of circle = 24 − 3.14 = 20.86 square units. Final answer: 20.86

7. A construction company needs to mix concrete using a ratio of 3 parts cement to 5 parts sand to 2 parts gravel. If they need to produce 15,000 kilograms of concrete for a foundation, how many kilograms of cement should they use? Answer: 4500 Solution: The ratio of cement : sand : gravel is 3 : 5 : 2. That means for every 3 parts of cement, we have 5 parts of sand and 2 parts of gravel. Total parts = 3 + 5 + 2 = 10 parts.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Understand the ratio** The ratio of cement : sand : gravel is 3 : 5 : 2. That means for every 3 parts of cement, we have 5 parts of sand and 2 parts of gravel. --- **Step 2: Find the total number of parts** Total parts = 3 + 5 + 2 = 10 parts. --- **Step 3: Relate parts to total concrete weight** The total concrete needed is 15,000 kg. So, 10 parts = 15,000 kg. Therefore, 1 part = 15,000 / 10 = 1,500 kg. --- **Step 4: Find the weight of cement** Cement is 3 parts. Weight of cement = 3 × 1,500 kg = 4,500 kg. --- **Step 5: Conclusion** They need **4,500 kg** of cement. --- **Final answer:** 4500

8. A car travels 180 km on 15 liters of fuel. How many liters are needed to travel 300 km? Answer: 25 Solution: Set up the proportion: distance/fuel = 180/15 = 300/x Simplify the known ratio: 180 ÷ 15 = 12 km per liter Write the equation: 180/15 = 300/x Cross multiply: 180 × x = 15 × 300 Calculate: 180x = 4500 Solve for x: x = 4500 ÷ 180 = 25 The car needs 25 liters of fuel to travel 300 km.
   Full step-by-step solution

   Step 1: Set up the proportion: distance/fuel = 180/15 = 300/x Step 2: Simplify the known ratio: 180 ÷ 15 = 12 km per liter Step 3: Write the equation: 180/15 = 300/x Step 4: Cross multiply: 180 × x = 15 × 300 Step 5: Calculate: 180x = 4500 Step 6: Solve for x: x = 4500 ÷ 180 = 25 The car needs 25 liters of fuel to travel 300 km.