Order Numbers

lessonbunny.com

Answer Key & Explanations

Order Numbers · Grade 8 · Worksheet 2

1. Tane is helping his family build a raised garden bed. He needs to cut a piece of wood to a length of √45 inches for one side and a piece of wood to a length of 7.5 inches for the other side. His sister Aroha says the √45 piece is longer, but Tane thinks 7.5 inches is longer. Who is correct, and how much longer is the longer piece? Round your answer to the nearest tenth if needed. Answer: Aroha is correct; √45 is approximately 6.7 inches, so 7.5 inches is longer by 0.8 inches. Solution: Estimate √45. The perfect squares near 45 are 36 (6²) and 49 (7²), so √45 is between 6 and 7. Since 45 is closer to 49 than to 36, √45 is closer to 7.
   Full step-by-step solution

   Step 1: Estimate √45. The perfect squares near 45 are 36 (6²) and 49 (7²), so √45 is between 6 and 7. Since 45 is closer to 49 than to 36, √45 is closer to 7. Step 2: Calculate more precisely: √45 = √(9×5) = 3√5. √5 ≈ 2.236, so 3 × 2.236 = 6.708 inches. Step 3: Compare 6.708 and 7.5. Since 7.5 > 6.708, the 7.5-inch piece is longer. Step 4: Find the difference: 7.5 - 6.708 = 0.792, which rounds to 0.8 inches. Final answer: Tane is incorrect; 7.5 inches is longer by 0.8 inches.

2. Charlotte is helping her family build a square patio. She has two options for the side length: Option A is 7 feet, and Option B is sqrt(47) feet. Charlotte wants the patio with the larger area. Which option has the larger area, and approximately how much larger is it in square feet? Round to the nearest tenth. Answer: Option A is larger by approximately 2.0 square feet. Solution: Find the area of Option A (side = 7 feet). Area = 7^2 = 49 square feet. Find the area of Option B (side = sqrt(47) feet).
   Full step-by-step solution

   Step 1: Find the area of Option A (side = 7 feet). Area = 7^2 = 49 square feet. Step 2: Find the area of Option B (side = sqrt(47) feet). Area = (sqrt(47))^2 = 47 square feet. Step 3: Compare the areas. 49 > 47, so Option A has the larger area. Step 4: Find the difference. 49 - 47 = 2 square feet. The answer is 2.0 square feet. Option A is larger by approximately 2.0 square feet.

3. √(49) + 3² - 2³ = ? Answer: 8 Solution: √(49) means the positive square root of 49. Since 7 × 7 = 49, we have √(49) = 7. Evaluate the exponent 3² 3² means 3 × 3 = 9.
   Full step-by-step solution

   Let's solve step by step. Step 1: Evaluate the square root √(49) means the positive square root of 49. Since 7 × 7 = 49, we have √(49) = 7. Step 2: Evaluate the exponent 3² 3² means 3 × 3 = 9. Step 3: Evaluate the exponent 2³ 2³ means 2 × 2 × 2 = 8. Step 4: Substitute back into the expression The expression is: √(49) + 3² − 2³ = 7 + 9 − 8 Step 5: Perform addition and subtraction from left to right First, 7 + 9 = 16 Then, 16 − 8 = 8 Final Answer: 8

4. Emma is designing a rectangular banner for a school project. The banner's length is 4√12 inches and its width is 3√27 inches. Her friend Noah says that if she simplifies both measurements to simplest radical form, the length and width will have the same radical part. Is Noah correct? If so, what is the common radical part? Answer: 4√3 Solution: Simplify the length 4√12 √12 = √(4 × 3) = √4 × √3 = 2√3 So 4√12 = 4 × 2√3 = 8√3 Simplify the width 3√27 √27 = √(9 × 3) = √9 × √3 = 3√3 So 3√27 = 3 × 3√3 = 9√3 Length: 8√3 Width: 9√3 Both measurements have the same radical part √3, so Noah is correct.
   Full step-by-step solution

   Step 1: Simplify the length 4√12 √12 = √(4 × 3) = √4 × √3 = 2√3 So 4√12 = 4 × 2√3 = 8√3 Step 2: Simplify the width 3√27 √27 = √(9 × 3) = √9 × √3 = 3√3 So 3√27 = 3 × 3√3 = 9√3 Step 3: Compare the simplified forms Length: 8√3 Width: 9√3 Both measurements have the same radical part √3, so Noah is correct. The common radical part is √3.

5. Noah is helping his family plan a square patio in their backyard. One design has a side length of √31 feet. Another design has a side length of 5.6 feet. A third design has a side length of 26/5 feet. Noah needs to place these three side lengths in order from smallest to largest on a number line to compare them. Which design has the smallest side length, and which has the largest? Answer: Smallest: 26/5 (5.2 feet); Largest: √31 (approximately 5.57 feet) Solution: Convert each value to a decimal for easy comparison. - √31: Since 5^2 = 25 and 6^2 = 36, √31 is between 5 and 6. 5.5^2 = 30.25, 5.6^2 = 31.36, so √31 is about 5.57.
   Full step-by-step solution

   Step 1: Convert each value to a decimal for easy comparison. - √31: Since 5^2 = 25 and 6^2 = 36, √31 is between 5 and 6. 5.5^2 = 30.25, 5.6^2 = 31.36, so √31 is about 5.57. - 5.6 is already a decimal: 5.6 - 26/5: Divide 26 by 5 = 5.2 Step 2: List the decimals: 5.57, 5.6, 5.2 Step 3: Order from smallest to largest: 5.2, 5.57, 5.6 Step 4: Match back to original numbers: Smallest: 26/5 (5.2 feet) Middle: √31 (≈5.57 feet) Largest: 5.6 feet The smallest side length is 26/5 feet, and the largest is 5.6 feet.

6. Aisha is comparing two different phone plans. Plan A costs $0.15 per minute with a $5 monthly fee. Plan B costs $0.10 per minute with a $10 monthly fee. Aisha estimates she'll use about 120 minutes per month. Which plan would be more cost-effective for her, and by how much? Answer: Plan B by $2 Solution: Monthly fee: $5 Cost per minute: $0.15 Minutes used: 120 Variable cost: 120 × $0.15 = $18 Total cost: $5 + $18 = $23 Monthly fee: $10 Cost per minute: $0.10 Minutes used: 120 Variable cost: 120 × $0.10 = $12 Total cost: $10 + $12 = $21 Plan A: $23 Plan B: $21 Difference: $23 - $21 = $2 Plan B is…
   Full step-by-step solution

   Step 1: Calculate total cost for Plan A Monthly fee: $5 Cost per minute: $0.15 Minutes used: 120 Variable cost: 120 × $0.15 = $18 Total cost: $5 + $18 = $23 Step 2: Calculate total cost for Plan B Monthly fee: $10 Cost per minute: $0.10 Minutes used: 120 Variable cost: 120 × $0.10 = $12 Total cost: $10 + $12 = $21 Step 3: Compare the costs Plan A: $23 Plan B: $21 Difference: $23 - $21 = $2 Plan B is more cost-effective by $2.

7. √(1.44) + (2/5)² = ? Answer: 1.36 Solution: Evaluate the square root: √(1.44) = 1.2 Evaluate the squared fraction: (2/5)² = 4/25 = 0.16 Add the results: 1.2 + 0.16 = 1.36 The answer is 1.36.
   Full step-by-step solution

   Step 1: Evaluate the square root: √(1.44) = 1.2 Step 2: Evaluate the squared fraction: (2/5)² = 4/25 = 0.16 Step 3: Add the results: 1.2 + 0.16 = 1.36 The answer is 1.36.