Order Numbers

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Answer Key & Explanations

Order Numbers · Grade 8 · Worksheet 1

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. 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.

3. Sophia draws a number line from 0 to 6. She marks the points for sqrt(11), 16/5, and sqrt(26). Place these three numbers in order from least to greatest. Answer: 16/5, sqrt(11), sqrt(26) Solution: Approximate sqrt(11). The perfect squares near 11 are 9 (3^2) and 16 (4^2), so sqrt(11) is between 3 and 4. Since 11 is closer to 9 than to 16, sqrt(11) ≈ 3.3.
   Full step-by-step solution

   Step 1: Approximate sqrt(11). The perfect squares near 11 are 9 (3^2) and 16 (4^2), so sqrt(11) is between 3 and 4. Since 11 is closer to 9 than to 16, sqrt(11) ≈ 3.3. Step 2: Approximate 16/5. Divide 16 by 5: 16/5 = 3.2 exactly. Step 3: Approximate sqrt(26). The perfect squares near 26 are 25 (5^2) and 36 (6^2), so sqrt(26) is between 5 and 6. Since 26 is very close to 25, sqrt(26) ≈ 5.1. Step 4: Compare the decimal approximations: 3.2, 3.3, 5.1. So from least to greatest: 16/5 (3.2), sqrt(11) (≈3.3), sqrt(26) (≈5.1). The answer is 16/5, sqrt(11), sqrt(26).

4. Mason is helping his uncle build a rectangular wooden frame for a large mirror. The frame's length is √75 inches, and its width is 8 inches. His cousin Charlotte says the diagonal of the frame should be √128 inches, but Mason thinks it should be about 13.2 inches. Who is correct? Show your reasoning by comparing the exact diagonal length to 13.2. Answer: Mason is correct; the diagonal is √139 ≈ 11.8 inches, which is less than 13.2. Solution: Simplify the length. √75 = √(25 × 3) = 5√3. So length = 5√3 inches, width = 8 inches.
   Full step-by-step solution

   Step 1: Simplify the length. √75 = √(25 × 3) = 5√3. So length = 5√3 inches, width = 8 inches. Step 2: Use the Pythagorean theorem: diagonal² = (5√3)² + 8² = 25 × 3 + 64 = 75 + 64 = 139. Step 3: Diagonal = √139. Now estimate √139: 11² = 121, 12² = 144, so √139 is between 11 and 12, closer to 12 since 139 is closer to 144. More precisely, 11.8² = 139.24, 11.7² = 136.89, so √139 ≈ 11.8. Step 4: Compare √139 ≈ 11.8 to 13.2. Since 11.8 < 13.2, Mason is correct that the diagonal is not 13.2 inches; it is about 11.8 inches.

5. Mason is building a ramp for his skateboard. He needs to compare the lengths of three different ramp designs he sketched. Ramp A has a length of √72 inches. Ramp B has a length of 8.5 inches. Ramp C has a length of 17/2 inches. Mason wants to order the ramps from shortest to longest to decide which one to build. What is the correct order? Answer: Ramp C (8.5 inches), Ramp B (8.5 inches), Ramp A (approximately 8.485 inches) or Ramp A, then Ramp B and Ramp C (tied) Solution: Simplify √72. √72 = √(36×2) = 6√2. Now estimate √2 ≈ 1.414, so 6 × 1.414 = 8.484.
   Full step-by-step solution

   Step 1: Simplify √72. √72 = √(36×2) = 6√2. Now estimate √2 ≈ 1.414, so 6 × 1.414 = 8.484. So Ramp A is about 8.484 inches. Step 2: Ramp B is given as 8.5 inches. Step 3: Ramp C is 17/2 = 8.5 inches. Step 4: Compare: 8.484 < 8.5, so Ramp A is shortest. Ramp B and Ramp C are equal at 8.5 inches. Step 5: Order from shortest to longest: Ramp A (√72 inches ≈ 8.484 inches), then Ramp B and Ramp C (both 8.5 inches, tied). The correct order is Ramp A, then Ramp B and Ramp C.

6. √(72) - 5√(2) = ? Answer: √2 Solution: Simplify √(72) by factoring: √(72) = √(36 × 2) = √(36) × √(2) = 6√(2) Substitute back into the expression: 6√(2) - 5√(2) Combine like terms: (6 - 5)√(2) = 1√(2) Simplify: 1√(2) = √(2) The answer is √2.
   Full step-by-step solution

   Step 1: Simplify √(72) by factoring: √(72) = √(36 × 2) = √(36) × √(2) = 6√(2) Step 2: Substitute back into the expression: 6√(2) - 5√(2) Step 3: Combine like terms: (6 - 5)√(2) = 1√(2) Step 4: Simplify: 1√(2) = √(2) The answer is √2.

7. Order: √(41), 6.4, 19/3, π² Answer: 6.4 < 19/3 < √(41) < π² Solution: Approximate each number as a decimal. - √(41): Since 6² = 36 and 7² = 49, √(41) is between 6 and 7. 6.4² = 40.96, so √(41) ≈ 6.403.
   Full step-by-step solution

   Step 1: Approximate each number as a decimal. - √(41): Since 6² = 36 and 7² = 49, √(41) is between 6 and 7. 6.4² = 40.96, so √(41) ≈ 6.403. - 6.4 is already a decimal: 6.4. - 19/3: Divide 19 by 3 to get 6.333... (repeating). - π²: π ≈ 3.1416, so π² ≈ 3.1416 × 3.1416 ≈ 9.8696. Step 2: List the decimal approximations: √(41) ≈ 6.403 6.4 = 6.4 19/3 ≈ 6.333 π² ≈ 9.870 Step 3: Order from smallest to largest: 6.333 (19/3) < 6.4 < 6.403 (√(41)) < 9.870 (π²) So the order is: 19/3, 6.4, √(41), π². The answer is 6.4 < 19/3 < √(41) < π².

8. Aroha draws a number line from 0 to 5 and marks the following five numbers on it: 7/3, sqrt(5), 2.6, sqrt(11), and 9/4. Which of these numbers is located farthest to the right on the number line? Answer: sqrt(11) Solution: Convert each number to a decimal or compare values. 7/3 = 2.333... sqrt(5) is between 2 and 3.
   Full step-by-step solution

   Step 1: Convert each number to a decimal or compare values. 7/3 = 2.333... sqrt(5) is between 2 and 3. Since 2^2=4 and 3^2=9, sqrt(5) ≈ 2.236 2.6 is exactly 2.6 sqrt(11) is between 3 and 4. Since 3^2=9 and 4^2=16, sqrt(11) ≈ 3.317 9/4 = 2.25 Step 2: Compare the decimal values: 2.236 (sqrt(5)), 2.25 (9/4), 2.333... (7/3), 2.6, 3.317 (sqrt(11)) Step 3: The largest value is sqrt(11) ≈ 3.317. The number farthest to the right on the number line is sqrt(11).