Order Rationals

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Answer Key & Explanations

Order Rationals · Grade 6 · Worksheet 1

1. A rectangular city park is drawn on a coordinate plane with corners at (10, 15), (60, 15), (60, 35), and (10, 35). A circular playground is placed exactly in the center of the park with a diameter equal to one-fifth of the park's length. What is the area of the playground? (Use π = 3.14) Answer: 78.5 Solution: Find the length of the rectangular park. The x-coordinates of the length are 10 and 60, so the length is 60 - 10 = 50 units.
   Full step-by-step solution

   Step 1: Find the length of the rectangular park. The x-coordinates of the length are 10 and 60, so the length is 60 - 10 = 50 units. Step 2: The diameter of the circular playground is one-fifth of the park's length, so diameter = 50 ÷ 5 = 10 units. Step 3: The radius is half the diameter, so radius = 10 ÷ 2 = 5 units. Step 4: The area of a circle is calculated using the formula: area = π × radius². Step 5: Substitute the values: area = 3.14 × 5² = 3.14 × 25 = 78.5. The area of the playground is 78.5 square units.

2. Liam is tracking the stock price of his favorite tech company. On Monday, the stock opened at $125.75 per share. During the day, it rose by $8.25, then fell by $12.50. On Tuesday, it gained $5.75 in the morning but dropped by $3.25 in the afternoon. What was the final stock price per share at the end of Tuesday? Answer: 124.00 Solution: Start with Monday's opening price: $125.75 Add Monday's gain: $125.75 + $8.25 = $134.00 Subtract Monday's loss: $134.00 - $12.50 = $121.50 Add Tuesday morning's gain: $121.50 + $5.75 = $127.25 Subtract Tuesday afternoon's loss: $127.25 - $3.25 = $124.00 The final stock price per share at the end…
   Full step-by-step solution

   Step 1: Start with Monday's opening price: $125.75 Step 2: Add Monday's gain: $125.75 + $8.25 = $134.00 Step 3: Subtract Monday's loss: $134.00 - $12.50 = $121.50 Step 4: Add Tuesday morning's gain: $121.50 + $5.75 = $127.25 Step 5: Subtract Tuesday afternoon's loss: $127.25 - $3.25 = $124.00 The final stock price per share at the end of Tuesday was $124.00.

3. Order: -3/7, 1.5, -2, 0.75, -1 1/3 Answer: -2, -1 1/3, -3/7, 0.75, 1.5 Solution: Convert all numbers to decimals for easy comparison. -3/7 ≈ -0.4286 1.5 = 1.5 -2 = -2 0.75 = 0.75 -1 1/3 = -1.3333 Order from least to greatest (most negative to most positive). -2 is the smallest.
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison. -3/7 ≈ -0.4286 1.5 = 1.5 -2 = -2 0.75 = 0.75 -1 1/3 = -1.3333 Step 2: Order from least to greatest (most negative to most positive). -2 is the smallest. Next is -1 1/3 (-1.3333). Then -3/7 (-0.4286). Then 0.75. Finally 1.5. Step 3: Write the ordered list: -2, -1 1/3, -3/7, 0.75, 1.5 The answer is -2, -1 1/3, -3/7, 0.75, 1.5.

4. Emma is tracking her bank account balance. On Monday, she had $1,250. On Tuesday, she deposited her paycheck of $850. On Wednesday, she paid her rent of $1,100 and then bought groceries for $95. On Thursday, she received a refund of $45. What was her final account balance on Thursday? Answer: $950 Solution: Start with Monday's balance: $1,250 Add Tuesday's paycheck: $1,250 + $850 = $2,100 Subtract Wednesday's rent: $2,100 - $1,100 = $1,000 Subtract Wednesday's groceries: $1,000 - $95 = $905 Add Thursday's refund: $905 + $45 = $950 Emma's final account balance on Thursday was $950.
   Full step-by-step solution

   Step 1: Start with Monday's balance: $1,250 Step 2: Add Tuesday's paycheck: $1,250 + $850 = $2,100 Step 3: Subtract Wednesday's rent: $2,100 - $1,100 = $1,000 Step 4: Subtract Wednesday's groceries: $1,000 - $95 = $905 Step 5: Add Thursday's refund: $905 + $45 = $950 Emma's final account balance on Thursday was $950.

5. Order: -3.6, 1/6, -1, 2.1, -5/6 Answer: -3.6, -1, -5/6, 1/6, 2.1 Solution: Convert all numbers to decimals for easy comparison. -3.6 is already a decimal. 1/6 = 0.1666...
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison. -3.6 is already a decimal. 1/6 = 0.1666... (approximately 0.167) -1 is already an integer. 2.1 is already a decimal. -5/6 = -0.8333... (approximately -0.833) Step 2: List the numbers with their decimal equivalents: -3.6, 0.167, -1, 2.1, -0.833 Step 3: Order from least to greatest (most negative to most positive): -3.6 is the smallest (most negative). Next is -1 (since -1 < -0.833). Next is -0.833 (-5/6). Next is 0.167 (1/6). Largest is 2.1. Step 4: Write the final order using the original forms: -3.6, -1, -5/6, 1/6, 2.1 The answer is -3.6, -1, -5/6, 1/6, 2.1.

6. Order: -3/5, 1.75, -2, 0.5, -1.25 Answer: -2, -1.25, -3/5, 0.5, 1.75 Solution: Convert all numbers to decimals for easy comparison: -3/5 = -0.6, 1.75 stays 1.75, -2 stays -2, 0.5 stays 0.5, -1.25 stays -1.25. Write the final order: -2, -1.25, -3/5, 0.5, 1.75.
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison: -3/5 = -0.6, 1.75 stays 1.75, -2 stays -2, 0.5 stays 0.5, -1.25 stays -1.25. Step 2: Order from least to greatest (left to right on number line): Start with the most negative: -2, then -1.25, then -0.6 (which is -3/5), then 0.5, then 1.75. Step 3: Write the final order: -2, -1.25, -3/5, 0.5, 1.75. The answer is -2, -1.25, -3/5, 0.5, 1.75.

7. Liam is tracking the temperature changes in his city over a week. On Monday morning, the temperature was -3°C. By noon, it had risen by 8°C. Overnight, it dropped by 5°C. On Tuesday, it increased by 4°C, and then dropped by 7°C by evening. What was the final temperature on Tuesday evening? Answer: -3 Solution: Monday morning: -3°C It rose by 8°C, so: -3 + 8 = 5°C Noon temperature: 5°C It dropped by 5°C, so: 5 - 5 = 0°C Monday overnight temperature: 0°C It increased by 4°C, so: 0 + 4 = 4°C Tuesday daytime temperature: 4°C It dropped by 7°C, so: 4 - 7 = -3°C Final temperature on Tuesday evening: -3°C
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Start with Monday morning temperature** Monday morning: -3°C **Step 2: Temperature change by Monday noon** It rose by 8°C, so: -3 + 8 = 5°C Noon temperature: 5°C **Step 3: Overnight drop (Monday night)** It dropped by 5°C, so: 5 - 5 = 0°C Monday overnight temperature: 0°C **Step 4: Tuesday increase** It increased by 4°C, so: 0 + 4 = 4°C Tuesday daytime temperature: 4°C **Step 5: Tuesday evening drop** It dropped by 7°C, so: 4 - 7 = -3°C --- **Final temperature on Tuesday evening:** -3°C

8. Emma is drawing a number line to order the following rational numbers: -3.5, 5/2, -1, 0.75, -2.25, 3. Plot each number on the number line, then list them in order from least to greatest. Answer: -3.5, -2.25, -1, 0.75, 5/2, 3 Solution: Convert all numbers to decimal form for easy comparison: -3.5 stays -3.5, 5/2 = 2.5, -1 stays -1, 0.75 stays 0.75, -2.25 stays -2.25, 3 stays 3. Identify the negatives: -3.5, -2.25, -1.
   Full step-by-step solution

   Step 1: Convert all numbers to decimal form for easy comparison: -3.5 stays -3.5, 5/2 = 2.5, -1 stays -1, 0.75 stays 0.75, -2.25 stays -2.25, 3 stays 3. Step 2: Identify the negatives: -3.5, -2.25, -1. On a number line, the more negative (smaller) numbers are to the left. Order negatives from least to greatest: -3.5, -2.25, -1. Step 3: Identify the positives: 0.75, 2.5, 3. Order from least to greatest: 0.75, 2.5, 3. Step 4: Combine the ordered lists: -3.5, -2.25, -1, 0.75, 2.5, 3. Step 5: Convert 2.5 back to fraction form: 5/2. Final answer: -3.5, -2.25, -1, 0.75, 5/2, 3.