Order Rationals

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Answer Key & Explanations

Order Rationals · Grade 6 · Worksheet 2

1. Order: -7/3, 3.5, -1, 9/4, -2.75 Answer: -2.75, -7/3, -1, 9/4, 3.5 Solution: Convert all numbers to decimals. -7/3 = -2.333... Compare negatives first: -2.75 < -2.333...
   Full step-by-step solution

   Step 1: Convert all numbers to decimals. -7/3 = -2.333... 3.5 = 3.5 -1 = -1 9/4 = 2.25 -2.75 = -2.75 Step 2: Order from least to greatest (left to right on number line). Compare negatives first: -2.75 < -2.333... < -1 Then positives: 2.25 < 3.5 Step 3: Write the ordered list: -2.75, -7/3, -1, 9/4, 3.5 The answer is -2.75, -7/3, -1, 9/4, 3.5.

2. 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.

3. (-3.5) + 2.75 - (-1.25) = ? Answer: 0.5 Solution: Write the original problem clearly. (-3.5) + 2.75 - (-1.25) Interpret the subtraction of a negative number.
   Full step-by-step solution

   Let's solve step-by-step: Step 1: Write the original problem clearly. (-3.5) + 2.75 - (-1.25) Step 2: Interpret the subtraction of a negative number. Subtracting a negative is the same as adding a positive: - (-1.25) = + 1.25 So the expression becomes: (-3.5) + 2.75 + 1.25 Step 3: Add the positive numbers first for simplicity. 2.75 + 1.25 = 4.00 Step 4: Now add the negative number to the result. (-3.5) + 4.00 = 0.5 Step 5: Final answer: 0.5

4. Order: -3.5, 7/3, -1.75, 0.6, -2.25 Answer: -3.5, -2.25, -1.75, 0.6, 7/3 Solution: Convert all numbers to decimals for easy comparison. -3.5 is already a decimal. 7/3 = 2.333...
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison. -3.5 is already a decimal. 7/3 = 2.333... (repeating) -1.75 is already a decimal. 0.6 is already a decimal. -2.25 is already a decimal. Step 2: List the numbers in decimal form: -3.5, 2.333..., -1.75, 0.6, -2.25 Step 3: Order from least to greatest (most negative to most positive). On a number line, the smallest number is the one farthest to the left. -3.5 is the smallest (most negative). Next is -2.25. Then -1.75. Then 0.6 (positive but less than 2.333...). Finally, 7/3 = 2.333... is the largest. Step 4: Write the ordered list: -3.5, -2.25, -1.75, 0.6, 7/3 The answer is -3.5, -2.25, -1.75, 0.6, 7/3.

5. Olivia is drawing a number line from -10 to 10. She needs to plot and order these rational numbers from least to greatest: -3.5, 15/4, -2, 0.25, -4.5, 5. List the numbers in order from least to greatest. Answer: -4.5, -3.5, -2, 0.25, 15/4, 5 Solution: Convert all numbers to decimal form for easy comparison. 15/4 = 3.75. The numbers become: -3.5, 3.75, -2, 0.25, -4.5, 5.
   Full step-by-step solution

   Step 1: Convert all numbers to decimal form for easy comparison. 15/4 = 3.75. The numbers become: -3.5, 3.75, -2, 0.25, -4.5, 5. Step 2: Separate negative and positive numbers. Negatives: -4.5, -3.5, -2. Positives: 0.25, 3.75, 5. Step 3: Order negatives from least (most negative) to greatest. On a number line, -4.5 is farthest left, then -3.5, then -2. So the order is: -4.5, -3.5, -2. Step 4: Order positives from least to greatest. 0.25 is smallest, then 3.75, then 5. So the order is: 0.25, 3.75, 5. Step 5: Combine the two ordered lists: -4.5, -3.5, -2, 0.25, 3.75, 5. Step 6: Convert 3.75 back to fraction form: 15/4. The final order is: -4.5, -3.5, -2, 0.25, 15/4, 5. The answer is -4.5, -3.5, -2, 0.25, 15/4, 5.

6. Tane is drawing a number line from -9 to 9. He needs to plot these numbers in order from least to greatest: -7/3, 3.75, -5, 1.5, -3 1/5, 0, 11/4. Help Tane arrange them in order from least to greatest. Answer: -5, -3 1/5, -7/3, 0, 1.5, 11/4, 3.75 Solution: Convert all numbers to decimal form. -7/3 = -2.333...
   Full step-by-step solution

   Step 1: Convert all numbers to decimal form. -7/3 = -2.333... (since 7 divided by 3 is 2.333...) 3.75 stays 3.75 -5 stays -5 1.5 stays 1.5 -3 1/5 = -3.2 (since 1/5 = 0.2) 0 stays 0 11/4 = 2.75 (since 11 divided by 4 is 2.75) Step 2: List all numbers in decimal form: -2.333..., 3.75, -5, 1.5, -3.2, 0, 2.75 Step 3: Order from least (most negative) to greatest (most positive). Most negative: -5 Next: -3.2 (which is -3 1/5) Next: -2.333... (which is -7/3) Next: 0 Next: 1.5 Next: 2.75 (which is 11/4) Greatest: 3.75 Step 4: Write the final order using the original forms: -5, -3 1/5, -7/3, 0, 1.5, 11/4, 3.75 The answer is -5, -3 1/5, -7/3, 0, 1.5, 11/4, 3.75.

7. (-3/4) + (1/2) = ? Answer: -1/4 Solution: Identify the problem. We are adding a negative fraction and a positive fraction: -3/4 + 1/2 Find a common denominator. The denominators are 4 and 2.
   Full step-by-step solution

   Let's solve: (-3/4) + (1/2) = ? Step 1: Identify the problem. We are adding a negative fraction and a positive fraction: -3/4 + 1/2 Step 2: Find a common denominator. The denominators are 4 and 2. The least common denominator is 4. Step 3: Rewrite the fractions with the common denominator. - The first fraction is already in terms of 4: -3/4 - The second fraction, 1/2, needs to be converted to have denominator 4. Multiply numerator and denominator by 2: (1 * 2) / (2 * 2) = 2/4 Step 4: Rewrite the problem with the common denominator. Now the problem is: (-3/4) + (2/4) Step 5: Combine the numerators. Since the denominators are the same, we add the numerators and keep the denominator. (-3) + (2) = -1 So the result is: -1/4 Step 6: Final Answer. -1/4 Therefore, (-3/4) + (1/2) = -1/4

8. Liam is tracking the stock prices of three tech companies over a week. TechCorp started at $125.50, dropped by $18.75, then rose by $12.25. DataSystems began at $98.80, rose by $15.40, then dropped by $8.60. CloudNet started at $142.20, dropped by $22.10, then rose by $9.85. What were the final stock prices for each company, and which one had the highest price at the end of the week? Answer: TechCorp: $119.00, DataSystems: $105.60, CloudNet: $129.95; CloudNet had the highest price Solution: Starting price: $125.50 After drop: $125.50 - $18.75 = $106.75 After rise: $106.75 + $12.25 = $119.00 Starting price: $98.80 After rise: $98.80 + $15.40 = $114.20 After drop: $114.20 - $8.60 = $105.60 Starting price: $142.20 After drop: $142.20 - $22.10 = $120.10 After rise: $120.10 + $9.85 =…
   Full step-by-step solution

   Step 1: Calculate TechCorp's final price Starting price: $125.50 After drop: $125.50 - $18.75 = $106.75 After rise: $106.75 + $12.25 = $119.00 Step 2: Calculate DataSystems' final price Starting price: $98.80 After rise: $98.80 + $15.40 = $114.20 After drop: $114.20 - $8.60 = $105.60 Step 3: Calculate CloudNet's final price Starting price: $142.20 After drop: $142.20 - $22.10 = $120.10 After rise: $120.10 + $9.85 = $129.95 Step 4: Compare the final prices TechCorp: $119.00 DataSystems: $105.60 CloudNet: $129.95 The highest price is CloudNet at $129.95.

9. Liam is tracking the stock price of his favorite tech company over four days. On Monday, the stock was at $152.75. On Tuesday, it dropped by $12.50. On Wednesday, it gained $8.25. On Thursday, it dropped again by $5.75. What was the final stock price on Thursday? Answer: $142.75 Solution: Start with Monday's price: $152.75 Subtract Tuesday's drop: $152.75 - $12.50 = $140.25 Add Wednesday's gain: $140.25 + $8.25 = $148.50 Subtract Thursday's drop: $148.50 - $5.75 = $142.75 The final stock price on Thursday was $142.75.
   Full step-by-step solution

   Step 1: Start with Monday's price: $152.75 Step 2: Subtract Tuesday's drop: $152.75 - $12.50 = $140.25 Step 3: Add Wednesday's gain: $140.25 + $8.25 = $148.50 Step 4: Subtract Thursday's drop: $148.50 - $5.75 = $142.75 Step 5: The final stock price on Thursday was $142.75.