Order Rationals

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Answer Key & Explanations

Order Rationals · Grade 6 · Worksheet 3

1. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A circular fountain is placed at the center of the garden with a radius that is exactly one-third the width of the rectangle. What is the area of the fountain? (Use π = 3.14) Answer: 12.56 Solution: Find the dimensions of the rectangle. The vertices are (2, 1), (8, 1), (8, 5), and (2, 5). The length along the x-axis from (2, 1) to (8, 1) is 8 - 2 = 6 units.
   Full step-by-step solution

   Step 1: Find the dimensions of the rectangle. The vertices are (2, 1), (8, 1), (8, 5), and (2, 5). The length along the x-axis from (2, 1) to (8, 1) is 8 - 2 = 6 units. The length along the y-axis from (2, 1) to (2, 5) is 5 - 1 = 4 units. So the rectangle has width = 6 and height = 4. Step 2: Identify which side is the width for the problem. The problem says the fountain's radius is one-third the width of the rectangle. Since width is 6, one-third of the width is 6 * 1/3 = 2. So the radius of the fountain is 2. Step 3: Calculate the area of the fountain. The fountain is a circle with radius r = 2. Area of a circle = π * r^2. Using π = 3.14: Area = 3.14 * (2^2) = 3.14 * 4 = 12.56. Step 4: Final answer. The area of the fountain is 12.56 square units.

2. Matiu is helping his class create a giant number line for the school hallway. He needs to plot the following elevations (in meters) from a geography field trip: -4.5, 2 3/4, -1.2, 3/2, and -3.25. Help Matiu order these elevations from least to greatest so he can place them correctly on the number line. Answer: -4.5, -3.25, -1.2, 3/2, 2 3/4 Solution: Convert all numbers to decimals for easy comparison. -4.5 is already a decimal. 2 3/4 = 2 + 3/4 = 2 + 0.75 = 2.75.
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison. -4.5 is already a decimal. 2 3/4 = 2 + 3/4 = 2 + 0.75 = 2.75. -1.2 is already a decimal. 3/2 = 1.5. -3.25 is already a decimal. Step 2: List all decimals: -4.5, 2.75, -1.2, 1.5, -3.25 Step 3: Order from least to greatest (most negative to most positive): -4.5 is the smallest (most negative). Next, compare -3.25 and -1.2. Since -3.25 is more negative than -1.2, it comes next. Then -1.2. Now for positives: 1.5, then 2.75. Step 4: Write the ordered list using the original forms: -4.5, -3.25, -1.2, 3/2, 2 3/4. The answer is -4.5, -3.25, -1.2, 3/2, 2 3/4.

3. Order: -7.25, 8/3, -5, 2.75, -11/4 Answer: -7.25, -5, -11/4, 8/3, 2.75 Solution: Convert all numbers to decimals for easy comparison. -7.25 is already a decimal. 8/3 = 2.666...
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison. -7.25 is already a decimal. 8/3 = 2.666... (since 8 ÷ 3 = 2.666...) -5 is already an integer. 2.75 is already a decimal. -11/4 = -2.75 (since 11 ÷ 4 = 2.75, and it is negative). Step 2: List the numbers as decimals: -7.25, 2.666..., -5, 2.75, -2.75. Step 3: Separate negative and positive numbers. Negatives: -7.25, -5, -2.75 Positives: 2.666..., 2.75 Step 4: Order the negatives from least to greatest (most negative first). -7.25 < -5 < -2.75 Step 5: Order the positives from least to greatest. 2.666... < 2.75 Step 6: Combine the ordered lists. -7.25, -5, -2.75, 2.666..., 2.75 Step 7: Write the final answer using the original forms. -7.25, -5, -11/4, 8/3, 2.75

4. Tane is helping his family prepare a weather report for their school project. They recorded the following outdoor temperatures (in degrees Celsius) at different times of the day: -3.25, 1 3/5, -1/2, 2.1, -2 3/4, and 0.9. Tane needs to order these temperatures from coldest to warmest to plot them correctly on a number line. What is the correct order from least to greatest? Answer: -3.25, -2 3/4, -1/2, 0.9, 1 3/5, 2.1 Solution: Convert all numbers to decimals for easy comparison. -3.25 is already a decimal. 1 3/5 = 1 + 3/5 = 1 + 0.6 = 1.6 -1/2 = -0.5 2.1 is already a decimal.
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison. -3.25 is already a decimal. 1 3/5 = 1 + 3/5 = 1 + 0.6 = 1.6 -1/2 = -0.5 2.1 is already a decimal. -2 3/4 = -(2 + 3/4) = -(2 + 0.75) = -2.75 0.9 is already a decimal. Step 2: List all decimals: -3.25, 1.6, -0.5, 2.1, -2.75, 0.9 Step 3: Order from least to greatest (coldest to warmest): -3.25 is the smallest (most negative, coldest). Next, compare -2.75 and -0.5. Since -2.75 is more negative than -0.5, it comes next. Then -0.5. Now for positives: 0.9, 1.6, 2.1. Order: 0.9, 1.6, 2.1. Step 4: Write the ordered list using the original forms: -3.25, -2 3/4, -1/2, 0.9, 1 3/5, 2.1. The answer is -3.25, -2 3/4, -1/2, 0.9, 1 3/5, 2.1.

5. A scientist is tracking temperature changes in a deep ocean trench. The temperatures recorded at different depths are: -2.75°C, -3.125°C, -2.5°C, and -3.25°C. Order these temperatures from warmest to coldest. Answer: -2.5,-2.75,-3.125,-3.25 Solution: Understand that these are negative temperatures. For negative numbers, the "warmest" means the highest number (closest to zero), and "coldest" means the lowest number (farthest from zero). -2.75 = -2.750 -3.125 = -3.125 -2.5 = -2.500 -3.25 = -3.250 Compare -2.5 and -2.75.
   Full step-by-step solution

   Let's order the temperatures from warmest to coldest: -2.75, -3.125, -2.5, -3.25. Step 1: Understand that these are negative temperatures. For negative numbers, the "warmest" means the highest number (closest to zero), and "coldest" means the lowest number (farthest from zero). So we need to sort them in descending order as numbers. Step 2: Compare the numbers by thinking about their values on a number line. Let's write them as decimals to compare easily: -2.75 = -2.750 -3.125 = -3.125 -2.5 = -2.500 -3.25 = -3.250 Step 3: Compare -2.5 and -2.75. -2.5 is greater than -2.75 because -2.5 is closer to 0. So -2.5 > -2.75. Step 4: Compare -2.75 and -3.125. -2.75 is greater than -3.125. So -2.75 > -3.125. Step 5: Compare -3.125 and -3.25. -3.125 is greater than -3.25 because -3.125 is closer to 0. So -3.125 > -3.25. Step 6: Now put them in order from greatest to least (warmest to coldest): -2.5 (warmest) -2.75 -3.125 -3.25 (coldest) Final order: -2.5, -2.75, -3.125, -3.25

6. Charlotte is organizing a set of rational numbers for her math poster. She needs to order the following numbers from least to greatest and then plot them on a number line: -8.3, 15/4, -9 1/2, 3.75, -7.6, 13/2. Help Charlotte determine the correct order from least to greatest. Answer: -9 1/2, -8.3, -7.6, 15/4, 3.75, 13/2 Solution: Convert all numbers to decimals for easy comparison. -8.3 is already a decimal. 15/4 = 3.75 -9 1/2 = -(9 + 1/2) = -(9 + 0.5) = -9.5 3.75 is already a decimal.
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison. -8.3 is already a decimal. 15/4 = 3.75 -9 1/2 = -(9 + 1/2) = -(9 + 0.5) = -9.5 3.75 is already a decimal. -7.6 is already a decimal. 13/2 = 6.5 Step 2: List all decimals: -8.3, 3.75, -9.5, 3.75, -7.6, 6.5 Step 3: Order from least to greatest (most negative to most positive): -9.5 is the smallest (most negative). Next, compare -8.3 and -7.6. Since -8.3 is more negative than -7.6, it comes next. Then -7.6. Now for positives: 3.75, 3.75, 6.5. Order: 3.75, 3.75, 6.5. Step 4: Write the ordered list using the original forms: -9 1/2, -8.3, -7.6, 15/4, 3.75, 13/2. The answer is -9 1/2, -8.3, -7.6, 15/4, 3.75, 13/2.

7. Sophia is plotting points on a number line that runs from -8 to 8. She needs to correctly order and place the following numbers: -3.5, 7/2, -2.75, 0, 1 1/4, -6. On the number line, what is the fourth number from the left? Answer: 0 Solution: Convert all numbers to decimals for easy comparison. -3.5 is already a decimal. 7/2 = 3.5 -2.75 is already a decimal.
   Full step-by-step solution

   Step 1: Convert all numbers to decimals for easy comparison. -3.5 is already a decimal. 7/2 = 3.5 -2.75 is already a decimal. 0 is already a decimal. 1 1/4 = 1.25 -6 is already an integer. Step 2: List the numbers from least to greatest (left to right on the number line). -6, -3.5, -2.75, 0, 1.25, 3.5 Step 3: Count the positions from the left. First: -6 Second: -3.5 Third: -2.75 Fourth: 0 The fourth number from the left is 0.

8. (-2.75) + 4.25 - (-1.5) = ? Answer: 3 Solution: Start with (-2.75) + 4.25 - (-1.5) First, add -2.75 and 4.25: -2.75 + 4.25 = 1.5 Now we have 1.5 - (-1.5) Subtracting a negative is the same as adding: 1.5 + 1.5 = 3 The final answer is 3
   Full step-by-step solution

   Step 1: Start with (-2.75) + 4.25 - (-1.5) Step 2: First, add -2.75 and 4.25: -2.75 + 4.25 = 1.5 Step 3: Now we have 1.5 - (-1.5) Step 4: Subtracting a negative is the same as adding: 1.5 + 1.5 = 3 Step 5: The final answer is 3