Negative Numbers

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Answer Key & Explanations

Negative Numbers · Grade 6 · Worksheet 1

1. A rectangular city park is drawn on a coordinate plane with corners at (-125, 80), (215, 80), (215, -60), and (-125, -60). Each unit on the grid represents 1 meter. What is the area of the park in square meters? Answer: 47600 Solution: Find the length of the rectangle using the x-coordinates. The left side is at x = -125 and the right side is at x = 215.
   Full step-by-step solution

   Step 1: Find the length of the rectangle using the x-coordinates. The left side is at x = -125 and the right side is at x = 215. Length = 215 - (-125) = 215 + 125 = 340 meters Step 2: Find the width of the rectangle using the y-coordinates. The top is at y = 80 and the bottom is at y = -60. Width = 80 - (-60) = 80 + 60 = 140 meters Step 3: Calculate the area using the formula: Area = length × width Area = 340 × 140 Step 4: Multiply 340 × 140: 340 × 100 = 34,000 340 × 40 = 13,600 Total = 34,000 + 13,600 = 47,600 The area of the park is 47,600 square meters.

2. -3 × (4 + (-9)) ÷ 5 = ? Answer: 3 Solution: Start with the expression: -3 × (4 + (-9)) ÷ 5 Simplify inside the parentheses first: 4 + (-9) = -5 The expression becomes: -3 × (-5) ÷ 5 Multiply -3 and -5: -3 × (-5) = 15 The expression becomes: 15 ÷ 5 Divide 15 by 5: 15 ÷ 5 = 3 The final answer is 3.
   Full step-by-step solution

   Step 1: Start with the expression: -3 × (4 + (-9)) ÷ 5 Step 2: Simplify inside the parentheses first: 4 + (-9) = -5 Step 3: The expression becomes: -3 × (-5) ÷ 5 Step 4: Multiply -3 and -5: -3 × (-5) = 15 Step 5: The expression becomes: 15 ÷ 5 Step 6: Divide 15 by 5: 15 ÷ 5 = 3 Step 7: The final answer is 3.

3. Liam's bank account balance is -$137. He deposits $95. What is his new balance? Answer: -$42 Solution: Start with the initial balance: -$137. Add the deposit of $95: -137 + 95. Since 137 is larger than 95, the result will be negative.
   Full step-by-step solution

   Step 1: Start with the initial balance: -$137. Step 2: Add the deposit of $95: -137 + 95. Step 3: Since 137 is larger than 95, the result will be negative. Subtract: 137 - 95 = 42. Step 4: The answer is -$42. The answer is -$42.

4. A submarine is exploring an underwater canyon. On a map of the ocean floor, the submarine's path forms a rectangle with corners at (-250, 180), (390, 180), (390, -320), and (-250, -320), where each unit on the grid represents 1 meter. The submarine starts at the surface (0 meters elevation) and dives to the deepest point of the rectangle. What is the elevation of the deepest point in meters, and what does the negative sign mean in this context? Answer: -320 meters; the negative sign means 320 meters below sea level Solution: Identify the y-coordinates of all four corners: 180, 180, -320, and -320. The deepest point has the smallest y-coordinate. Step 2: The smallest y-coordinate is -320.
   Full step-by-step solution

   Step 1: Identify the y-coordinates of all four corners: 180, 180, -320, and -320. The deepest point has the smallest y-coordinate. Step 2: The smallest y-coordinate is -320. This means the submarine dives to an elevation of -320 meters. Step 3: The negative sign means that the elevation is 320 meters below sea level (the surface is at 0 meters). Final Answer: The elevation of the deepest point is -320 meters. The negative sign means it is 320 meters below sea level.

5. -3 × (4 + (-7)) - 5 = ? Answer: 4 Solution: Start with the expression: -3 × (4 + (-7)) - 5 Calculate inside the parentheses first: 4 + (-7) = -3 Now the expression is: -3 × (-3) - 5 Multiply next: -3 × (-3) = 9 (because a negative times a negative is positive) Now the expression is: 9 - 5 Subtract: 9 - 5 = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Start with the expression: -3 × (4 + (-7)) - 5 Step 2: Calculate inside the parentheses first: 4 + (-7) = -3 Step 3: Now the expression is: -3 × (-3) - 5 Step 4: Multiply next: -3 × (-3) = 9 (because a negative times a negative is positive) Step 5: Now the expression is: 9 - 5 Step 6: Subtract: 9 - 5 = 4 The answer is 4.

6. Liam is monitoring the temperature in a science experiment. The temperature starts at 12°C at 9:00 AM. By 10:00 AM, it drops by 8 degrees. Then, over the next hour, it rises by 15 degrees. What is the final temperature at 11:00 AM? Answer: 19 Solution: Start with the initial temperature at 9:00 AM. Initial temperature = 12°C. By 10:00 AM, the temperature drops by 8 degrees.
   Full step-by-step solution

   Let's go step by step. Step 1: Start with the initial temperature at 9:00 AM. Initial temperature = 12°C. Step 2: By 10:00 AM, the temperature drops by 8 degrees. Dropping means subtract: 12 - 8 = 4°C at 10:00 AM. Step 3: Over the next hour (from 10:00 AM to 11:00 AM), the temperature rises by 15 degrees. Rising means add: 4 + 15 = 19°C at 11:00 AM. Final answer: 19°C.

7. A rectangular garden is drawn on a coordinate plane with corners at (-3, 2), (5, 2), (5, -1), and (-3, -1). What is the perimeter of this garden? Answer: 22 Solution: Identify the vertices of the rectangle. A = (-3, 2) B = (5, 2) C = (5, -1) D = (-3, -1) Determine the length of the horizontal sides.
   Full step-by-step solution

   Step 1: Identify the vertices of the rectangle. The points are: A = (-3, 2) B = (5, 2) C = (5, -1) D = (-3, -1) Step 2: Determine the length of the horizontal sides. Points A and B have the same y-coordinate (y = 2), so the distance between them is the difference in x-coordinates: Length AB = 5 - (-3) = 5 + 3 = 8 Step 3: Determine the length of the vertical sides. Points B and C have the same x-coordinate (x = 5), so the distance between them is the difference in y-coordinates: Length BC = 2 - (-1) = 2 + 1 = 3 Step 4: Confirm the shape is a rectangle. In a rectangle, opposite sides are equal. So: AB = CD = 8 BC = DA = 3 Step 5: Calculate the perimeter. Perimeter = 2 * (length + width) Perimeter = 2 * (8 + 3) Perimeter = 2 * 11 Perimeter = 22 Final Answer: 22

8. Liam is monitoring the temperature in a science experiment. The liquid starts at 15°C. During the first phase, the temperature drops by 22 degrees. In the second phase, it rises by 18 degrees. What is the final temperature of the liquid? Answer: 11 Solution: Start with the initial temperature. The liquid starts at 15°C. During the first phase, the temperature drops by 22 degrees.
   Full step-by-step solution

   Let's go step-by-step. **Step 1: Start with the initial temperature.** The liquid starts at 15°C. **Step 2: Apply the first change (temperature drop).** During the first phase, the temperature drops by 22 degrees. That means: 15 − 22 = −7°C So after the first phase, the temperature is −7°C. **Step 3: Apply the second change (temperature rise).** In the second phase, it rises by 18 degrees. That means: −7 + 18 = 11°C **Step 4: State the final temperature.** The final temperature is 11°C. **Final Answer:** 11

9. -25 + (-13) - 7 = ? Answer: -45 Solution: Start with -25 + (-13) Adding a negative is the same as subtracting: -25 - 13 = -38 Now subtract 7 from -38: -38 - 7 = -45 The final answer is -45.
   Full step-by-step solution

   Step 1: Start with -25 + (-13) Step 2: Adding a negative is the same as subtracting: -25 - 13 = -38 Step 3: Now subtract 7 from -38: -38 - 7 = -45 Step 4: The final answer is -45.