Absolute Value

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Answer Key & Explanations

Absolute Value · Grade 6 · Worksheet 3

1. Tane is monitoring the altitude of his paraglider during a flight. He starts at a launch point that is 145 meters above sea level. First, he climbs up another 213 meters to catch a thermal. Then, he descends 189 meters to avoid some clouds. Finally, he climbs up another 97 meters to reach a stable altitude. What is the total vertical distance Tane traveled during this part of his flight? Answer: 499 Solution: Identify each movement and its magnitude (absolute value). First climb: 213 meters up (magnitude = 213) First descent: 189 meters down (magnitude = 189) Second climb: 97 meters up (magnitude = 97) Add the magnitudes of all movements to find the total vertical distance traveled.
   Full step-by-step solution

   Step 1: Identify each movement and its magnitude (absolute value). First climb: 213 meters up (magnitude = 213) First descent: 189 meters down (magnitude = 189) Second climb: 97 meters up (magnitude = 97) Step 2: Add the magnitudes of all movements to find the total vertical distance traveled. Total distance = 213 + 189 + 97 Step 3: Calculate the sum. 213 + 189 = 402 402 + 97 = 499 The total vertical distance Tane traveled is 499 meters.

2. |−3400 + 2600| − |−1800 ÷ 12| = ? Answer: 650 Solution: Simplify inside the first absolute value: −3400 + 2600 = −800 Find the absolute value: |−800| = 800 Simplify inside the second absolute value: −1800 ÷ 12 = −150 Find the absolute value: |−150| = 150 Subtract: 800 − 150 = 650 The answer is 650.
   Full step-by-step solution

   Step 1: Simplify inside the first absolute value: −3400 + 2600 = −800 Step 2: Find the absolute value: |−800| = 800 Step 3: Simplify inside the second absolute value: −1800 ÷ 12 = −150 Step 4: Find the absolute value: |−150| = 150 Step 5: Subtract: 800 − 150 = 650 The answer is 650.

3. Emma is exploring a cave system. She starts at an elevation of 0 meters at the entrance. She descends 45 meters to reach an underground river, then climbs up 20 meters to a dry ledge, and finally descends another 35 meters to the deepest chamber. What is the total vertical distance Emma traveled during her descent and ascent, considering only the magnitude of each movement? Answer: 100 Solution: Identify each movement and its magnitude (absolute value). First descent: 45 meters down → magnitude = 45 First ascent: 20 meters up → magnitude = 20 Second descent: 35 meters down → magnitude = 35 Add the magnitudes of all movements to find the total vertical distance traveled.
   Full step-by-step solution

   Step 1: Identify each movement and its magnitude (absolute value). First descent: 45 meters down → magnitude = 45 First ascent: 20 meters up → magnitude = 20 Second descent: 35 meters down → magnitude = 35 Step 2: Add the magnitudes of all movements to find the total vertical distance traveled. Total distance = 45 + 20 + 35 Step 3: Calculate the sum. 45 + 20 = 65 65 + 35 = 100 The total vertical distance Emma traveled is 100 meters.

4. Liam is tracking the temperature changes in his city. On Monday morning, the temperature was -5°C. By noon, it had risen by 8 degrees. On Tuesday morning, it was 3°C, but then a cold front came through and dropped the temperature by 11 degrees. What was the final temperature on Tuesday after the cold front, and how does its distance from zero compare to Monday's morning temperature? Answer: -8°C, and Tuesday's final temperature is 3 degrees farther from zero than Monday's morning temperature. Solution: Find Monday's temperature at noon. Monday morning: -5°C Temperature rose by 8 degrees: -5 + 8 = 3°C at noon on Monday. Find Tuesday's temperature after the cold front.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Find Monday's temperature at noon.** Monday morning: -5°C Temperature rose by 8 degrees: -5 + 8 = 3°C at noon on Monday. --- **Step 2: Find Tuesday's temperature after the cold front.** Tuesday morning: 3°C Temperature dropped by 11 degrees: 3 - 11 = -8°C after the cold front. So the final temperature on Tuesday is **-8°C**. --- **Step 3: Compare distances from zero.** Distance from zero means the absolute value of the temperature. Monday morning: |-5| = 5 Tuesday final: |-8| = 8 Difference in distances: 8 - 5 = 3 So Tuesday's final temperature is **3 degrees farther from zero** than Monday's morning temperature. --- **Final Answer:** -8°C, and Tuesday's final temperature is 3 degrees farther from zero than Monday's morning temperature.

5. |−2500 + 1800| ÷ |−25 + 15| = ? Answer: 70 Solution: Calculate -2500 + 1800 = -700 Apply absolute value: |-700| = 700 Calculate -25 + 15 = -10 Apply absolute value: |-10| = 10 Divide the results: 700 ÷ 10 = 70 The answer is 70.
   Full step-by-step solution

   Step 1: Calculate -2500 + 1800 = -700 Step 2: Apply absolute value: |-700| = 700 Step 3: Calculate -25 + 15 = -10 Step 4: Apply absolute value: |-10| = 10 Step 5: Divide the results: 700 ÷ 10 = 70 The answer is 70.

6. Liam is tracking the daily temperature changes in his city. On Monday morning, it was -3°C. By afternoon, the temperature rose by 8°C. Then overnight, it dropped by 5°C. What was the final temperature on Tuesday morning? Answer: 0 Solution: Start with Monday morning temperature. Monday morning: -3°C Temperature rose by 8°C by afternoon. -3 + 8 = 5°C So Monday afternoon temperature = 5°C Overnight, temperature dropped by 5°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 rose by 8°C by afternoon. -3 + 8 = 5°C So Monday afternoon temperature = 5°C Step 3: Overnight, temperature dropped by 5°C. 5 - 5 = 0°C Step 4: This overnight change leads to Tuesday morning. So Tuesday morning temperature = 0°C Final answer: 0

7. |−1272 + 872| − |−42| = ? Answer: 358 Solution: Simplify inside the first absolute value: −1272 + 872 = −400 Apply absolute value: |−400| = 400 Simplify inside the second absolute value: |−42| = 42 Subtract: 400 − 42 = 358 The answer is 358.
   Full step-by-step solution

   Step 1: Simplify inside the first absolute value: −1272 + 872 = −400 Step 2: Apply absolute value: |−400| = 400 Step 3: Simplify inside the second absolute value: |−42| = 42 Step 4: Subtract: 400 − 42 = 358 The answer is 358.

8. A submarine is exploring a deep ocean trench. The submarine starts at sea level (0 meters). It descends to 1250 meters below sea level, then ascends 480 meters, then descends another 320 meters. What is the submarine's final position relative to sea level? Answer: -1090 Solution: The submarine starts at 0 meters relative to sea level. Descending means going below sea level, so we subtract from the current position.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Start at sea level** The submarine starts at 0 meters relative to sea level. --- **Step 2: First descent** Descending means going below sea level, so we subtract from the current position. Descend 1250 m: 0 - 1250 = -1250 meters --- **Step 3: First ascent** Ascending means moving upward toward sea level, so we add to the current position. Ascend 480 m: -1250 + 480 = -770 meters --- **Step 4: Second descent** Descending again means subtracting from the current position. Descend 320 m: -770 - 320 = -1090 meters --- **Step 5: Interpret the result** -1090 meters means 1090 meters below sea level. --- **Final answer:** -1090