Absolute Value

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Answer Key & Explanations

Absolute Value · Grade 6 · Worksheet 1

1. Maria is tracking the altitude of a submarine during a research mission. The submarine starts at sea level (0 meters). It first descends to -250 meters to study deep-sea life. Then it ascends 180 meters to avoid a strong current. Finally, it descends another 120 meters to reach its target location. What is the total distance the submarine traveled from its starting point, considering only the magnitude of the movements? Answer: 550 Solution: Identify the individual movements and their magnitudes (absolute values). First descent: 250 meters down (magnitude = 250) First ascent: 180 meters up (magnitude = 180) Second descent: 120 meters down (magnitude = 120) Add the magnitudes of all movements to find the total distance traveled.
   Full step-by-step solution

   Step 1: Identify the individual movements and their magnitudes (absolute values). First descent: 250 meters down (magnitude = 250) First ascent: 180 meters up (magnitude = 180) Second descent: 120 meters down (magnitude = 120) Step 2: Add the magnitudes of all movements to find the total distance traveled. Total distance = 250 + 180 + 120 Step 3: Calculate the sum. 250 + 180 = 430 430 + 120 = 550 The total distance the submarine traveled is 550 meters.

2. Noah is a pilot for a hot air balloon company. He takes tourists on scenic rides. During one ride, the balloon starts at ground level (0 meters). First, it ascends 1,200 meters to give passengers a great view of the mountains. Then, to avoid some low clouds, it descends 450 meters. Later, it ascends another 700 meters to catch a favorable wind. What is the total vertical distance the balloon traveled during the ride? Answer: 2350 Solution: The balloon first ascends 1,200 meters. Distance traveled = 1,200 meters. Step 2: The balloon then descends 450 meters.
   Full step-by-step solution

   Step 1: The balloon first ascends 1,200 meters. Distance traveled = 1,200 meters. Step 2: The balloon then descends 450 meters. Distance traveled = 450 meters. Step 3: The balloon finally ascends 700 meters. Distance traveled = 700 meters. Step 4: Add all the distances together: 1,200 + 450 + 700 = 2,350 meters. The total vertical distance traveled is 2,350 meters.

3. Mere is a park ranger tracking the elevation of a hiking trail. The trail starts at the visitor center at 0 meters. Hikers first descend to a river that is 240 meters below the visitor center. Then they climb up to a lookout that is 160 meters above the river. Finally, they descend 80 meters to reach a campsite. What is the total vertical distance (magnitude) the hikers travel from the visitor center to the campsite? Answer: 480 Solution: The hikers first descend 240 meters from the visitor center to the river. The magnitude of this movement is | -240 | = 240 meters. From the river, they climb up 160 meters to the lookout.
   Full step-by-step solution

   Step 1: The hikers first descend 240 meters from the visitor center to the river. The magnitude of this movement is | -240 | = 240 meters. Step 2: From the river, they climb up 160 meters to the lookout. The magnitude of this movement is | 160 | = 160 meters. Step 3: From the lookout, they descend 80 meters to the campsite. The magnitude of this movement is | -80 | = 80 meters. Step 4: Add the magnitudes of all movements to find the total vertical distance: 240 + 160 + 80 = 480 meters. The answer is 480 meters.

4. Noah is tracking the altitude of his drone during a flight. He starts at ground level (0 meters). First, he flies the drone up to 125 meters to take photos of a building. Then he descends 85 meters to avoid birds. Finally, he ascends another 60 meters to get above some trees. What is the total distance Noah's drone traveled vertically during this flight? Answer: 270 Solution: The drone first ascends 125 meters. Distance traveled = 125 meters The drone then descends 85 meters. Distance traveled = 85 meters The drone finally ascends 60 meters.
   Full step-by-step solution

   Step 1: The drone first ascends 125 meters. Distance traveled = 125 meters Step 2: The drone then descends 85 meters. Distance traveled = 85 meters Step 3: The drone finally ascends 60 meters. Distance traveled = 60 meters Step 4: Add all the distances: 125 + 85 + 60 = 270 meters Step 5: The total vertical distance traveled is 270 meters.

5. Charlotte is exploring a cave system. She starts at an elevation of 0 meters at the entrance. She descends 127 meters to reach the first chamber, then climbs up 72 meters to a ledge, and finally descends another 57 meters to reach the bottom of the cave. What is the total vertical distance Charlotte traveled during her descent and ascent? Answer: 256 Solution: Identify each movement and its distance (absolute value). First descent: 127 meters (distance = 127) Second ascent: 72 meters (distance = 72) Third descent: 57 meters (distance = 57) Add all the distances to find the total vertical distance traveled.
   Full step-by-step solution

   Step 1: Identify each movement and its distance (absolute value). First descent: 127 meters (distance = 127) Second ascent: 72 meters (distance = 72) Third descent: 57 meters (distance = 57) Step 2: Add all the distances to find the total vertical distance traveled. Total distance = 127 + 72 + 57 Step 3: Calculate the sum. 127 + 72 = 199 199 + 57 = 256 The total vertical distance Charlotte traveled is 256 meters.

6. |−1725 + 975| ÷ |−35 + 15| = ? Answer: 37.5 Solution: Simplify inside the first absolute value: −1725 + 975 = −750 Find the absolute value: |−750| = 750 Simplify inside the second absolute value: −35 + 15 = −20 Find the absolute value: |−20| = 20 Divide the results: 750 ÷ 20 = 37.5 The answer is 37.5.
   Full step-by-step solution

   Step 1: Simplify inside the first absolute value: −1725 + 975 = −750 Step 2: Find the absolute value: |−750| = 750 Step 3: Simplify inside the second absolute value: −35 + 15 = −20 Step 4: Find the absolute value: |−20| = 20 Step 5: Divide the results: 750 ÷ 20 = 37.5 The answer is 37.5.