Middle School Math Quiz: Understand Additive Inverses
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Understand Additive InversesQuestion 1 of 20

A student deposits $35 into a school store account and later withdraws $35. What is the net change in the account balance?

$-35
$35
$0
$70
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Middle School Math Quiz

Middle School Math Quiz: Understand Additive Inverses

Practice Understand Additive Inverses in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Understand Additive Inverses, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A student deposits $35 into a school store account and later withdraws $35. What is the net change in the account balance?

  1. $-35
  2. $35
  3. $0 (correct answer)
  4. $70
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like a deposit of $50 and withdrawal of $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip the sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, a deposit of $35 is modeled as +35, and a withdrawal of $35 as -35, combined: 35 + (-35) = 0 net (back to starting balance). In this case, the net change in the account balance is $0, as the deposit and withdrawal are additive inverses that cancel each other out. A common error might be thinking the net is $70 by adding the absolute values or claiming -35, confusing the withdrawal as the final balance. To find the additive inverse, simply flip the sign (5→-5, -3→3, 0→0), and verify by checking if a + b = 0 (if 7 + (-7) = 0, yes inverses; if 7 + 5 = 12 ≠ 0, not inverses). In real-world contexts like bank accounts, a deposit of $100 and withdrawal of $100 leave the balance unchanged (opposites neutralize), and remember the property that the inverse is unique (only one number adds with 35 to give 0: must be -35).

Question 2

A hiker climbs 200200 meters up a trail and then goes 200200 meters back down to the starting elevation. Which expression shows the net change in elevation?

  1. 200+(−200)=0200 + (-200) = 0 (correct answer)
  2. −200+(−200)=−400-200 + (-200) = -400
  3. 200+200=400200 + 200 = 400
  4. 200×(−200)=−40,000200 \times (-200) = -40{,}000
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like depositing $50 and withdrawing $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, climbing +200 m and descending -200 m: 200 + (-200) = 0 net change (back to start). The correct expression is 200 + (-200) = 0, showing the net elevation change as zero due to additive inverses. A common error is using multiplication like 200 × (-200) = -40,000 (confusing operations) or adding positives to 400 (ignoring signs), but additive inverses involve addition to zero. Finding the additive inverse involves flipping the sign (200→-200), and verifying means checking if they sum to zero (200 + (-200) = 0, yes). In hiking contexts, up and down are opposites that cancel when equal.

Question 3

Are −9-9 and 99 additive inverses? (In other words, do they add to 00?)

  1. Yes, because −9-9 is the reciprocal of 99.
  2. Yes, because −9+9=0-9 + 9 = 0. (correct answer)
  3. No, because −9+9=18-9 + 9 = 18.
  4. No, because −9×9=−1-9 \times 9 = -1.
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like a deposit of $50 and withdrawal of $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip the sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, -9 and 9 sum to -9 + 9 = 0, so they are additive inverses. Yes, -9 and 9 are additive inverses because -9 + 9 = 0. A common error is confusing with multiplication like -9 × 9 = -81 or claiming the sum is 18 by adding absolutes, or mixing with reciprocal where 1/9 × 9 = 1. To find the additive inverse, simply flip the sign (5→-5, -3→3, 0→0), and verify by checking if a + b = 0 (if -9 + 9 = 0, yes inverses; if -9 + (-9) = -18 ≠ 0, not). Avoid mistakes like confusing additive and multiplicative inverses (a + (-a) = 0 vs. a × (1/a) = 1 are different), or claiming the sum is non-zero.

Question 4

Which statement correctly defines an additive inverse?

  1. The additive inverse of aa is the number that makes a+(inverse)=0a + (\text{inverse}) = 0. (correct answer)
  2. The additive inverse of aa is the number that makes a×(inverse)=1a \times (\text{inverse}) = 1.
  3. Only 00 has an additive inverse.
  4. The additive inverse of aa is always ∣a∣|a|.
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like depositing $50 and withdrawing $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, for any a, its inverse b satisfies a + b = 0, so b = -a. The correct statement is that the additive inverse of a is the number that makes a + (inverse) = 0. A common error is confusing it with multiplicative inverse (a × inverse = 1) or wrongly saying only 0 has one (all numbers do), or claiming it's |a| (which is positive and doesn't work for positives). Finding the additive inverse involves flipping the sign (5→-5), and verifying means checking if a + b = 0. Mistakes include using the wrong operation or limiting inverses incorrectly.

Question 5

A hiker climbs 200200 meters up a trail and then walks 200200 meters back down. What is the hiker's net change in elevation?

  1. 00 meters (correct answer)
  2. −200-200 meters
  3. 200200 meters
  4. 400400 meters
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like a deposit of $50 and withdrawal of $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip the sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, climbing +200 m and descending -200 m: 200 + (-200) = 0 net change (back to start). The hiker's net change in elevation is 0 meters, as the up and down movements are additive inverses that cancel. A common error might be adding to 400 meters or claiming 200 meters by ignoring the direction. To find the additive inverse, simply flip the sign (5→-5, -3→3, 0→0), and verify by checking if a + b = 0 (if 200 + (-200) = 0, yes inverses; if 200 + 200 = 400 ≠ 0, not). In contexts like elevation, identify opposites (up ↔ down), and they combine to zero net effect; the inverse is unique (only -200 adds with 200 to 0).

Question 6

A robot moves forward 2.52.5 meters and then moves backward 2.52.5 meters. Which equation best represents the net change in position?

  1. 2.5×(−2.5)=02.5\times(-2.5) = 0
  2. 2.5+(−2.5)=52.5 + (-2.5) = 5
  3. 2.5+2.5=02.5 + 2.5 = 0
  4. 2.5+(−2.5)=02.5 + (-2.5) = 0 (correct answer)
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like a deposit of $50 and withdrawal of $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, forward +2.5 m and backward -2.5 m: 2.5 + (-2.5) = 0 net. The equation 2.5 + (-2.5) = 0 best represents the net change of zero. A common error is using same signs (2.5 + 2.5 = 5) or multiplication, but additive inverses involve opposite signs summing to zero. To find an additive inverse, flip the sign (2.5 → -2.5). Verifying: 2.5 + (-2.5) = 0, yes; in robotics, opposite movements cancel position change.

Question 7

Are 55 and −5-5 additive inverses? (Additive inverses add to 00.)

  1. No, because 5+(−5)=105 + (-5) = 10.
  2. Yes, because 5+(−5)=05 + (-5) = 0. (correct answer)
  3. Yes, because 5×(−5)=−15\times(-5) = -1.
  4. No, because only 00 has an additive inverse.
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like a deposit of $50 and withdrawal of $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, 5 and -5 sum to 5 + (-5) = 0, so they are inverses. Yes, 5 and -5 are additive inverses because 5 + (-5) = 0. A common error is confusing with multiplication (5 × (-5) = -25, not relevant) or miscalculating the sum as 10. To find an additive inverse, flip the sign (5 → -5). Verifying: check if 5 + (-5) = 0, yes; mistakes include thinking only 0 has an inverse, but all numbers do.

Question 8

A robot moves forward 2.5 meters and then moves backward 2.5 meters. Which equation both models this situation and correctly shows the net change in position?

  1. 2.5+(−2.5)=02.5 + (-2.5) = 0 (correct answer)
  2. 2.5×(−2.5)=02.5 \times (-2.5) = 0
  3. 2.5+2.5=02.5 + 2.5 = 0
  4. −2.5+(−2.5)=0-2.5 + (-2.5) = 0
Explanation: Moving forward 2.5 meters is a positive change, and moving backward 2.5 meters is a negative change, so the situation is modeled by 2.5 + (-2.5). Adding these opposite values gives a net change of 0, which matches this being the correct equation both for the situation and for the arithmetic. Choice B is wrong because multiplying does not represent combining forward and backward movements. Choice C is wrong because it adds two positive movements, which would mean the robot moved 5 meters forward only. Choice D is wrong because it adds two negative movements, which would mean the robot moved 5 meters backward only.

Question 9

A hiker's elevation changes during a hike as follows: +120+120 ft, −45-45 ft, +90+90 ft, −165-165 ft. At the end of the hike, the hiker is back at the same elevation as the start. Which statement correctly explains this using additive inverses?

  1. The elevation returned to start because the two positive changes (120120 ft and 9090 ft) are additive inverses of each other.
  2. The elevation returned to start because −45-45 ft and −165-165 ft are additive inverses of each other.
  3. The elevation returned to start because the largest gain (120120 ft) and largest loss (165165 ft) are additive inverses of each other.
  4. The elevation returned to start because the sum of all four elevation changes equals zero, so the complete set of changes acts as its own additive inverse. (correct answer)
Explanation: Adding all four changes: 120+(-45)+90+(-165)=0. Since the total is zero, the full set of changes are additive inverses of zero, matching choice D. Choice A is false because both 120 and 90 are positive; two positive numbers are never additive inverses of each other. Choice B is false for the same reason: -45 and -165 are both negative. Choice C pairs the largest gain and largest loss, but 120+(-165)=-45, not zero, so these two values are not additive inverses of each other.

Question 10

In a card game, positive cards add points and negative cards subtract points. Jamie draws cards with values +8+8, −12-12, +15+15, −6-6, and +7+7. To end the round with exactly 00 points, Jamie needs to draw one more card. What value must this card have, and why does this demonstrate additive inverses?

  1. +12+12, because the additive inverse must match the total's magnitude and sign
  2. +6+6, because it's the additive inverse of the −6-6 card already in the hand
  3. −12-12, because it's the additive inverse of the current total, making the final sum zero (correct answer)
  4. −15-15, because it's the additive inverse of the largest card drawn (+15+15)
Explanation: The current total is 8-12+15-6+7=12. To reach zero, Jamie needs a card worth -12, since 12+(-12)=0, the additive inverse of the total. This matches choice C. Choice A incorrectly assumes the inverse must share the same sign as the original number; an additive inverse always has the opposite sign. Choice B only considers one card already drawn (-6) rather than the running total of all five cards. Choice D uses only the largest single card (+15) instead of the total of all cards drawn so far.

Question 11

An elevator starts at the ground floor (Floor 0). It goes up 77 floors, then down 44 floors, then up 1111 floors, then down 1414 floors. At this point, the elevator is at Floor 0 again. A passenger observes that the elevator's movement demonstrates additive inverses. Which explanation best describes what the passenger noticed?

  1. Going up 77 floors and down 77 floors are additive inverses, as are other equal up-down pairs
  2. The total upward movement and total downward movement are equal and opposite, making them additive inverses (correct answer)
  3. The net displacement of 00 floors shows that all movements combined equal the additive inverse of the starting position
  4. Each upward movement has a corresponding downward movement that serves as its additive inverse
Explanation: Up movements: +7 + 11 = +18 floors. Down movements: -4 + (-14) = -18 floors. The total up (+18) and total down (-18) are additive inverses because they sum to zero. Choice A incorrectly pairs movements that aren't equal. Choice C misuses the concept - the starting position is 0, and its additive inverse is also 0. Choice D is incorrect because the movements don't pair up as described (7 up, 4 down, 11 up, 14 down).

Question 12

A submarine's depth changes are recorded as: dives 4545 feet, rises 2020 feet, dives 3535 feet, rises 1515 feet, then dives 2525 feet. After these movements, the submarine is 7070 feet below its starting depth. The captain wants to return to the starting depth in one movement. This situation illustrates additive inverses in which way?

  1. The total diving distance and total rising distance are additive inverses of each other
  2. Each dive and rise movement has an additive inverse that cancels its effect completely
  3. The required movement (+70+70 feet) is the additive inverse of the current displacement (−70-70 feet) (correct answer)
  4. The starting depth and ending depth are additive inverses that sum to the required movement
Explanation: When you see problems involving opposite movements and returning to a starting point, you're working with additive inverses - numbers that add up to zero and cancel each other out. Let's track the submarine's movements. Diving means going down (negative), and rising means going up (positive): −45+20−35+15−25=−70-45 + 20 - 35 + 15 - 25 = -70 feet. The submarine is now 70 feet below its starting depth, which we represent as −70-70 feet. To return to the starting depth, the submarine needs to move up 70 feet, or +70+70 feet. Notice that −70+70=0-70 + 70 = 0, which brings the submarine back to its starting point (zero displacement). This demonstrates additive inverses perfectly: the current displacement (−70-70) and the required movement (+70+70) are opposites that sum to zero. Answer A is incorrect because the total diving distance (105 feet) and total rising distance (35 feet) don't add to zero - they're not additive inverses. Answer B misunderstands the concept; individual movements don't have inverses within this problem. Answer D incorrectly suggests that depths themselves are additive inverses, when actually it's the displacement and correction movement that form the additive inverse pair. Remember: additive inverses always sum to zero. When you need to "undo" a displacement or return to a starting point, look for the number that, when added to your current position, gives you zero change overall.

Question 13

Sarah's science experiment involves adding chemicals to a solution. She adds 2.52.5 mL of acid, then 1.81.8 mL of base, then 3.23.2 mL of acid, then some amount of base. After all additions, the solution has the same acidity level as when she started. If acids and bases neutralize each other in a 1:11:1 ratio, how much base did she add in the final step?

  1. 3.93.9 mL, because this amount makes the total acid equal the total base added (correct answer)
  2. 5.75.7 mL, because this creates additive inverses with the total acid amount
  3. 1.81.8 mL, because this balances the 3.23.2 mL of acid added in the third step
  4. 2.52.5 mL, because this balances the 2.52.5 mL of acid added in the first step
Explanation: Total acid added: 2.5 + 3.2 = 5.7 mL. Base already added: 1.8 mL. To neutralize all acid, total base needed: 5.7 mL. Additional base needed: 5.7 - 1.8 = 3.9 mL. The total acid (5.7 mL) and total base (5.7 mL) are additive inverses in terms of their effect on acidity. Choice B confuses the final amount with the total. Choices C and D only consider partial neutralization rather than complete neutralization.

Question 14

A weather station records wind speed changes each hour: up 55 mph, down 88 mph, up 1212 mph, down 33 mph, up 66 mph, then down 1212 mph. After these changes, the wind speed is the same as it was at the start. Which statement best explains how this shows additive inverses?

  1. Each increase has a corresponding decrease of equal magnitude that serves as its additive inverse
  2. The increases (+5,+12,+6+5, +12, +6) total +23+23 mph, which equals the decreases (−8,−3,−12-8, -3, -12) total of −23-23 mph (correct answer)
  3. The final wind speed and initial wind speed are additive inverses because their difference is zero
  4. There are exactly three increases and three decreases, so the changes are additive inverses of each other
Explanation: The increases are +5, +12, and +6 mph, totaling +23 mph. The decreases are -8, -3, and -12 mph, totaling -23 mph. Since +23 and -23 are opposite in sign and equal in magnitude, they are additive inverses, and their sum is zero, confirming the wind speed returns to its starting value. This matches choice B. Choice A is incorrect because individual increases don't pair with equal individual decreases (+5 does not pair with -5, for example); the additive inverse relationship exists between the totals, not individual pairs. Choice C misunderstands additive inverses: the initial and final wind speeds are equal to each other, not additive inverses of each other. Choice D incorrectly assumes that having an equal number of increases and decreases is what makes them additive inverses; what matters is that the total magnitudes are equal and opposite, not how many individual changes occurred.

Question 15

What is the additive inverse (opposite) of −12-12?

  1. 00
  2. 1212 (correct answer)
  3. 112\frac{1}{12}
  4. −12-12
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like depositing $50 and withdrawing $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, the number -12 has an inverse of 12 since -12 + 12 = 0. The correct inverse here is 12, as it flips the sign of -12 to make the sum zero. A common error is claiming the inverse of -12 is still -12 (not flipping the sign), or confusing it with the reciprocal 1/12, which is the multiplicative inverse where -12 × (1/-12) = 1, not additive. Finding the additive inverse involves flipping the sign (-12→12), and verifying means checking if they sum to zero (-12 + 12 = 0, yes). Properties include the inverse being unique (only 12 adds with -12 to give 0), and avoiding mistakes like claiming the sum is not zero.

Question 16

A thermometer shows a temperature increase of 8∘8^\circC in the morning and a decrease of 8∘8^\circC in the afternoon. What is the total change in temperature for the day?

  1. 0∘0^\circC (correct answer)
  2. −16∘-16^\circC
  3. 8∘8^\circC
  4. 16∘16^\circC
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like depositing $50 and withdrawing $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, a temperature rise of 8°C is +8, a fall of 8°C is -8, and combined: 8 + (-8) = 0 net change (back to starting temperature). The total change is 0°C because the increase and decrease are additive inverses that cancel. A common error might be adding absolute values to get 16°C or halving to 8°C, or wrongly claiming -16°C by doubling the negative, but opposites sum to zero. Finding the additive inverse involves flipping the sign (+8→-8), and verifying means checking if they sum to zero (8 + (-8) = 0, yes). In contexts like weather, opposite changes neutralize, and the property is commutative: +8 + (-8) = (-8) + 8 = 0.

Question 17

A student deposits $25 into a school store account and later withdraws $25. What is the net change in the account balance? (Think of withdrawal as a negative change.)

  1. 2525
  2. −25-25
  3. 00 (correct answer)
  4. 5050
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like a deposit of $50 and withdrawal of $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, a deposit of $25 is modeled as +25, and a withdrawal of $25 as -25, combined: 25 + (-25) = 0 net (back to starting balance). In this case, the net change is $0, as the deposit and withdrawal are additive inverses that cancel each other out. A common error is thinking the net is $50 by adding magnitudes without signs (25 + 25 = 50) or confusing with multiplication, but actually, opposites sum to zero. To find an additive inverse, flip the sign (like 25 → -25). You can verify by checking if they sum to zero: 25 + (-25) = 0, confirming they are inverses; in real-world contexts like bank accounts, a deposit and equal withdrawal leave the balance unchanged.

Question 18

A student deposits $30 into a school store account and later withdraws $30. What is the net change in the account balance?

  1. $30
  2. $60
  3. $0 (correct answer)
  4. $-60
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like depositing $50 and withdrawing $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, a deposit of $30 is modeled as +30, a withdrawal of $30 as -30, and combined: 30+(-30)=0 net (back to starting balance). In this case, the net change is $0 because the deposit and withdrawal are additive inverses that cancel each other out. A common error might be thinking the net is $60 by adding the absolute values without considering signs, or confusing it with $30 by halving, but actually, opposites cancel to zero. Finding the additive inverse involves flipping the sign (like +30→-30), and verifying means checking if they sum to zero (30 + (-30) = 0, yes). In real-world contexts, such as bank accounts, a deposit and equal withdrawal leave the balance unchanged, demonstrating how opposites neutralize each other.

Question 19

What is the value of −(−9)-(-9)?

  1. −18-18
  2. 00
  3. 99 (correct answer)
  4. −9-9
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like depositing $50 and withdrawing $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, the additive inverse of 9 is -9, so -(-9) applies the negative sign to -9, flipping it back to 9, since -(-9) = 9. The value is 9, as taking the negative of a negative number gives the positive (the additive inverse of -9 is 9). A common error is keeping it negative like -9 (not flipping) or doubling to -18 (confusing with multiplication), or claiming 0 (misunderstanding), but -(-a) = a. Finding the additive inverse involves flipping the sign (-9→9), and verifying means checking if -9 + 9 = 0, yes. Properties include the inverse being unique, and this shows double negation returns the original.

Question 20

Which number must be added to 34\dfrac{3}{4} to make 00?

  1. −34-\dfrac{3}{4} (correct answer)
  2. −43-\dfrac{4}{3}
  3. 43\dfrac{4}{3}
  4. 34\dfrac{3}{4}
Explanation: This question tests understanding of additive inverses: opposite quantities (a and -a) that combine to zero (a+(-a)=0), occurring when opposite transactions, changes, or directions cancel. The additive inverse of a is -a: a number with the opposite sign that sums to zero (7 and -7: 7+(-7)=0, mutually inverse); in contexts like a deposit of $50 and withdrawal of $50 summing to $0 net change (opposite transactions cancel), temperature rising 8° and falling 8° giving 0° net change (opposite changes cancel), or gaining 15 yards and losing 15 yards resulting in 0 net yards (opposite directions cancel); every number has an additive inverse: 5→-5, -3→3 (flip the sign), 2.5→-2.5, even 0→0 (zero is its own inverse). For example, the number 3/4 has an inverse of -3/4 since 3/4 + (-3/4) = 0. The number that must be added to 3/4 to make 0 is -3/4, as it is the additive inverse. A common error is confusing with the reciprocal 4/3, which is the multiplicative inverse (3/4 × 4/3 = 1), not additive, or claiming 3/4 itself where the sum is not zero. To find the additive inverse, simply flip the sign (5→-5, -3→3, 0→0), and verify by checking if a + b = 0 (if 3/4 + (-3/4) = 0, yes; if 3/4 + 4/3 ≠ 0, not). Properties include commutativity (3/4 + (-3/4) = (-3/4) + 3/4 = 0), and avoid claiming the sum is non-zero or using the wrong sign.