📝 Negative numbers and opposites (35 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 35 questions available

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What is Negative numbers and opposites?

Definition:
Negative numbers are numbers less than zero, represented with a minus sign ( $-$ ), while the opposite of a number is the number that is the same distance from zero on the number line but in the opposite direction, so the opposite of $a$ is $-a$ , and the sum of a number and its opposite is zero.

Working:
To find the opposite, change the sign of the number: the opposite of $5$ is $-5$ , and the opposite of $-3$ is $3$ ; on a number line, negative numbers lie to the left of zero, and the absolute value of a number is its distance from zero, ignoring sign.

Example:
Find the opposite of $-7$ and compare $-4$ and $-9$ .
Solution: Opposite of $-7$ is $7$ ; on number line, $-4$ is to the right of $-9$ , so $-4 > -9$ .

Reason:
Understanding negatives and opposites is essential for operations with integers, solving equations like $x + 5 = 0$ , and interpreting directed quantities in physics, finance, and everyday contexts like debt or temperature below zero.

Easy

Medium

Hard

📝 All Negative numbers and opposites MCQs

Q1. A number line shows a point at $-7$ . Its opposite is represented by moving the same distance from zero in the other direction. Which value represents the opposite, and why?

-14

, because the distance doubles

7

, because opposites are equally distant from zero on opposite sides ✅

-7

, because the point does not change

0

, because zero is between all opposites

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The opposite of a number has the same distance from zero but lies on the other side. Since $-7$ is seven units left of zero, its opposite is $7$ , seven units right of zero.

Q2. A student says, The opposite of $-12$ is $-(-12)=-12$ , because two negative signs should keep the result negative." Which response best evaluates the student's reasoning?"

A.Correct, because negative signs always produce a negative result

B.Correct, because opposites must have the same sign

C.Incorrect, because applying the opposite operation changes

-12

12

✅

D.Incorrect, because the opposite of every number is zero

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The student's mistake is treating two negative signs as automatically producing a negative value. The expression $-(-12)$ means the opposite of $-12$ , which is $12$ .

Q3. A submarine is $18$ meters below sea level, represented by $-18$ . It rises $7$ meters and then moves to the position opposite its current position relative to sea level. What is its final position?

A.11 meters below sea level

B.11 meters above sea level ✅

C.25 meters above sea level

D.25 meters below sea level

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Starting at $-18$ , rising $7$ meters gives $-11$ . The position opposite to $-11$ is $11$ , so the submarine's final position is $11$ meters above sea level.

Q4. A student evaluates $-(-5)+(-3)$ as $-8$ . Which step contains the error, and what is the correct result?

A.The first term should remain

-5

; the result is

-8

B.The opposite of

-5

5

; therefore the result is

2

✅

C.The second term should become

3

; therefore the result is

8

D.Both negatives cancel, so the result is

0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The expression begins with $-(-5)$ , which means the opposite of $-5$ , giving $5$ . Then $5+(-3)=2$ . The error was failing to interpret the nested negative correctly.

Q5. On a number line, point $P$ is at $-6$ , while point $Q$ is at $4$ . A third point $R$ is placed at the opposite of $P$ . Which statement correctly compares $Q$ and $R$ ?

Q

is 10 units left of

R

Q

and

R

are at the same position

Q

is 2 units left of

R

Q

is 2 units right of

R

✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The opposite of $P=-6$ is $R=6$ . Since $Q=4$ , it lies two units to the left of $R=6$ . Therefore $Q$ is not at the same location and is not to the right.

Q6. A temperature changes from $-9^\circ\text{C}$ to its opposite value. It then decreases by $4^\circ\text{C}$ . Which expression and result correctly model the situation?

-(-9)-4=5

✅

-9-(-9)-4=-4

-(-9)+4=13

9+(-4)=5

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The opposite of $-9$ is $9$ . Decreasing by $4$ gives $9-4=5$ . Thus $-(-9)-4=5$ correctly models both transformations and produces the final temperature.

Q7. A game awards the opposite of a player's penalty. A player receives a penalty of $-x$ points, where $x>0$ . The player then loses $3$ more points. Which expression represents the final score change, and what happens when $x=8$ ?

-x-3=-11

x-3=5

✅

-x+3=-5

x+3=11

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The opposite of the penalty $-x$ is $x$ , so the score change becomes $x$ . Losing $3$ points gives $x-3$ . When $x=8$ , the result is $5$ , requiring both symbolic reasoning and substitution.

Q8. A bank account has a balance of $-\<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 3: 45\̲)̲. The owner dep…" style="color:#cc0000">45\). The owner deposits $\</span>70$ and then withdraws $\$30$ . What is the final balance, and what does its sign indicate?

-\$5

, meaning the account still owes money ✅

-\$5

, meaning the account has positive savings

\$5

, meaning the account has money available

\$155

, meaning all transactions increase the balance

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Starting with $-45$ , adding $70$ gives $25$ . Subtracting $30$ gives $25-30=-5$ , not $5$ . Therefore, the correct balance is actually $-\$5$ , indicating money is still owed. None of the listed choices correctly states both the calculation and interpretation, so the intended correct choice is A.

Q9. Which statement best explains why $-8$ is greater than $-12$ , even though $8$ is smaller than $12$ ?

A.Because negative numbers become greater as they move farther left

B.Because

-8

is closer to zero and therefore represents a greater value ✅

C.Because the larger absolute value is always the greater negative number

D.Because subtracting two negative signs makes every negative number positive

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: On a number line, values increase as they move to the right. Since $-8$ lies to the right of $-12$ , it is greater. Its smaller distance from zero also helps explain why $-8$ represents the greater negative value.

Q10. A temperature graph shows the temperature changing from $-6^\circ\text{C}$ at 6 a.m. to $-1^\circ\text{C}$ at noon, then to $-4^\circ\text{C}$ at 6 p.m. During which interval did the temperature increase the most?

A.6 a.m. to noon ✅

B.Noon to 6 p.m.

C.Both intervals increased equally

D.Neither interval showed an increase

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: From 6 a.m. to noon, the temperature changes from $-6$ to $-1$ , an increase of $5^\circ\text{C}$ . From noon to 6 p.m., it decreases by $3^\circ\text{C}$ . Therefore, the first interval has the greatest increase.

Q11. A student claims that $-15+9=-24$ because both numbers should be combined using addition. Which explanation identifies the error and gives the correct result?

A.The student should add absolute values; the result is

24

B.The positive number must be ignored; the result is

-15

C.The numbers have opposite signs, so their absolute values are compared; the result is

-6

✅

D.Both signs cancel automatically; the result is

6

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When adding numbers with opposite signs, compare their absolute values and subtract the smaller from the larger. Since $15>9$ , the result has the sign of $-15$ , giving $-6$ .

Q12. A hiker begins at an elevation of $-20$ meters relative to a reference level. She climbs $13$ meters, descends $8$ meters, and then climbs $6$ meters. What is her final elevation?

-9

meters ✅

-3

meters

1

meter

7

meters

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Starting at $-20$ , climbing $13$ gives $-7$ . Descending $8$ gives $-15$ , and climbing $6$ gives $-9$ . Therefore, the mathematically correct final elevation is $-9$ meters, making option A correct.

Q13. On a number line, point $A$ is at $-11$ , point $B$ is at $-3$ , and point $C$ is at $5$ . Which comparison is correct?

A>B>C

C>B>A

✅

B>C>A

A>C>B

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Number-line values increase from left to right. Therefore, $-11$ is the smallest, $-3$ is next, and $5$ is the greatest. The correct ordering from greatest to least is $5>-3>-11$ .

Q14. Three consecutive integers have a sum of $-18$ . Which set could they be?

-8,-6,-4

-7,-6,-5

✅

-6,-5,-4

-9,-6,-3

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Let the middle integer be $n$ . Three consecutive integers are $n-1,n,n+1$ , whose sum is $3n$ . Setting $3n=-18$ gives $n=-6$ , so the integers are $-7,-6,-5$ . This requires modeling rather than simple calculation.

Q15. Which statement correctly describes the relationship between $-9$ and $4$ on a number line?

-9

is greater because

9>4

-9

is less because it lies farther left than

4

✅

C.Both numbers are equal because they are integers

4

is less because it is closer to zero

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Integers are ordered by their positions on the number line, with values increasing from left to right. Since $-9$ lies to the left of $4$ , $-9$ is less than $4$ , regardless of its larger absolute value.

Q16. A player starts a game with $6$ points, loses $11$ points, gains $8$ points, and then loses $5$ points. What is the final score?

A.2

-2

✅

C.18

-18

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The score changes can be modeled as $6-11+8-5$ . First, $6-11=-5$ , then $-5+8=3$ , and finally $3-5=-2$ . Therefore, the player's final score is $-2$ .

Q17. A student calculates $(-7)+(-5)=2$ and explains that the two negative signs cancel. What is the best evaluation of this reasoning?

A.Correct, because any two negative integers produce a positive sum

B.Correct, because addition always removes negative signs

C.Incorrect, because numbers with the same negative sign combine to give

-12

✅

D.Incorrect, because the answer must always be zero when both numbers are negative

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: When adding two negative integers, their distances from zero combine and the result remains negative. Thus $(-7)+(-5)=-(7+5)=-12$ . Negative signs do not automatically cancel during addition.

Q18. A number line records four positions: $A=-8$ , $B=-2$ , $C=3$ , and $D=7$ . A student claims the distance from $A$ to $B$ is $6$ , while the distance from $C$ to $D$ is $4$ . Which conclusion is correct?

A.Only the first distance is correct

B.Only the second distance is correct

C.Both distances are correct ✅

D.Both distances are incorrect because negative integers cannot have distance

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The distance between two integer positions is the absolute difference. For $A$ and $B$ , $|-8-(-2)|=6$ . For $C$ and $D$ , $|3-7|=4$ . Therefore, both calculations are correct.

Q19. A temperature is $-4^\circ\text{C}$ . It falls $7^\circ\text{C}$ , rises $12^\circ\text{C}$ , and then falls $5^\circ\text{C}$ . What temperature is recorded at the end?

-4^\circ\text{C}

✅

0^\circ\text{C}

4^\circ\text{C}

6^\circ\text{C}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The changes can be represented by $-4-7+12-5$ . Starting with $-4$ , falling $7$ gives $-11$ , rising $12$ gives $1$ , and falling $5$ gives $-4$ . The final temperature is $-4^\circ\text{C}$ .

Q20. Three consecutive integers have a sum of $15$ . Which set of integers satisfies the condition?

3,4,5

✅

4,5,6

2,5,8

1,6,8

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Three consecutive integers can be represented as $n-1,n,n+1$ . Their sum is $3n$ , so $3n=15$ gives $n=5$ . Therefore, the consecutive integers are $4,5,6$ , making option B the correct set.

Q21. A student compares $(-3)(-4)+(-2)$ with $(-3)[(-4)+(-2)]$ and says they must be equal because both expressions contain the same integers. Which comparison is correct?

A.Both equal

10

B.Both equal

-18

C.The first equals

10

, while the second equals

18

✅

D.The first equals

10

, while the second equals

18

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The first expression is $(-3)(-4)+(-2)=12-2=10$ . The second is $(-3)[(-4)+(-2)]=(-3)(-6)=18$ . The grouping changes which operations are performed together, so the expressions are not equal.

Q22. Which ordering lists the integers from least to greatest?

-2,-7,0,4

-7,-2,0,4

✅

4,0,-2,-7

0,-2,-7,4

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: On a number line, values increase as you move from left to right. Therefore, the integer farthest left is the smallest. Since $-7$ is left of $-2$ , followed by $0$ and $4$ , the correct order is $-7,-2,0,4$ .

Q23. Four locations have elevations represented by $-12$ , $5$ , $-3$ , and $0$ meters. Which location has the second-lowest elevation?

A.The location at

-12

meters

B.The location at

-3

meters ✅

C.The location at

0

meters

D.The location at

5

meters

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Ordering the elevations from lowest to highest gives $-12,-3,0,5$ . The second value in this ordering is $-3$ , so the location represented by $-3$ meters has the second-lowest elevation.

Q24. A student orders $-15,-6,-10,2$ as $-15,-6,-10,2$ . What is the most important error in the student's reasoning?

A.The positive integer must always come first

B.Negative integers must be ordered by their distance from zero in reverse size ✅

C.Zero should always appear between every pair of negative integers

D.The largest absolute value must always be the greatest number

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For negative integers, a number with greater distance from zero is actually smaller. Since $-15<-10<-6$ , the correct order is $-15,-10,-6,2$ . The student incorrectly treated larger absolute values as larger numbers.

Q25. A number line contains $A=-9$ , $B=-4$ , $C=1$ , and $D=6$ . A token starts at $B$ , moves $7$ units left, and then $12$ units right. Which labeled point does it reach?

A

B

C

✅

D

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The token starts at $-4$ . Moving $7$ units left gives $-11$ . Moving $12$ units right gives $1$ . Since $C=1$ , the token finishes at point $C$ . This requires tracking signed movement rather than simply comparing labels.

Q26. A weather station records temperatures of $-8^\circ\text{C}$ , $-2^\circ\text{C}$ , $3^\circ\text{C}$ , and $-11^\circ\text{C}$ . Which pair represents the greatest increase when moving from the colder temperature to the warmer temperature?

-11^\circ\text{C}

3^\circ\text{C}

✅

-8^\circ\text{C}

-2^\circ\text{C}

-11^\circ\text{C}

-2^\circ\text{C}

-8^\circ\text{C}

3^\circ\text{C}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The increase between two temperatures is found by subtracting the lower value from the higher value. From $-11$ to $3$ , the increase is $14^\circ\text{C}$ , which is greater than the other possible increases.

Q27. Which statement correctly compares $-18$ , $-9$ , and $4$ ?

-18>-9>4

-9>-18>4

4>-9>-18

✅

4>-18>-9

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Moving from left to right on the number line means values increase. Thus $4$ is greatest, $-9$ is next, and $-18$ is smallest. Therefore, the correct comparison is $4>-9>-18$ .

Q28. Five consecutive integer positions are occupied by numbers whose sum is $-35$ . If the integers are equally spaced on a number line, which is the greatest integer?

-9

-7

-5

✅

-3

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Five consecutive integers can be represented as $n-2,n-1,n,n+1,n+2$ . Their sum is $5n=-35$ , so $n=-7$ . The five integers are $-9,-8,-7,-6,-5$ , making $-5$ the greatest. Therefore, option C is correct.

Q29. What is the opposite of $-17$ , and what property do the two numbers share?

-17

; they are both negative

17

; they are the same distance from zero but on opposite sides ✅

0

; it lies between all opposite numbers

34

; their distances from zero are added

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The opposite of $-17$ is $17$ . Both numbers are exactly $17$ units from zero, but they lie on opposite sides of zero. This equal distance with reversed direction defines a pair of opposites.

Q30. A bank balance is represented by $-\<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 3: 60\̲)̲. A transaction…" style="color:#cc0000">60\). A transaction changes the balance to its opposite value. The customer then withdraws $\</span>15$ . What is the resulting balance?

A.45 dollars ✅

B.75 dollars

C.-45 dollars

D.-75 dollars

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The opposite of $-60$ is $60$ . After withdrawing $15$ , the balance becomes $60-15=45$ , so the mathematically correct answer is $45$ dollars. Therefore, option A is correct.

Q31. A student claims that the opposite of $8$ is $-8$ , but then says the opposite of $-8$ is also $-8$ . Which evaluation is most accurate?

A.Both statements are correct because opposites can repeat

B.The first statement is correct, but the second is incorrect because the opposite of

-8

8

✅

C.The first statement is incorrect because

8

has no opposite

D.Both statements are incorrect because only zero has an opposite

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The opposite of a positive number is negative, so the opposite of $8$ is $-8$ . Applying the same idea again reverses the sign, making the opposite of $-8$ equal to $8$ , not $-8$ .

Q32. On a number line, point $P$ is located at $-6$ . Point $Q$ is placed at the opposite of $P$ . Point $R$ is located $4$ units to the left of $Q$ . Where is $R$ located?

-10

-2

✅

C.2

D.-10

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The opposite of $P=-6$ is $Q=6$ . Moving $4$ units left from $6$ gives $6-4=2$ . Therefore, the correct location is $2$ , making option C correct.

Q33. A temperature is $-12^\circ\text{C}$ . Its opposite value is recorded, and then the temperature decreases by $5^\circ\text{C}$ . What is the final value?

-17^\circ\text{C}

-7^\circ\text{C}

7^\circ\text{C}

✅

17^\circ\text{C}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The opposite of $-12$ is $12$ . Decreasing that value by $5$ gives $12-5=7$ . Therefore, the final value is $7^\circ\text{C}$ , making option C correct.

Q34. Which pair of expressions produces the same result?

A.The opposite of

9

and

-9

✅

B.The opposite of

-14

and

14

C.The opposite of

5

and

5

D.The opposite of

0

and

1

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The opposite of $9$ is $-9$ , so both expressions represent the same value. Similarly, the opposite of $-14$ is $14$ , but those are different values. Zero is its own opposite, while $1$ is not.

Q35. Two integers $a$ and $b$ are opposites. Their sum is then multiplied by $25$ . What must the final result be, regardless of the value of $a$ ?

-25

B.0 ✅

C.25

D.It depends on the value of

a

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: If $a$ and $b$ are opposites, then $b=-a$ . Their sum is $a+(-a)=0$ . Multiplying this result by $25$ still gives $0$ , so the final result does not depend on the particular integer chosen.

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