Signed Numbers Lesson Test

Answer: A

Correct answer: −5. The signs differ, so subtract the absolute values:

The number with the larger absolute value, −9, is negative, so the sum is −5.

• A (correct): 9 − 4 = 5, and the answer takes the sign of −9, the number with the larger absolute value.
• B: This adds the absolute values as if both numbers were negative.
• C: This adds the absolute values and ignores both signs.
• D: The size is right, but the sum takes the sign of −9, which is negative.

Answer: D

Correct answer: −11. On the number line the number farthest left is smallest. Among negatives, the one with the bigger absolute value is smaller, so −11 is less than −6 and −1. The value |−14| equals 14, which is positive.

• A: −6 is to the right of −11, so it is greater than −11.
• B: −1 is closest to zero of the negatives, so it is the greatest of them.
• C: Absolute value is never negative; |−14| = 14, which is greater than every negative number.
• D (correct): It lies farthest left on the number line, so it is the least value.

Answer: D

Correct answer: 42. Multiply the absolute values:

The two factors have the same sign, so the product is positive: 42.

• A: This adds the absolute values instead of multiplying.
• B: This adds the two numbers instead of multiplying them.
• C: The size is right, but like signs give a positive product, not a negative one.
• D (correct): 6 × 7 = 42, and two negative factors give a positive product.

Answer: C

Correct answer: 7. The two negative signs cancel, so

• A: This treats −(−12) as −12, missing that the double negative is positive.
• B: This gives the right size with the wrong sign; 12 − 5 is positive.
• C (correct): −(−12) is 12, and 12 − 5 = 7.
• D: This adds 5 instead of subtracting it after rewriting −(−12) as 12.

Answer: D

Correct answer: -7 ÷ 0. Division by zero has no answer, so −7 ÷ 0 is undefined. The other expressions are fine:

• A: Any number times zero equals 0, which is defined.
• B: Zero divided by a nonzero number is simply 0.
• C: Subtracting −7 means adding 7, so the value is 7.
• D (correct): Dividing any number by zero is undefined.

Answer: B

Correct answer: 2. Rewrite subtraction of a negative as addition: 5 + 8 + (−11). Then

• A: This drops all the signs and adds 5 + 8 + 11.
• B (correct): 5 + 8 = 13, and 13 + (−11) = 2.
• C: This swaps the signs of both terms, computing 5 − 8 + 11.
• D: This subtracts 8 instead of adding it, giving 5 − 8 − 11.

Answer: A

Correct answer: −30. Count the negative factors first: there are three, an odd count, so the product is negative. The size is

so the answer is −30.

• A (correct): Three negative factors make the product negative, and the size is 30.
• B: This adds the factors, −3 + 2 − 5 − 1, instead of multiplying them.
• C: The size is right, but three negative factors is an odd count, so the sign is negative.
• D: This adds the numbers and also loses the sign.

Answer: C

Correct answer: −7. Without parentheses only 4 is squared, so

With parentheses the base is −3, so

• A: This makes both squares negative, using −16 + (−9).
• B: This treats $-4^2$ as $(-4{)}^2=16$ and adds 9.
• C (correct): $-4^2=-16$ and $(-3{)}^2=9$, so the sum is −7.
• D: This gets the size right but treats $-4^2$ as 16 and $(-3{)}^2$ as −9.

Answer: B

Correct answer: -1/8. Use the common denominator 8:

negative because the negative term has the larger absolute value.

• A: This adds the sizes 6/8 and 5/8 as if both numbers were negative.
• B (correct): −6/8 + 5/8 = −1/8; the negative term is larger in size.
• C: This adds numerators and denominators separately, (−3 + 5)/(4 + 8) = 2/12, which is not how fractions add.
• D: The size is right, but the sum takes the sign of −6/8, which is negative.

Answer: C

Correct answer: −0.6. Multiply first: two negative factors give a positive product, (−2.5)

• A: This reverses the subtraction, computing 1.6 − 1.
• B: This makes the product −1 instead of +1, then subtracts 1.6.
• C (correct): The product is +1, and 1 − 1.6 = −0.6.
• D: This adds 1.6 to the product instead of subtracting it.

Answer: D

Correct answer: 50 feet below sea level. Write below sea level and descents as negatives: −42 + 17 + (−25). Then

so she is 50 feet below sea level.

• A: The size is right, but the result is negative, so she is below sea level.
• B: This treats the 17-foot rise as another descent, giving −42 − 17 − 25.
• C: This reverses the moves, subtracting the 17-foot rise and adding the 25-foot descent.
• D (correct): −42 + 17 − 25 = −50, which is 50 feet below sea level.

Answer: C

Correct answer: −28. Exponent first:

so the first term is

The fraction is

• A: This gives the fraction a negative value, so it adds 10 instead of subtracting it.
• B: This treats $(-3{)}^2$ as −9, making the first term +18 instead of −18.
• C (correct): −2 × 9 = −18 and the fraction equals 10, so −18 − 10 = −28.
• D: This multiplies −2 by −3 before squaring, getting 36 instead of −18.

Answer: A

Correct answer: A drop of 6°F. The midnight reading is

Change = final − start =

so the temperature fell 6°F overall.

• A (correct): Midnight is −19°F, and −19 − (−13) = −6.
• B: This is only the change from dawn to noon; the evening drop is left out.
• C: The size is right, but final minus start is negative, so the temperature dropped.
• D: This subtracts 13 instead of subtracting −13, computing −19 − 13.

Answer: A

Correct answer: -(-2)³. Evaluate each one with care.

because only 3 is squared.

The greatest is 8.

• A (correct): $(-2{)}^3=-8$, and its opposite is 8, the largest value here.
• B: The absolute value is 10, and the minus sign outside makes it −10.
• C: This equals 7, which is positive but less than 8.
• D: Only 3 is squared, so this equals −9, not 9.

Answer: B

Correct answer: −1. Divide by multiplying by the reciprocal:

Subtracting −1/2 means adding 1/2:

• A: This drops the negative sign of the quotient, computing 3/2 + 1/2.
• B (correct): The quotient is −3/2, and −3/2 + 1/2 = −1.
• C: This subtracts 1/2 instead of adding it, missing that minus a negative is plus.
• D: This multiplies by 4/9 instead of its reciprocal, getting −8/27 + 1/2.

Answer: B

Correct answer: −19. When both numbers have the same sign, add their absolute values and keep the common sign:

• A: The sizes are added correctly, but two losses make a larger loss, so the sign must stay negative.
• B (correct): Both numbers are negative, so add 7 and 12 to get 19 and keep the negative sign.
• C: This subtracts the sizes, which is the rule for unlike signs, not like signs.
• D: This subtracts the sizes and also drops the negative sign.

Answer: B

Correct answer: 5. Change subtracting a negative into adding a positive:

Then

• A: This treats subtracting −15 as subtracting 15, so every term ends up negative.
• B (correct): Rewrite as −4 + 15 + (−6); then −4 + 15 = 11 and 11 − 6 = 5.
• C: This changes the sign of the −6 as well as the −15; only the subtracted number gets its sign changed.
• D: This subtracts 15 instead of adding it and turns the −6 into +6.

Answer: A

Correct answer: −8. Divide the absolute values,

The signs are different, so the quotient is negative: −8 feet per stage.

• A (correct): 48 ÷ 6 = 8, and a negative divided by a positive is negative.
• B: The size is right, but unlike signs give a negative quotient.
• C: This subtracts 6 from 48 instead of dividing.
• D: This adds the numbers instead of dividing them.

Answer: A

Correct answer: −36. With parentheses the base is −3. An odd power stays negative:

An even power is positive:

Then

• A (correct): (−3)³ = −27 and (−3)² = 9, so −27 − 9 = −36.
• B: This takes (−3)² as −9; a negative base with an even exponent is positive.
• C: This takes (−3)³ as +27, giving 27 − 9; an odd power of a negative number stays negative.
• D: This multiplies the base by the exponent, (−3)(3) − (−3)(2) = −9 + 6, instead of raising to a power.

Answer: D

Correct answer: −18. Powers first: in $-2^4$ only the 2 is raised, so it equals −16. Then multiply and divide from left to right: (−3)

Finally

• A: This treats −2⁴ as (−2)⁴ = 16; without parentheses only the 2 is raised to the power.
• B: This gets −16 but makes 12 ÷ (−6) positive; unlike signs give a negative quotient.
• C: This makes both mistakes: it squares the sign into +16 and makes the quotient +2.
• D (correct): −2⁴ = −16 and (−3)(−4) ÷ (−6) = 12 ÷ (−6) = −2, so −16 + (−2) = −18.

Answer: C

Correct answer: −3.2, −1.5, −0.4, 0.6. Negative numbers are less than positive ones, and among negatives the one farthest from zero is smallest. So

• A: This orders the negatives by size without signs; among negatives, a bigger absolute value is a smaller number.
• B: This is the order from greatest to least, the reverse of what is asked.
• C (correct): Moving left to right on the number line gives −3.2, then −1.5, then −0.4, then 0.6.
• D: This orders the numbers by absolute value, ignoring their signs.

Answer: A

Correct answer: −1.8. Count the negatives first: three, an odd number, so the answer is negative. Then multiply the sizes:

The product is −1.8.

• A (correct): Three negative factors give a negative product, and 0.6 × 1.5 × 2 = 1.8.
• B: The size is right, but an odd number of negative factors makes the product negative.
• C: The sign is right, but the decimal point is misplaced; 0.6 × 1.5 = 0.9, and 0.9 × 2 = 1.8.
• D: The sign is right, but the decimal places were dropped; the product is 1.8, not 18.

Answer: C

Correct answer: −9. Work inside the bars first: |−9| = 9 because −9 is 9 units from zero. The minus sign in front then takes the opposite: −9.

• A: This finds |−9| = 9 but ignores the minus sign written outside the bars.
• B: This treats the two minus signs as canceling to zero; −|−9| means the opposite of 9.
• C (correct): |−9| = 9, and the minus sign outside the bars makes the result −9.
• D: This squares 9; absolute value is a distance, not a power.

Answer: B

Correct answer: $37. Write deposits as positive and withdrawals as negative: −35 + 120 + (−48).

so the balance is $37.

• A: This treats the starting balance as +$35 instead of −$35.
• B (correct): −35 + 120 − 48 = 85 − 48 = 37 dollars.
• C: This adds the $48 check instead of subtracting it.
• D: The size is right, but −35 + 120 = 85 is positive, so the final balance is positive.

Answer: C

Correct answer: A rise of 255 feet. Depths below sea level are negative: start = −340, final = −85. Change = final − start =

A positive change is a rise of 255 feet.

• A: The size is right, but moving from −340 to −85 is moving up toward the surface.
• B: This adds 340 and 85 as if the sub crossed sea level; both positions are below the surface.
• C (correct): Change = final − start = −85 − (−340) = 255, a positive change, so it rose.
• D: This adds the depths and gets the direction wrong as well.

Answer: A

Correct answer: -7/12. Subtracting a negative means adding its opposite: $-\frac{5}{6}+\frac{1}{4}$. The common denominator is 12:

• A (correct): With denominator 12: −10/12 + 3/12 = −7/12.
• B: This subtracts 1/4 instead of adding it: −10/12 − 3/12.
• C: The size is right, but the negative part (10/12) is larger, so the sum is negative.
• D: This adds numerators and denominators directly, (−5 + 1)/(6 + 4), without a common denominator.

Answer: D

Correct answer: a²b. Settle each sign. Squaring a negative gives a positive, so $a^2$ is positive, and multiplying by positive b keeps it positive. The other expressions all combine one negative with one positive and come out negative.

• A: A negative times a positive is negative.
• B: Subtracting a positive from a negative moves farther left, so the result is negative.
• C: A negative divided by a positive is negative.
• D (correct): a² is positive because a negative times a negative is positive, and a positive times positive b stays positive.

Answer: B

Correct answer: −3. Parentheses:

Power:

Multiply and divide left to right:

Finally

• A: This squares −5 as −25, which makes the quotient −5 instead of 5.
• B (correct): (4 − 9)² = 25, −3 × 25 = −75, −75 ÷ (−15) = 5, and 5 + (−8) = −3.
• C: This gets 5 but takes (−2)³ as +8; an odd power of a negative stays negative.
• D: This gets −5 from the division and +8 from the cube, two sign errors.

Answer: D

Correct answer: 0 ÷ (−6). Zero divided by a nonzero number is zero, so

Dividing by zero, as in −6 ÷ 0, is undefined rather than zero.

• A: Division by zero is undefined, so this has no value at all.
• B: This equals −12; subtracting 6 from −6 moves farther left.
• C: (−6)(0) = 0, but adding 1 gives 1.
• D (correct): Zero divided by any nonzero number is zero.

Answer: A

Correct answer: −4°F. Rises are positive and drops are negative. Total rise =

so the evening temperature was −4°F.

• A (correct): −11 + 4 × 5 − 13 = −11 + 20 − 13 = −4.
• B: This starts at +11 instead of −11: 11 + 20 − 13 = 18.
• C: This adds only one hour's rise of 4 degrees instead of 4 × 5 = 20.
• D: This treats the rise of 20 degrees as a drop: −11 − 20 − 13.

Answer: A

Correct answer: 23. A number plus its opposite equals zero. The opposite of −23 is 23, so

• A (correct): 23 is the opposite of −23, and a number plus its opposite is zero.
• B: −23 + (−23) = −46, not 0.
• C: Adding 0 leaves −23 unchanged.
• D: −23 + 46 = 23, which overshoots zero by 23.

Answer: B

Correct answer: 5. Substitute with parentheses:

Then

• A: This subtracts 8 instead of subtracting −8; y³ is −8, so −y³ is +8.
• B (correct): x² = 9, 2xy = 12 and y³ = −8, so 9 − 12 − (−8) = 5.
• C: This takes 2xy as −12; two negatives multiply to a positive, so 2xy = 12.
• D: This takes x² as −9; squaring a negative number gives a positive.

Answer: D

Correct answer: −54. Multiply the sizes,

then apply the sign rule: different signs give a negative product, so the change is −54 gallons.

• A: The size is right, but unlike signs give a negative product.
• B: This adds −9 and −6 instead of multiplying.
• C: This adds −9 and 6 instead of multiplying.
• D (correct): 9 × 6 = 54, and a negative times a positive is negative.

Answer: D

Correct answer: −0.05. All four numbers are negative, so the greatest is the one closest to zero. Since 0.05 is the smallest absolute value, −0.05 is the greatest.

• A: It is to the left of −0.05 because it is farther from zero.
• B: Its larger absolute value makes it smaller, not larger, than −0.05.
• C: It is the farthest left of all, so it is the least of the four.
• D (correct): It is the closest to zero of the four negatives, so it lies farthest right on the number line.

Answer: C

Correct answer: 2/3. Divide by multiplying by the reciprocal of the divisor:

Both numbers are negative, so the quotient is positive.

• A: The size is right, but like signs give a positive quotient.
• B: This multiplies by 9/10 instead of by its reciprocal, 10/9.
• C (correct): 3/5 × 10/9 = 30/45 = 2/3, and two negatives give a positive quotient.
• D: This is the reciprocal of the correct answer; it flips the first fraction instead of the divisor.

Answer: D

Correct answer: −16. An odd power of −1 is −1, and an even power of −2 is positive:

The product is

• A: This makes (−1)⁷ positive; a negative base with an odd exponent stays negative.
• B: This multiplies −2 by 4 instead of raising −2 to the fourth power.
• C: This multiplies −2 by 4 and also loses the sign of (−1)⁷.
• D (correct): (−1)⁷ = −1 because the exponent is odd, (−2)⁴ = 16, and −1 × 16 = −16.

Answer: C

Correct answer: −2 dollars. Add the gains and losses:

so the total is

The average is

per day.

• A: The size is right, but the losses outweigh the gains, so the average is negative.
• B: This is the total change for all five days, not the average per day.
• C (correct): The changes sum to −10, and −10 ÷ 5 = −2.
• D: This adds all the sizes as losses (26) and divides by 5, ignoring the two gains.

Answer: B

Correct answer: -2⁴ × (-1)³. Find each sign before any arithmetic. In $-2^4\times (-1{)}^3$,

two negative factors give a positive product, 16. Each of the other expressions ends with an odd count of negatives.

• A: (−2)³ = −8 and (−1)⁴ = 1, so the product is −8.
• B (correct): −2⁴ = −16 and (−1)³ = −1, and two negatives multiply to +16.
• C: (−3)² = 9, and the minus sign in front makes it −9.
• D: Three negative factors give a negative product, −6.

Answer: C

Correct answer: −1.4. Rewrite the subtraction of a negative as addition: −4.75 + 2.3 + 1.05. Then

• A: This subtracts 1.05 instead of adding it.
• B: The size is right, but −2.45 + 1.05 is still below zero.
• C (correct): −4.75 + 2.3 = −2.45, and subtracting −1.05 adds 1.05, giving −1.4.
• D: This treats every term as negative: −4.75 − 2.3 − 1.05.