Middle School Math Quiz: Add And Subtract Rational Numbers
20 questions · exam conditions
0:00
Add And Subtract Rational NumbersQuestion 1 of 20

A submarine's depth changes during a mission: starts at −125-125 feet, goes down 7878 feet, up 4545 feet, down 156156 feet, and up 6767 feet. Using properties of operations, which calculation method would be most efficient for finding the final depth?

Calculate −125+(−78+45)+(−156+67)-125 + (-78 + 45) + (-156 + 67) by grouping each pair of opposite movements together
Calculate −125+(−78−156)+(45+67)-125 + (-78 - 156) + (45 + 67) by separating all downward and upward movements completely
Calculate (−125−78−156)+(45+67)(-125 - 78 - 156) + (45 + 67) by combining initial depth with all downward movements first
Calculate −125+(45+67)+(−78−156)-125 + (45 + 67) + (-78 - 156) by using commutative property to group upward movements together
← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Add And Subtract Rational Numbers

Practice Add And Subtract Rational Numbers in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Add And Subtract Rational Numbers, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

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.

All questions

Question 1

A submarine's depth changes during a mission: starts at −125-125 feet, goes down 7878 feet, up 4545 feet, down 156156 feet, and up 6767 feet. Using properties of operations, which calculation method would be most efficient for finding the final depth?

  1. Calculate −125+(−78+45)+(−156+67)-125 + (-78 + 45) + (-156 + 67) by grouping each pair of opposite movements together
  2. Calculate −125+(−78−156)+(45+67)-125 + (-78 - 156) + (45 + 67) by separating all downward and upward movements completely
  3. Calculate (−125−78−156)+(45+67)(-125 - 78 - 156) + (45 + 67) by combining initial depth with all downward movements first
  4. Calculate −125+(45+67)+(−78−156)-125 + (45 + 67) + (-78 - 156) by using commutative property to group upward movements together (correct answer)
Explanation: Choice D uses the commutative property to rearrange terms and group upward movements (45+67=112)(45 + 67 = 112) and downward movements (−78−156=−234)(-78 - 156 = -234), giving −125+112+(−234)=−247-125 + 112 + (-234) = -247 feet. This is most efficient for mental calculation. Choice A groups pairs but isn't as strategic. Choice B has incorrect notation. Choice C incorrectly combines the starting depth with movements.

Question 2

Jenny is working with the expression 34−56+23−12\frac{3}{4} - \frac{5}{6} + \frac{2}{3} - \frac{1}{2}. She wants to use properties of operations to group terms strategically. Which regrouping would be most efficient for mental calculation?

  1. (34−12)+(23−56)(\frac{3}{4} - \frac{1}{2}) + (\frac{2}{3} - \frac{5}{6}), because it pairs fractions with related denominators for easier computation (correct answer)
  2. (34+23)−(56+12)(\frac{3}{4} + \frac{2}{3}) - (\frac{5}{6} + \frac{1}{2}), because it separates positive and negative terms into distinct groups
  3. (34−56)+(23−12)(\frac{3}{4} - \frac{5}{6}) + (\frac{2}{3} - \frac{1}{2}), because it maintains the original order while creating manageable pairs
  4. (34+56)−(23+12)(\frac{3}{4} + \frac{5}{6}) - (\frac{2}{3} + \frac{1}{2}), because it groups fractions with similar numerators for simplified calculation
Explanation: Choice A pairs 34−12=34−24=14\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4} and 23−56=46−56=−16\frac{2}{3} - \frac{5}{6} = \frac{4}{6} - \frac{5}{6} = -\frac{1}{6}, making mental calculation easier with related denominators. Choice B incorrectly changes subtraction to addition. Choice C keeps the original problematic order. Choice D incorrectly changes the signs and doesn't create easier computations.

Question 3

A phone battery percentage changes by decimals. Evaluate: −3.5+2.8-3.5+2.8.

  1. 6.36.3
  2. −0.7-0.7 (correct answer)
  3. 0.70.7
  4. −6.3-6.3
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). Subtraction as addition: p-q=p+(-q) (7-4=7+(-4)=3, 5-(-2)=5+2=7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+a), group (associative: (a+b)+c=a+(b+c)), strategically (47+3+(-18) group as (47+3)+(-18)=50-18=32 easier mental math). Fractions: common denominator (1/2+1/3=3/6+2/6=5/6). For example, calculate -8+15-5, rewrite: -8+15+(-5) (subtraction as addition), rearrange: 15+(-8)+(-5) (positive first), group negatives: 15+(-8-5)=15+(-13)=2; or fractions: 1/2-3/4=1/2+(-3/4)=2/4+(-3/4)=-1/4; or decimals: 5.2-(-1.5)=5.2+1.5=6.7. Here, add -3.5 + 2.8: subtract magnitudes 3.5-2.8=0.7, sign of larger (-3.5) gives -0.7. A common error is sign confusion, like adding as positive to get 6.3, or reversing signs to get 0.7. Process: (1) rewrite subtractions as additions (p-q→p+(-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (1/2=3/6, 1/3=2/6, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: -97+100-3=100+(-97)+(-3) group: 100+(-100)=0; or 27+(-18)+3=(27+3)+(-18)=30-18=12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.

Question 4

A temperature changes by −3.5∘C-3.5^\circ\text{C}, then by +2.8∘C+2.8^\circ\text{C}. Evaluate −3.5+2.8-3.5 + 2.8.

  1. 0.70.7
  2. 6.36.3
  3. −6.3-6.3
  4. −0.7-0.7 (correct answer)
Explanation: This problem tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). For −3.5+2.8-3.5 + 2.8, we have negative+positive with different signs. Subtract the smaller magnitude from the larger: 3.5−2.8=0.73.5 - 2.8 = 0.7. Since ∣−3.5∣>∣2.8∣|-3.5| > |2.8| and the larger magnitude is negative, the result is −0.7-0.7. A student might incorrectly add the magnitudes to get 6.3 or -6.3, or make a subtraction error. Process: (1) identify signs (negative+positive), (2) find magnitudes (3.5 and 2.8), (3) apply rules (subtract smaller from larger magnitude), (4) determine sign (larger magnitude was negative), (5) calculate: 3.5-2.8=0.7, so answer is -0.7.

Question 5

A recipe uses fractional cups of ingredients. Calculate: 12+(−34)\frac{1}{2}+\left(-\frac{3}{4}\right).

  1. −14-\frac{1}{4} (correct answer)
  2. 14\frac{1}{4}
  3. −54-\frac{5}{4}
  4. 25\frac{2}{5}
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=83+5=8), negative+negative (add magnitudes, negative result: −3+(−5)=−8-3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(−5)=38+(-5)=3, −8+5=−3-8+5=-3). Subtraction as addition: p−q=p+(−q)p-q=p+(-q) (7−4=7+(−4)=37-4=7+(-4)=3, 5−(−2)=5+2=75-(-2)=5+2=7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+aa+b=b+a), group (associative: (a+b)+c=a+(b+c)(a+b)+c=a+(b+c)), strategically (47+3+(−18)47+3+(-18) group as (47+3)+(−18)=50−18=32(47+3)+(-18)=50-18=32 easier mental math). Fractions: common denominator (1/2+1/3=3/6+2/6=5/61/2+1/3=3/6+2/6=5/6). For example, calculate −8+15−5-8+15-5, rewrite: −8+15+(−5)-8+15+(-5) (subtraction as addition), rearrange: 15+(−8)+(−5)15+(-8)+(-5) (positive first), group negatives: 15+(−8−5)=15+(−13)=215+(-8-5)=15+(-13)=2; or fractions: 1/2−3/4=1/2+(−3/4)=2/4+(−3/4)=−1/41/2-3/4=1/2+(-3/4)=2/4+(-3/4)=-1/4; or decimals: 5.2−(−1.5)=5.2+1.5=6.75.2-(-1.5)=5.2+1.5=6.7. Here, convert to common denominator 4: 1/2=2/41/2=2/4, +(−3/4)=2/4−3/4=−1/4+(-3/4)=2/4-3/4=-1/4. A common error is adding without common denominator, like 1/2+(−3/4)1/2 + (-3/4) as (1−3)/(2+4)=−2/6=−1/3(1-3)/(2+4)=-2/6=-1/3, or sign error to get 1/41/4. Process: (1) rewrite subtractions as additions (p−q→p+(−q)p-q\to p+(-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (1/2=3/61/2=3/6, 1/3=2/61/3=2/6, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: −97+100−3=100+(−97)+(−3)-97+100-3=100+(-97)+(-3) group: 100+(−100)=0100+(-100)=0; or 27+(−18)+3=(27+3)+(−18)=30−18=1227+(-18)+3=(27+3)+(-18)=30-18=12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.

Question 6

Use the associative and commutative properties to group terms and evaluate: −8+12−5+3-8+12-5+3.

  1. −2-2
  2. −18-18
  3. 22 (correct answer)
  4. 1818
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). Subtraction as addition: p-q=p+(-q) (7-4=7+(-4)=3, 5-(-2)=5+2=7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+a), group (associative: (a+b)+c=a+(b+c)), strategically (47+3+(-18) group as (47+3)+(-18)=50-18=32 easier mental math). Fractions: common denominator (1/2+1/3=3/6+2/6=5/6). For example, calculate -8+15-5, rewrite: -8+15+(-5) (subtraction as addition), rearrange: 15+(-8)+(-5) (positive first), group negatives: 15+(-8-5)=15+(-13)=2; or fractions: 1/2-3/4=1/2+(-3/4)=2/4+(-3/4)=-1/4; or decimals: 5.2-(-1.5)=5.2+1.5=6.7. To solve -8 + 12 - 5 + 3, rewrite -5 as +(-5), group using associative and commutative: (12 + 3) + (-8 -5) = 15 - 13 = 2. A common error is strategic rearranging wrong like -8 + 12 = 4, then 4 -5 = -1, -1 +3=2 but missing grouping, or sign wrong like all positive 18. Process: (1) rewrite subtractions as additions (p-q→p+(-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (1/2=3/6, 1/3=2/6, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: -97+100-3=100+(-97)+(-3) group: 100-100=0. Or: 27+(-18)+3=(27+3)+(-18)=30-18=12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.

Question 7

During a card game, Alex's score changes are: +24+24, −18-18, +15+15, −31-31, +8+8. He wants to calculate his total score change by grouping positive and negative values: (24+15+8)+(−18+(−31))(24 + 15 + 8) + (-18 + (-31)). What properties of operations justify this regrouping, and what is the total change?

  1. Only commutative property is used; total change is −2-2 points because terms were reordered strategically
  2. Only associative property is used; total change is +2+2 points because grouping doesn't affect the sum
  3. Both commutative and associative properties are used; total change is −2-2 points from the regrouped calculation (correct answer)
  4. Both commutative and associative properties are used; total change is +2+2 points from the strategic regrouping
Explanation: When you encounter problems involving reordering and regrouping numbers in addition, you need to identify which properties of operations allow these manipulations. Let's trace through Alex's regrouping step by step. The original expression is +24+(−18)+(+15)+(−31)+(+8)+24 + (-18) + (+15) + (-31) + (+8). Alex wants to group it as (24+15+8)+(−18+(−31))(24 + 15 + 8) + (-18 + (-31)). First, he reordered the terms to put positives together and negatives together - this uses the commutative property, which allows you to change the order of addends. Then he grouped them with parentheses - this uses the associative property, which allows you to change how terms are grouped. So both properties are needed. Now for the calculation: (24+15+8)+(−18+(−31))=47+(−49)=−2(24 + 15 + 8) + (-18 + (-31)) = 47 + (-49) = -2 Let's check why the other answers fail. Choice A claims only the commutative property is used, ignoring that parentheses represent regrouping (associative property). Choice B claims only the associative property is used, missing that terms were reordered first. Choice D gets the properties right but incorrectly calculates the total as +2+2 instead of −2-2. The correct answer is C because both properties are used and the total change is −2-2 points. Study tip: Remember that reordering terms requires the commutative property, while regrouping with parentheses requires the associative property. Most real problems involve both when you're strategically rearranging expressions.

Question 8

Rewrite subtraction as addition, then evaluate: 12−(−5)12 - (-5).

  1. 77
  2. 1717 (correct answer)
  3. −17-17
  4. −7-7
Explanation: This problem tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). For 12−(−5)12 - (-5), we apply the rule that subtracting a negative is the same as adding a positive: 12−(−5)=12+512 - (-5) = 12 + 5. Now we have positive+positive, so we add magnitudes and keep the positive sign: 12+5=1712 + 5 = 17. A common error is to think subtracting makes things smaller and get 7, or to mishandle the double negative and get -17. Process: (1) rewrite subtraction of negative as addition (−(−5)-(-5) becomes +5+5), (2) identify signs (both positive now), (3) apply rules (same signs: add magnitudes keep sign), (4) calculate: 12+5=17.

Question 9

Maria calculates her weekly budget changes: she spends $32.75 on groceries, earns $15.50 from babysitting, spends $8.25 on gas, and receives $12.00 as a gift. She rearranges the calculation as $(15.50+12.00)+(−32.75+(−8.25))(15.50 + 12.00) + (-32.75 + (-8.25)) $ to group income and expenses separately. What is her net change, and which property allows this rearrangement?

  1. Net change is −$13.50-\$13.50; commutative property allows rearranging the order of the rational number terms
  2. Net change is −$13.50-\$13.50; associative property allows regrouping additions without changing the final sum result (correct answer)
  3. Net change is +$13.50+\$13.50; commutative and associative properties work together to enable this strategic rearrangement
  4. Net change is +$13.50+\$13.50; distributive property allows factoring out common elements from the grouped expressions
Explanation: The calculation is (15.50+12.00)+(−32.75+(−8.25))=27.50+(−41.00)=−13.50(15.50 + 12.00) + (-32.75 + (-8.25)) = 27.50 + (-41.00) = -13.50. The associative property allows regrouping additions (treating subtraction as adding negative numbers). Choice A has the correct answer but wrong property. Choices C and D have the wrong sign for the net change.

Question 10

A runner's elevation change is recorded as −3.5+2.8-3.5+2.8 (in meters). What is the result?

  1. −6.3-6.3
  2. 6.36.3
  3. −0.7-0.7 (correct answer)
  4. 0.70.7
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). Subtraction as addition: p-q=p+(-q) (7-4=7+(-4)=3, 5-(-2)=5+2=7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+a), group (associative: (a+b)+c=a+(b+c)), strategically (47+3+(-18) group as (47+3)+(-18)=50-18=32 easier mental math). Fractions: common denominator (1/2+1/3=3/6+2/6=5/6). For example, calculate -8+15-5, rewrite: -8+15+(-5) (subtraction as addition), rearrange: 15+(-8)+(-5) (positive first), group negatives: 15+(-8-5)=15+(-13)=2; or fractions: 1/2-3/4=1/2+(-3/4)=2/4+(-3/4)=-1/4; or decimals: 5.2-(-1.5)=5.2+1.5=6.7. To solve -3.5 + 2.8, subtract magnitudes 3.5 - 2.8 = 0.7 and use the negative sign from the larger magnitude, resulting in -0.7. A common error is decimal arithmetic error like -3.5 + 2.8 = -6.3 or sign wrong like positive 0.7. Process: (1) rewrite subtractions as additions (p-q→p+(-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (1/2=3/6, 1/3=2/6, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: -97+100-3=100+(-97)+(-3) group: 100-100=0. Or: 27+(-18)+3=(27+3)+(-18)=30-18=12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.

Question 11

Simplify the mixed-form expression by converting 34\dfrac{3}{4} to a decimal (or −1.5-1.5 to a fraction): −1.5−34-1.5-\dfrac{3}{4}.

  1. 2.252.25
  2. −0.75-0.75
  3. −2.25-2.25 (correct answer)
  4. −1.25-1.25
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). Subtraction as addition: p-q=p+(-q) (7-4=7+(-4)=3, 5-(-2)=5+2=7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+a), group (associative: (a+b)+c=a+(b+c)), strategically (47+3+(-18) group as (47+3)+(-18)=50-18=32 easier mental math). Fractions: common denominator (1/2+1/3=3/6+2/6=5/6). For example, calculate -8+15-5, rewrite: -8+15+(-5) (subtraction as addition), rearrange: 15+(-8)+(-5) (positive first), group negatives: 15+(-8-5)=15+(-13)=2; or fractions: 1/2-3/4=1/2+(-3/4)=2/4+(-3/4)=-1/4; or decimals: 5.2-(-1.5)=5.2+1.5=6.7. To solve -1.5 - 3/4, convert 3/4 to 0.75, then -1.5 - 0.75 = -2.25, or convert -1.5 to -3/2 = -6/4, -6/4 - 3/4 = -9/4 = -2.25. A common error is sign wrong like -1.5 + 0.75 = -0.75, or arithmetic error like -1.5 - 0.75 = -1.25. Process: (1) rewrite subtractions as additions (p-q→p+(-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (1/2=3/6, 1/3=2/6, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: -97+100-3=100+(-97)+(-3) group: 100-100=0. Or: 27+(-18)+3=(27+3)+(-18)=30-18=12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.

Question 12

A student tracks points in a game. Use properties of addition to calculate efficiently: 47+(−18)+347+(-18)+3.​

  1. 3232 (correct answer)
  2. −32-32
  3. 2626
  4. 6868
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). Subtraction as addition: p-q=p+(-q) (7-4=7+(-4)=3, 5-(-2)=5+2=7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+a), group (associative: (a+b)+c=a+(b+c)), strategically (47+3+(-18) group as (47+3)+(-18)=50-18=32 easier mental math). Fractions: common denominator (1/2+1/3=3/6+2/6=5/6). For example, calculate -8+15-5, rewrite: -8+15+(-5) (subtraction as addition), rearrange: 15+(-8)+(-5) (positive first), group negatives: 15+(-8-5)=15+(-13)=2; or fractions: 1/2-3/4=1/2+(-3/4)=2/4+(-3/4)=-1/4; or decimals: 5.2-(-1.5)=5.2+1.5=6.7. Here, group 47+3 first to get 50, then 50+(-18)=32, using associative property for efficiency. A common error is not grouping strategically, like adding 47+(-18)=29 then +3=32, which works but is less efficient, or sign error like treating -18 as +18 to get 68. Process: (1) rewrite subtractions as additions (p-q→p+(-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (1/2=3/6, 1/3=2/6, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: -97+100-3=100+(-97)+(-3) group: 100+(-100)=0; or 27+(-18)+3=(27+3)+(-18)=30-18=12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.

Question 13

A stock's value changes over five days: −$2.75-\$2.75, +$4.25+\$4.25, −$1.50-\$1.50, −$3.80-\$3.80, +$2.15+\$2.15. An investor calculates the net change as (−2.75−1.50−3.80)+(4.25+2.15)(-2.75 - 1.50 - 3.80) + (4.25 + 2.15). The result is −$1.65-\$1.65. Which statement about this calculation is correct?

  1. The calculation is incorrect because the commutative property doesn't allow changing negative terms to subtraction
  2. The calculation violates the associative property because positive and negative terms cannot be grouped separately
  3. The calculation correctly applies properties of operations and the final answer of −$1.65-\$1.65 is accurate (correct answer)
  4. The calculation is correct in method but the arithmetic is wrong; the actual net change should be +$1.65+\$1.65
Explanation: When working with positive and negative numbers, you can rearrange and group terms using the properties of operations to make calculations easier. This problem tests whether you understand these properties and can verify arithmetic with signed numbers. The investor's approach is mathematically sound. By grouping negative changes (−2.75−1.50−3.80)(-2.75 - 1.50 - 3.80) and positive changes (4.25+2.15)(4.25 + 2.15) separately, they're applying the commutative and associative properties correctly. The negative changes total −$8.05-\$8.05, and the positive changes total +$6.40+\$6.40. Adding these gives −8.05+6.40=−$1.65-8.05 + 6.40 = -\$1.65. You can verify this by adding the original sequence in order: −2.75+4.25+(−1.50)+(−3.80)+2.15=−$1.65-2.75 + 4.25 + (-1.50) + (-3.80) + 2.15 = -\$1.65. Both methods yield the same result, confirming the calculation is correct. Choice A is wrong because the commutative property absolutely allows rearranging terms, including changing +(−1.50)+(-1.50) to −1.50-1.50. Choice B incorrectly claims the associative property is violated—grouping positive and negative terms separately is a perfectly valid application of this property. Choice D suggests the arithmetic is wrong and the answer should be positive $1.65, but this would mean the gains exceeded the losses, which isn't the case here. Remember: when working with signed numbers, you can rearrange and group terms to simplify calculations, but always double-check your arithmetic. Grouping like signs together often makes the math cleaner and reduces errors.

Question 14

Evaluate: 12−(−5)12-(-5).

  1. 1717 (correct answer)
  2. 77
  3. −17-17
  4. −7-7
Explanation: Subtracting a negative number is the same as adding its positive: 12-(-5)=12+5=17, matching choice A. Choice B mistakenly simplifies -(-5) to just -5, computing 12-5=7 instead of 12+5. Choice C keeps the correct magnitude but applies the wrong sign, giving -17. Choice D combines both errors, computing 12-5 and then applying the wrong sign to get -7.

Question 15

A hiker begins the day at an elevation of 1212 meters above sea level. A cave entrance nearby is at an elevation of −5-5 meters (5 meters below sea level). What is the difference in elevation between the hiker's starting point and the cave entrance? Evaluate 12−(−5)12-(-5) to find out.

  1. 77
  2. −17-17
  3. 1717 (correct answer)
  4. −7-7
Explanation: To find the difference in elevation, evaluate 12-(-5). Subtracting a negative number is the same as adding its positive: 12-(-5)=12+5=17 meters, matching choice C. Choice A mistakenly simplifies -(-5) to just -5, computing 12-5=7 instead of 12+5. Choice B keeps the correct magnitude but applies the wrong sign, giving -17. Choice D combines both errors, computing 12-5 and then applying the wrong sign to get -7.

Question 16

In a chemistry lab, a solution's pH changes through several reactions. Starting at pH 7.27.2, it decreases by 1.81.8, increases by 0.60.6, decreases by 2.42.4, and increases by 1.11.1. A student writes the calculation as 7.2+(−1.8+0.6)+(−2.4+1.1)7.2 + (-1.8 + 0.6) + (-2.4 + 1.1). What property justifies this grouping, and what is the final pH?

  1. Commutative property; final pH is 4.74.7 because the order of operations has been rearranged appropriately
  2. Associative property; final pH is 4.74.7 because the grouping of additions can be changed without affecting the result (correct answer)
  3. Distributive property; final pH is 5.55.5 because negative values are distributed across the grouped terms correctly
  4. Associative property; final pH is 5.55.5 because regrouping the terms allows for more efficient calculation methods
Explanation: The associative property allows changing the grouping of additions. The calculation becomes: 7.2+(−1.8+0.6)+(−2.4+1.1)=7.2+(−1.2)+(−1.3)=7.2−1.2−1.3=4.77.2 + (-1.8 + 0.6) + (-2.4 + 1.1) = 7.2 + (-1.2) + (-1.3) = 7.2 - 1.2 - 1.3 = 4.7. Choice A incorrectly identifies the property as commutative. Choices C and D give the wrong final answer of 5.5.

Question 17

A runner's elevation changes are recorded as integers. Evaluate: −6−9-6-9 (rewrite subtraction as addition if helpful).​

  1. −3-3
  2. 33
  3. −15-15 (correct answer)
  4. 1515
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). Subtraction as addition: p-q=p+(-q) (7-4=7+(-4)=3, 5-(-2)=5+2=7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+a), group (associative: (a+b)+c=a+(b+c)), strategically (47+3+(-18) group as (47+3)+(-18)=50-18=32 easier mental math). Fractions: common denominator (1/2+1/3=3/6+2/6=5/6). For example, calculate -8+15-5, rewrite: -8+15+(-5) (subtraction as addition), rearrange: 15+(-8)+(-5) (positive first), group negatives: 15+(-8-5)=15+(-13)=2; or fractions: 1/2-3/4=1/2+(-3/4)=2/4+(-3/4)=-1/4; or decimals: 5.2-(-1.5)=5.2+1.5=6.7. Here, rewrite -6-9 as -6+(-9), add magnitudes 6+9=15, keep negative sign to get -15. A common error is treating negative+negative as subtraction, like -6-9 as 6-9=-3, or forgetting the negative result to get 15. Process: (1) rewrite subtractions as additions (p-q→p+(-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (1/2=3/6, 1/3=2/6, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: -97+100-3=100+(-97)+(-3) group: 100+(-100)=0; or 27+(-18)+3=(27+3)+(-18)=30-18=12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.

Question 18

A student records a change in a bank of points in a classroom game. Evaluate: −1.5−34-1.5 - \frac{3}{4} (convert to decimals or fractions).

  1. 2.252.25
  2. −0.75-0.75
  3. −2.25-2.25 (correct answer)
  4. −1.25-1.25
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive + positive (add magnitudes, positive result: 3+5=83+5=8), negative + negative (add magnitudes, negative result: −3+(−5)=−8-3 + (-5) = -8), positive + negative or negative + positive (subtract smaller magnitude from larger, sign of larger: 8+(−5)=38 + (-5) = 3, −8+5=−3-8 + 5 = -3). Subtraction as addition: p−q=p+(−q)p - q = p + (-q) (7−4=7+(−4)=37 - 4 = 7 + (-4) = 3, 5−(−2)=5+2=75 - (-2) = 5 + 2 = 7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+aa + b = b + a), group (associative: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)), strategically (47+3+(−18)47 + 3 + (-18) group as (47+3)+(−18)=50−18=32(47 + 3) + (-18) = 50 - 18 = 32 easier mental math). Fractions: common denominator (12+13=36+26=56\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}). For example, calculate −8+15−5-8 + 15 - 5, rewrite: −8+15+(−5)-8 + 15 + (-5) (subtraction as addition), rearrange: 15+(−8)+(−5)15 + (-8) + (-5) (positive first), group negatives: 15+(−8−5)=15+(−13)=215 + (-8 - 5) = 15 + (-13) = 2; or fractions: 12−34=12+(−34)=24+(−34)=−14\frac{1}{2} - \frac{3}{4} = \frac{1}{2} + (-\frac{3}{4}) = \frac{2}{4} + (-\frac{3}{4}) = -\frac{1}{4}; or decimals: 5.2−(−1.5)=5.2+1.5=6.75.2 - (-1.5) = 5.2 + 1.5 = 6.7. Here, convert −1.5−34-1.5 - \frac{3}{4} as −1.5+(−0.75)=−2.25-1.5 + (-0.75) = -2.25, or fractions: −32−34=−64−34=−94=−2.25-\frac{3}{2} - \frac{3}{4} = -\frac{6}{4} - \frac{3}{4} = -\frac{9}{4} = -2.25. A common error is converting incorrectly, like 34\frac{3}{4} as 0.340.34 to get −1.84-1.84, or sign error to get 2.252.25. Process: (1) rewrite subtractions as additions (p−q→p+(−q)p - q \to p + (-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (12=36\frac{1}{2} = \frac{3}{6}, 13=26\frac{1}{3} = \frac{2}{6}, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: −97+100−3=100+(−97)+(−3)-97 + 100 - 3 = 100 + (-97) + (-3) group: 100+(−100)=0100 + (-100) = 0; or 27+(−18)+3=(27+3)+(−18)=30−18=1227 + (-18) + 3 = (27 + 3) + (-18) = 30 - 18 = 12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.

Question 19

A swimmer's time improves by subtracting a negative amount. Evaluate: 5.2−(−1.5)5.2 - (-1.5).

  1. −3.7-3.7
  2. 6.76.7 (correct answer)
  3. −6.7-6.7
  4. 3.73.7
Explanation: This problem tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). For 5.2−(−1.5)5.2 - (-1.5), we apply the rule that subtracting a negative equals adding a positive: 5.2−(−1.5)=5.2+1.55.2 - (-1.5) = 5.2 + 1.5. Now we have positive+positive, so add magnitudes: 5.2+1.5=6.75.2 + 1.5 = 6.7. A student might forget the double negative rule and compute 5.2−1.5=3.75.2 - 1.5 = 3.7, or make a sign error. Process: (1) rewrite subtraction of negative as addition (−(−1.5)=+1.5-(-1.5) = +1.5), (2) identify signs (both positive), (3) apply rules (add magnitudes, keep positive), (4) calculate: 5.2+1.5=6.7.

Question 20

A student records a change in a bank of points in a classroom game. Evaluate: −1.5−34-1.5-\frac{3}{4} (convert to decimals or fractions).​

  1. 2.252.25
  2. −0.75-0.75
  3. −1.25-1.25
  4. −2.25-2.25 (correct answer)
Explanation: This question tests adding and subtracting rational numbers (integers, fractions, decimals, positive/negative) using properties of operations strategically. Sign rules: positive+positive (add magnitudes, positive result: 3+5=8), negative+negative (add magnitudes, negative result: -3+(-5)=-8), positive+negative or negative+positive (subtract smaller magnitude from larger, sign of larger: 8+(-5)=3, -8+5=-3). Subtraction as addition: p-q=p+(-q) (7-4=7+(-4)=3, 5-(-2)=5+2=7 subtracting negative adds). Properties: rearrange (commutative: a+b=b+a), group (associative: (a+b)+c=a+(b+c)), strategically (47+3+(-18) group as (47+3)+(-18)=50-18=32 easier mental math). Fractions: common denominator (1/2+1/3=3/6+2/6=5/6). For example, calculate -8+15-5, rewrite: -8+15+(-5) (subtraction as addition), rearrange: 15+(-8)+(-5) (positive first), group negatives: 15+(-8-5)=15+(-13)=2; or fractions: 1/2-3/4=1/2+(-3/4)=2/4+(-3/4)=-1/4; or decimals: 5.2-(-1.5)=5.2+1.5=6.7. Here, convert -1.5 - 3/4 as -1.5 + (-0.75) = -2.25, or fractions: -3/2 - 3/4 = -6/4 - 3/4 = -9/4 = -2.25. A common error is converting incorrectly, like 3/4 as 0.34 to get -1.84, or sign error to get 2.25. Process: (1) rewrite subtractions as additions (p-q→p+(-q), makes all same operation), (2) identify signs (which positive, which negative), (3) apply rules (same signs: add magnitudes keep sign, different signs: subtract magnitudes use larger's sign), (4) for fractions: common denominators first (1/2=3/6, 1/3=2/6, then add/subtract), (5) for mixed forms: convert to one type (decimals or fractions), (6) use properties strategically (rearrange to make easier: group round numbers, opposites). Strategic examples: -97+100-3=100+(-97)+(-3) group: 100+(-100)=0; or 27+(-18)+3=(27+3)+(-18)=30-18=12. Mistakes: sign errors most common, fraction operations without denominators, not using properties for efficiency, arithmetic errors tracking negatives.