7th Grade Math Quiz: Add And Subtract Rational Numbers

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 7th Grade 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.

Question 1

1. Only commutative property is used; total change is $-2$ points because terms were reordered strategically
2. Only associative property is used; total change is $+2$ points because grouping doesn't affect the sum
3. Both commutative and associative properties are used; total change is $-2$ points from the regrouped calculation (correct answer)
4. Both commutative and associative properties are used; total change is $+2$ 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 $+24 + (-18) + (+15) + (-31) + (+8)$. Alex wants to group it as $(24 + 15 + 8) + (-18 + (-31))$. 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: $(24 + 15 + 8) + (-18 + (-31)) = 47 + (-49) = -2$ 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 $+2$ instead of $-2$. The correct answer is C because both properties are used and the total change is $-2$ 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 2

1. Calculate $-125 + (-78 + 45) + (-156 + 67)$ by grouping each pair of opposite movements together
2. Calculate $-125 + (-78 - 156) + (45 + 67)$ by separating all downward and upward movements completely
3. Calculate $(-125 - 78 - 156) + (45 + 67)$ by combining initial depth with all downward movements first
4. Calculate $-125 + (45 + 67) + (-78 - 156)$ 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 $(45 + 67 = 112)$ and downward movements $(-78 - 156 = -234)$, giving $-125 + 112 + (-234) = -247$ 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 3

1. $(\frac{3}{4} - \frac{1}{2}) + (\frac{2}{3} - \frac{5}{6})$, because it pairs fractions with related denominators for easier computation (correct answer)
2. $(\frac{3}{4} + \frac{2}{3}) - (\frac{5}{6} + \frac{1}{2})$, because it separates positive and negative terms into distinct groups
3. $(\frac{3}{4} - \frac{5}{6}) + (\frac{2}{3} - \frac{1}{2})$, because it maintains the original order while creating manageable pairs
4. $(\frac{3}{4} + \frac{5}{6}) - (\frac{2}{3} + \frac{1}{2})$, because it groups fractions with similar numerators for simplified calculation

Explanation: Choice A pairs $\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}$ and $\frac{2}{3} - \frac{5}{6} = \frac{4}{6} - \frac{5}{6} = -\frac{1}{6}$, 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 4

1. Commutative property; final pH is $4.7$ because the order of operations has been rearranged appropriately
2. Associative property; final pH is $4.7$ because the grouping of additions can be changed without affecting the result (correct answer)
3. Distributive property; final pH is $5.5$ because negative values are distributed across the grouped terms correctly
4. Associative property; final pH is $5.5$ because regrouping the terms allows for more efficient calculation methods

Explanation: The associative property allows changing the grouping of additions. The calculation becomes: $7.2 + (-1.8 + 0.6) + (-2.4 + 1.1) = 7.2 + (-1.2) + (-1.3) = 7.2 - 1.2 - 1.3 = 4.7$. Choice A incorrectly identifies the property as commutative. Choices C and D give the wrong final answer of 5.5.

Question 5

1. $-3$
2. $3$
3. $-15$ (correct answer)
4. $15$

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 6

1. $2.25$
2. $-0.75$
3. $-2.25$ (correct answer)
4. $-1.25$

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 ($\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$). 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: $\frac{1}{2} - \frac{3}{4} = \frac{1}{2} + (-\frac{3}{4}) = \frac{2}{4} + (-\frac{3}{4}) = -\frac{1}{4}$; or decimals: $5.2 - (-1.5) = 5.2 + 1.5 = 6.7$. Here, convert $-1.5 - \frac{3}{4}$ as $-1.5 + (-0.75) = -2.25$, or fractions: $-\frac{3}{2} - \frac{3}{4} = -\frac{6}{4} - \frac{3}{4} = -\frac{9}{4} = -2.25$. A common error is converting incorrectly, like $\frac{3}{4}$ as $0.34$ to get $-1.84$, or sign error to get $2.25$. Process: (1) rewrite subtractions as additions ($p - q \to p + (-q)$, 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 ($\frac{1}{2} = \frac{3}{6}$, $\frac{1}{3} = \frac{2}{6}$, 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: $-97 + 100 - 3 = 100 + (-97) + (-3)$ 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

1. $-3.7$
2. $6.7$ (correct answer)
3. $-6.7$
4. $3.7$

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 $5.2 - (-1.5)$, we apply the rule that subtracting a negative equals adding a positive: $5.2 - (-1.5) = 5.2 + 1.5$. Now we have positive+positive, so add magnitudes: $5.2 + 1.5 = 6.7$. A student might forget the double negative rule and compute $5.2 - 1.5 = 3.7$, or make a sign error. Process: (1) rewrite subtraction of negative as addition ($-(-1.5) = +1.5$), (2) identify signs (both positive), (3) apply rules (add magnitudes, keep positive), (4) calculate: 5.2+1.5=6.7.

Question 8

1. $2.25$
2. $-0.75$
3. $-1.25$
4. $-2.25$ (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.

Question 9

1. $6.3$
2. $-0.7$ (correct answer)
3. $0.7$
4. $-6.3$

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 10

1. $\frac{1}{4}$
2. $-\frac{1}{4}$ (correct answer)
3. $\frac{2}{5}$
4. $-\frac{2}{5}$

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 $\frac{1}{2} + (-\frac{3}{4})$, we need a common denominator: $\frac{1}{2} = \frac{2}{4}$. Now we have $\frac{2}{4} + (-\frac{3}{4})$, which is positive+negative with different signs. We subtract the smaller magnitude from the larger: $\frac{3}{4} - \frac{2}{4} = \frac{1}{4}$, and since $\frac{3}{4} > \frac{2}{4}$ and the larger term is negative, the result is $-\frac{1}{4}$. A student might forget to find a common denominator and incorrectly compute $\frac{1}{2} + (-\frac{3}{4}) = -\frac{2}{6}$ or make a sign error. Process: (1) find common denominator (4), (2) convert fractions ($\frac{1}{2} = \frac{2}{4}$), (3) identify signs (positive+negative), (4) apply rules (subtract magnitudes, use sign of larger), (5) calculate: $\frac{2}{4} - \frac{3}{4} = -\frac{1}{4}$.

Question 11

1. $0.7$
2. $6.3$
3. $-6.3$
4. $-0.7$ (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 $-3.5 + 2.8$, we have negative+positive with different signs. Subtract the smaller magnitude from the larger: $3.5 - 2.8 = 0.7$. Since $|-3.5| > |2.8|$ and the larger magnitude is negative, the result is $-0.7$. 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 12

1. $-2.25$ (correct answer)
2. $-0.75$
3. $2.25$
4. $-1.25$

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 $-1.5 - \frac{3}{4}$, first rewrite as addition: $-1.5 + (-\frac{3}{4})$. Convert $\frac{3}{4}$ to decimal: $\frac{3}{4} = 0.75$. Now we have $-1.5 + (-0.75)$, which is negative+negative. Add magnitudes: $1.5 + 0.75 = 2.25$, and keep the negative sign: $-2.25$. A student might make a conversion error ($\frac{3}{4} \neq 0.75$) or a sign error. Process: (1) rewrite subtraction as addition, (2) convert to common form ($\frac{3}{4} = 0.75$), (3) identify signs (both negative), (4) apply rules (add magnitudes, keep negative), (5) calculate: $1.5 + 0.75 = 2.25$, so answer is $-2.25$.

Question 13

1. $32$ (correct answer)
2. $26$
3. $68$
4. $-32$

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 $47 + (-18) + 3$, we can use the commutative property to rearrange as $47 + 3 + (-18)$, then use the associative property to group as $(47 + 3) + (-18) = 50 + (-18)$. Since we have positive+negative with different signs, we subtract magnitudes: $50 - 18 = 32$, and since 50 > 18, the result is positive: $32$. A student might incorrectly add all magnitudes getting 68, or make a sign error getting -32, but the correct answer using strategic grouping is 32. Process: (1) rewrite subtractions as additions (already done), (2) identify signs (47 positive, -18 negative, 3 positive), (3) rearrange strategically to make mental math easier (group 47+3=50), (4) apply rules (50+(-18) means subtract 18 from 50), (5) verify: 50-18=32.

Question 14

1. $-\frac{1}{4}$ (correct answer)
2. $\frac{1}{4}$
3. $-\frac{5}{4}$
4. $\frac{2}{5}$

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: $3+5=8$), 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 to common denominator 4: $1/2=2/4$, $+(-3/4)=2/4-3/4=-1/4$. A common error is adding without common denominator, like $1/2 + (-3/4)$ as $(1-3)/(2+4)=-2/6=-1/3$, or sign error to get $1/4$. Process: (1) rewrite subtractions as additions ($p-q\to p+(-q)$, 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 ($1/2=3/6$, $1/3=2/6$, 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 15

1. $7$ (correct answer)
2. $-7$
3. $-23$
4. $23$

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 + 15, subtract magnitudes 15 - 8 = 7 and use the positive sign from the larger magnitude, resulting in 7. A common error is adding magnitudes with negative sign like - (8 + 15) = -23, or sign wrong like -15 + 8 = -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 16

1. $-2$
2. $-18$
3. $2$ (correct answer)
4. $18$

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 17

1. $\frac{1}{6}$
2. $-\frac{1}{2}$
3. $-\frac{5}{6}$ (correct answer)
4. $\frac{5}{6}$

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 $-\frac{2}{3} - \frac{1}{6}$, first rewrite as addition: $-\frac{2}{3} + (-\frac{1}{6})$. Find common denominator: $-\frac{2}{3} = -\frac{4}{6}$. Now we have $-\frac{4}{6} + (-\frac{1}{6})$, which is negative+negative. Add magnitudes: $\frac{4}{6} + \frac{1}{6} = \frac{5}{6}$, and keep the negative sign: $-\frac{5}{6}$. A student might forget the common denominator or make a sign error thinking negative minus negative becomes positive. Process: (1) rewrite subtraction as addition, (2) find common denominator (6), (3) convert fractions ($-\frac{2}{3} = -\frac{4}{6}$), (4) identify signs (both negative), (5) apply rules (add magnitudes, keep negative sign), (6) calculate: $-\frac{4}{6} + (-\frac{1}{6}) = -\frac{5}{6}$.

Question 18

1. $7$
2. $17$ (correct answer)
3. $-17$
4. $-7$

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 $12 - (-5)$, we apply the rule that subtracting a negative is the same as adding a positive: $12 - (-5) = 12 + 5$. Now we have positive+positive, so we add magnitudes and keep the positive sign: $12 + 5 = 17$. 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 ($-(-5)$ becomes $+5$), (2) identify signs (both positive now), (3) apply rules (same signs: add magnitudes keep sign), (4) calculate: 12+5=17.

Question 19

1. Yes, because $12-(-5)=7$.
2. Yes, because $12-(-5)=12-5$.
3. No, because $12-(-5)= -17$.
4. No, because $12-(-5)=17$. (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, verify 12 - (-5) by rewriting as 12 + 5 = 17, which is not equal to 7. The correct calculation is 12 + 5 = 17, so the statement is false because it equals 17, not 7. A common error is confusing subtraction of negative as subtraction without flipping, like 12 - 5 = 7, but that's incorrect here. 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 as 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 20

1. $-\dfrac{1}{6}$
2. $-\dfrac{1}{2}$
3. $-\dfrac{5}{6}$ (correct answer)
4. $\dfrac{5}{6}$

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 -2/3 - 1/6, rewrite as -2/3 + (-1/6), use common denominator 6 to get -4/6 + (-1/6) = -5/6. A common error is fraction addition without common denominator like -2/3 - 1/6 = -3/9 = -1/3, or sign wrong like positive 5/6. 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.