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Add and Subtract Rational Numbers - 7th Grade Math

  1. During a science lab, the temperature changed by −8∘-8^\circ then +15∘+15^\circ. What is −8+15-8+15?
    Answer: 77 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.
  2. 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).
    Answer: −2.25-2.25 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.
  3. Use properties to simplify and evaluate: −1.5−34-1.5 - \frac{3}{4}. (Hint: rewrite subtraction as addition and/or convert 34\frac{3}{4} to a decimal.)
    Answer: −2.25-2.25 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 −1.5−34-1.5 - \frac{3}{4}, first rewrite as addition: −1.5+(−34)-1.5 + (-\frac{3}{4}). Convert 34\frac{3}{4} to decimal: 34=0.75\frac{3}{4} = 0.75. Now we have −1.5+(−0.75)-1.5 + (-0.75), which is negative+negative. Add magnitudes: 1.5+0.75=2.251.5 + 0.75 = 2.25, and keep the negative sign: −2.25-2.25. A student might make a conversion error (34≠0.75\frac{3}{4} \neq 0.75) or a sign error. Process: (1) rewrite subtraction as addition, (2) convert to common form (34=0.75\frac{3}{4} = 0.75), (3) identify signs (both negative), (4) apply rules (add magnitudes, keep negative), (5) calculate: 1.5+0.75=2.251.5 + 0.75 = 2.25, so answer is −2.25-2.25.
  4. Rewrite subtraction as addition, then evaluate: 12−(−5)12 - (-5).
    Answer: 1717 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.
  5. A student says: "−6−9=3-6 - 9 = 3 because subtracting makes it bigger." Evaluate −6−9-6 - 9 to check the claim (rewrite subtraction as addition if helpful).
    Answer: −15-15 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 −6−9-6 - 9, we rewrite subtraction as addition: −6+(−9)-6 + (-9). Now we have negative+negative, so we add magnitudes (6 + 9 = 15) and keep the negative sign: −15-15. The student's claim that "subtracting makes it bigger" resulting in 3 is incorrect - when subtracting a positive from a negative, we move further negative, not toward positive. Process: (1) rewrite subtraction as addition (−9-9 becomes +(−9)+(-9)), (2) identify signs (both negative), (3) apply rules (same signs: add magnitudes keep sign), (4) calculate: -6+(-9)=-15.
  6. A student tracks points in a game. Use properties of addition to calculate efficiently: 47+(−18)+347+(-18)+3.​
    Answer: 3232 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.
  7. 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).​
    Answer: −2.25-2.25 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.
  8. 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}.
    Answer: −2.25-2.25 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.
  9. A swimmer's time improves by subtracting a negative amount. Evaluate: 5.2−(−1.5)5.2 - (-1.5).
    Answer: 6.76.7 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.
  10. A recipe uses fractional cups of ingredients. Calculate: 12+(−34)\frac{1}{2}+\left(-\frac{3}{4}\right)
    Answer: −14-\frac{1}{4} Adding 1/2 and negative 3/4 requires a common denominator, so 1/2 is rewritten as 2/4, and 2/4 plus negative 3/4 equals negative 1/4, matching choice B. Choice A, negative 5/4, comes from adding the numerators and denominators separately instead of finding a common denominator first. Choice C, 2/5, and choice D, 1/4, both drop the negative sign somewhere in the process, which changes the sign of the final answer. Rewriting subtraction as addition of a negative, and always finding a common denominator before combining fractions, are the two steps that keep this kind of problem from going wrong.
  11. Verify whether the statement is true. Evaluate the left side using subtraction as addition: Is 12−(−5)=712-(-5)=7?
    Answer: No, because 12−(−5)=1712-(-5)=17. 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.
  12. A runner's elevation changes are recorded as integers. Evaluate: −6−9-6-9 (rewrite subtraction as addition if helpful).
    Answer: −15-15 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.
  13. A student tracks points in a game. Use properties of addition to calculate efficiently: 47+(−18)+347+(-18)+3.
    Answer: 3232 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 47 + (-18) + 3, use commutative and associative properties to group 47 + 3 first as 50, then 50 + (-18) = 32. A common error is not grouping strategically, like adding 47 + (-18) = 29 then +3=32 but missing the efficiency, or sign error like treating -18 as +18 giving 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.
  14. 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?
    Answer: Associative property; final pH is 4.74.7 because the grouping of additions can be changed without affecting the result 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.
  15. 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?
    Answer: Both commutative and associative properties are used; total change is −2-2 points from the regrouped calculation 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.
  16. A recipe uses fractional cups of ingredients. Calculate: 12+(−34)\frac{1}{2}+\left(-\frac{3}{4}\right).
    Answer: −14-\frac{1}{4} 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.
  17. A runner's elevation changes are recorded as integers. Evaluate: −6−9-6-9 (rewrite subtraction as addition if helpful).​
    Answer: −15-15 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.
  18. A student measures two changes in a water level in fractions of a meter. Evaluate: −23−16-\frac{2}{3}-\frac{1}{6}.
    Answer: −56-\frac{5}{6} Rewriting the subtraction as addition, negative 2/3 minus 1/6 becomes negative 2/3 plus negative 1/6, and converting both fractions to a common denominator of 6 gives negative 4/6 plus negative 1/6, which equals negative 5/6, matching choice B. Choice A, positive 5/6, has the correct value but the wrong sign, likely from dropping the negative sign during the calculation. Choice C, negative 1/2, and choice D, positive 1/2, both come from subtracting the numerators and denominators directly instead of finding a common denominator first. Since both terms being combined are negative, the result must also be negative, and adding their magnitudes together after finding a common denominator gives the correct value.
  19. A student measures two changes in a water level in fractions of a meter. Evaluate: −23−16-\frac{2}{3}-\frac{1}{6}.​
    Answer: −56-\frac{5}{6} 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 -2/3 -1/6 as -2/3 + (-1/6), common denominator 6: -4/6 + (-1/6)=-5/6. A common error is subtracting without common denominator, like -2/3 -1/6 as -(2-1)/(3-6)= -1/-3=1/3, or missing negative sign to get 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.
  20. Simplify using a common denominator: 12+(−34)\frac{1}{2} + \left(-\frac{3}{4}\right).
    Answer: −14-\frac{1}{4} 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=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). For 12+(−34)\frac{1}{2} + (-\frac{3}{4}), we need a common denominator: 12=24\frac{1}{2} = \frac{2}{4}. Now we have 24+(−34)\frac{2}{4} + (-\frac{3}{4}), which is positive+negative with different signs. We subtract the smaller magnitude from the larger: 34−24=14\frac{3}{4} - \frac{2}{4} = \frac{1}{4}, and since 34>24\frac{3}{4} > \frac{2}{4} and the larger term is negative, the result is −14-\frac{1}{4}. A student might forget to find a common denominator and incorrectly compute 12+(−34)=−26\frac{1}{2} + (-\frac{3}{4}) = -\frac{2}{6} or make a sign error. Process: (1) find common denominator (4), (2) convert fractions (12=24\frac{1}{2} = \frac{2}{4}), (3) identify signs (positive+negative), (4) apply rules (subtract magnitudes, use sign of larger), (5) calculate: 24−34=−14\frac{2}{4} - \frac{3}{4} = -\frac{1}{4}.

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