Year 8 Mathematics | Victorian Curriculum 2.0
Four operations with rationals
Topic 04 | Number & Algebra | Answer key

Year 8 core - answers

Fluency

Integer × and ÷

    1. −5-5
    2. 66 (negative times negative)
    3. −42-42
    4. 4040 (two negatives give a positive)
    5. −36-36
    6. −27-27 (three negative factors: odd, so negative)
    7. 1616 (four negative factors: even, so positive)
    8. −7-7
    9. 99 (two negatives cancel)
    10. −24-24 (three negative factors: odd)
    11. 11 (even power of −1-1)
    12. 44
Fluency

Negative fractions and decimals

    1. −16-\dfrac{1}{6}. Method: common denominator 66; −36+26-\dfrac{3}{6} + \dfrac{2}{6}.
    2. −710-\dfrac{7}{10}. Method: −410−310-\dfrac{4}{10} - \dfrac{3}{10}.
    3. −16-\dfrac{1}{6}. Method: −1272-\dfrac{12}{72}.
    4. 34\dfrac{3}{4}. Method: −56×−910=4560=34-\dfrac{5}{6} \times -\dfrac{9}{10} = \dfrac{45}{60} = \dfrac{3}{4}.
    5. −2-2
    6. −1.2-1.2
    7. 44
    8. 0.250.25
Reasoning

Order of operations

    1. −11-11.
    2. 2626.
    3. 33. Method: −14×8=−2-\dfrac{1}{4} \times 8 = -2; −2+5=3-2 + 5 = 3.
    4. 44. Method: −8−2=4\dfrac{-8}{-2} = 4.
    5. −32-32. Method: (3−7)2=16(3 - 7)^2 = 16; −2×16=−32-2 \times 16 = -32.
    6. 54\dfrac{5}{4}. Method: −12×(−4)=2-\dfrac{1}{2} \times (-4) = 2; −34+2=54-\dfrac{3}{4} + 2 = \dfrac{5}{4}.
Reasoning

Explain and spot the mistake

    1. −32=−(3×3)=−9-3^2 = -(3 \times 3) = -9, not 99. The power applies only to the 33, so the minus sign stays in front. (−3)2=9(-3)^2 = 9; that’s the version Sam was thinking of.
    2. Negative. There are three negative factors; three is odd.
    3. Dividing asks “how many of the second fit into the first”. Two negatives cancel because flipping the sign of both the “how many” and the “of what” gives the same answer.
    4. Correct. −6+2=−4-6 + 2 = -4; −4−2=2\dfrac{-4}{-2} = 2.
Problem solving

Applications

    1. Net change: 2.5×20−1.8×30=50−54=−42.5 \times 20 - 1.8 \times 30 = 50 - 54 = -4 m. The balloon is 44 m below the starting altitude.
    2. 00 °C. Method: sum =−4−1+3+5+2−2−3=0= -4 - 1 + 3 + 5 + 2 - 2 - 3 = 0; 0÷70 \div 7.
    3. Not back to original. Starting at 100100: drop gives 9696; then rise 4%4\% gives 99.8499.84. Net loss.
    4. 00. Method: −3+5+2−4=0-3 + 5 + 2 - 4 = 0.

Challenge - answers

Reasoning

Harder reasoning

    1. 1212. Method: top =(−8)×9=−72= (-8) \times 9 = -72; −72÷−6=12-72 \div -6 = 12.
    2. x=−12x = -12. Method: multiply both sides by −32-\dfrac{3}{2}.
    3. n=3n = 3. Method: (−n)3=−n3=−27(-n)^3 = -n^3 = -27, so n3=27n^3 = 27, so n=3n = 3.
    4. −415-\dfrac{4}{15}. Method: bracket =109−69=49= \dfrac{10}{9} - \dfrac{6}{9} = \dfrac{4}{9}; −35×49=−1245=−415-\dfrac{3}{5} \times \dfrac{4}{9} = -\dfrac{12}{45} = -\dfrac{4}{15}.
Year 8 Mathematics study companion | Answer key