Hard GCSE Substitution Questions

Challenging, exam-style GCSE Substitution questions with worked solutions. Stretch yourself on the hardest substitution, powers, order of operations, negatives problems.

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GCSE Foundation34 questionsStep-by-step solutions
Question 1
6 markschallenging
Work out the value of 2a2−bc+d\frac{2a^2 - b}{c + d} when a=3a = 3, b=4b = 4, c=5c = 5 and d=2d = 2.
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Worked solution

  1. Write down the expression

    2a2−bc+d\frac{2a^2 - b}{c + d}

    Top and bottom each act as a bracket.

  2. Write down the values

    a=3, b=4, c=5, d=2a = 3,\ b = 4,\ c = 5,\ d = 2

    Substitute all four.

  3. Substitute

    2×32−45+2\frac{2 \times 3^2 - 4}{5 + 2}

    Replace every letter.

  4. Square a

    32=93^2 = 9

    Index on the top first.

  5. Multiply by 2

    2×9=182 \times 9 = 18

    The first part of the numerator.

  6. Finish the top

    18−4=1418 - 4 = 14

    Numerator is 14.

  7. Work out the bottom

    5+2=75 + 2 = 7

    Denominator is 7.

  8. Rewrite

    147\frac{14}{7}

    Now divide.

  9. Divide

    147=2\frac{14}{7} = 2

    Fourteen over seven is 2.

  10. Check

    2×7=142 \times 7 = 14

    Multiply back to confirm.

  11. Interpret

    the value is a whole number\text{the value is a whole number}

    The division is exact.

  12. Common mistake

    2×32≠622 \times 3^2 \neq 6^2

    Square before multiplying by 2.

  13. Order note

    top and bottom separately, then divide\text{top and bottom separately, then divide}

    The bar groups each part.

  14. Estimate

    147=2\frac{14}{7} = 2

    Exactly 2.

  15. State the answer

    22

    The value is 2.

Answer
22
Question 2
6 markschallenging
Work out the value of 2+3x22 + 3x^2 when x=4x = 4.
Show worked solution

Worked solution

  1. Read the expression

    2+3x22 + 3x^2

    A constant add three times x squared.

  2. Write down the value

    x=4x = 4

    Substitute for x.

  3. Substitute

    2+3×422 + 3 \times 4^2

    Replace x with 4.

  4. Decide the order

    index→×→+\text{index} \to \times \to +

    BIDMAS: power first, then multiply, then add.

  5. Work out the power

    42=164^2 = 16

    Square the 4 first.

  6. Rewrite

    2+3×162 + 3 \times 16

    Now the multiplication.

  7. Multiply

    3×16=483 \times 16 = 48

    Three lots of 16.

  8. Rewrite

    2+482 + 48

    Finally add.

  9. Add

    2+48=502 + 48 = 50

    Fifty.

  10. Rule out adding first

    (2+3)×16=80 is wrong(2 + 3) \times 16 = 80 \text{ is wrong}

    You must not add the 2 and 3 first.

  11. Rule out squaring the sum

    (2+3×4)2=196 is wrong(2 + 3 \times 4)^2 = 196 \text{ is wrong}

    The power sits on x only.

  12. Rule out 4 times 3 for the square

    3×12=36 uses the wrong square3 \times 12 = 36 \text{ uses the wrong square}

    x squared is 16, not 4 times 3.

  13. Check

    50−2=48=3×1650 - 2 = 48 = 3 \times 16

    Working backwards confirms it.

  14. Reflect

    powers bind before× and +\text{powers bind before} \times \text{ and } +

    Order of operations is everything here.

  15. State the answer

    5050

    The value is 50.

Answer
5050
Question 3
5 markschallenging
A taxi fare in pounds is C=3+2mC = 3 + 2m for a journey of mm miles. Work out the fare for a 77 mile journey.
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Worked solution

  1. Write down the formula

    C=3+2mC = 3 + 2m

    Cost in pounds for m miles.

  2. Identify the value

    m=7m = 7

    The journey is 7 miles.

  3. State what 2m means

    2m=2×m2m = 2 \times m

    A charge of 2 pounds per mile.

  4. Substitute

    C=3+2×7C = 3 + 2 \times 7

    Replace m with 7.

  5. Work out 2m

    2×7=142 \times 7 = 14

    Multiplication before addition.

  6. Rewrite

    C=3+14C = 3 + 14

    Now add the fixed charge.

  7. Add

    3+14=173 + 14 = 17

    Seventeen.

  8. Rewrite

    C=17C = 17

    Collect the result.

  9. Add units

    £17\pounds 17

    The fare is 17 pounds.

  10. Check

    17−3=14=2×717 - 3 = 14 = 2 \times 7

    Working backwards confirms it.

  11. Interpret the 3

    the 3 is a fixed charge\text{the } 3 \text{ is a fixed charge}

    It is paid whatever the distance.

  12. Common mistake

    (3+2)×7=35 is wrong(3 + 2) \times 7 = 35 \text{ is wrong}

    You must not add before multiplying.

  13. Estimate

    3+14=173 + 14 = 17

    The estimate matches.

  14. Reflect

    multiply the per-mile part first\text{multiply the per-mile part first}

    BIDMAS keeps the fixed charge separate.

  15. State the answer

    1717

    The fare is 17 pounds.

Answer
1717
Question 4
6 markschallenging
Work out the value of x3−xx2−1\frac{x^3 - x}{x^2 - 1} when x=3x = 3.
Show worked solution

Worked solution

  1. Write down the expression

    x3−xx2−1\frac{x^3 - x}{x^2 - 1}

    Top and bottom each act as a bracket.

  2. Write down the value

    x=3x = 3

    Substitute for x.

  3. Substitute

    33−332−1\frac{3^3 - 3}{3^2 - 1}

    Replace each x with 3.

  4. Cube on top

    33=273^3 = 27

    Three cubed is 27.

  5. Finish the top

    27−3=2427 - 3 = 24

    Numerator is 24.

  6. Square on the bottom

    32=93^2 = 9

    Three squared is 9.

  7. Finish the bottom

    9−1=89 - 1 = 8

    Denominator is 8.

  8. Rewrite

    248\frac{24}{8}

    Now divide.

  9. Divide

    248=3\frac{24}{8} = 3

    Twenty-four over eight is 3.

  10. Check

    3×8=243 \times 8 = 24

    Multiply back to confirm.

  11. Interpret

    the value is a whole number\text{the value is a whole number}

    The division is exact.

  12. Common mistake

    x3−xx2−1≠x3−x−x2+1\frac{x^3 - x}{x^2 - 1} \neq x^3 - x - x^2 + 1

    You cannot split a fraction bar like that.

  13. Estimate

    248=3\frac{24}{8} = 3

    Exactly 3.

  14. Reflect

    evaluate top and bottom, then divide\text{evaluate top and bottom, then divide}

    Treat the bar as grouping.

  15. State the answer

    33

    The value is 3.

Answer
33
Question 5
5 markschallenging
Kinetic energy is E=12mv2E = \frac{1}{2}mv^2. Work out EE when m=4m = 4 and v=3v = 3.
Show worked solution

Worked solution

  1. Write down the formula

    E=12mv2E = \frac{1}{2}mv^2

    Kinetic energy formula. Only v is squared.

  2. Write down the values

    m=4, v=3m = 4,\ v = 3

    Substitute both.

  3. Substitute

    E=12×4×32E = \frac{1}{2} \times 4 \times 3^2

    Replace m and v.

  4. Square v

    32=93^2 = 9

    Index first.

  5. Rewrite

    E=12×4×9E = \frac{1}{2} \times 4 \times 9

    Now the multiplications.

  6. Multiply m by v squared

    4×9=364 \times 9 = 36

    Four nines are 36.

  7. Take half

    12×36=18\frac{1}{2} \times 36 = 18

    Half of 36 is 18.

  8. Rewrite

    E=18E = 18

    Collect the result.

  9. Add units

    18 J18 \text{ J}

    Energy in joules.

  10. Common mistake

    (12×4×3)2≠18(\tfrac{1}{2} \times 4 \times 3)^2 \neq 18

    Only v is squared, not the whole product.

  11. Check

    18×2=36=4×918 \times 2 = 36 = 4 \times 9

    Doubling back confirms it.

  12. Order note

    index, then ×, then ×12\text{index, then } \times, \text{ then } \times \tfrac{1}{2}

    BIDMAS order used.

  13. Estimate

    about 12×40=20\text{about } \tfrac{1}{2} \times 40 = 20

    18 is close to the estimate.

  14. Reflect

    square only what the power sits on\text{square only what the power sits on}

    The power applies to v alone.

  15. State the answer

    1818

    So E=18E = 18.

Answer
1818

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