[discia] GCSE Maths › Algebra › Simplifying Algebra › Practise
  1. Either paper 2 marks

    Simplify \(8s + 8t + 4 - 6s - 3 +t\)

    Show answer
    \(2s + 9t + 1\)

    Collect each kind of term separately. Only terms with exactly the same letter can be added together.

    \(s\) terms: \(8s - 6s = 2s\)

    \(t\) terms: \(8t +t = 9t\)

    Numbers: \(4 - 3 = 1\)

    So the expression simplifies to \(2s + 9t + 1\)

    • M1collects two kinds of term correctly, e.g. \(2s\), \(9t\) or \(1\)
    • A1fully simplified \(2s + 9t + 1\)
  2. Either paper 2 marks
    (a)

    Simplify \(3a \times 6b\)

    (b)

    Simplify \(2a \times 6a\)

    Show answer
    (a) \(18ab\)
    (b) \(12a^2\)

    Multiply the numbers, then write the letters next to each other.

    \(3 \times 6 = 18\), so \(3a \times 6b = 18ab\)

    \(2 \times 6 = 12\), and \(a \times a = a^2\).

    \(2a \times 6a = 12a^2\)

  3. Either paper 3 marks

    Simplify fully.

    (a)

    \(c \times c \times c\)

    (b)

    \(p + p\)

    (c)

    \(b + b + b + b + 3 \times a \times a\)

    Show answer
    (a) \(c^{3}\)
    (b) \(2p\)
    (c) \(4b + 3a^{2}\)

    Multiplying repeats the letter as a factor, so 3 copies of c multiplied make c to the power 3: \(c^{3}\).

    Adding repeats the letter, so 2 copies of p added make 2 lots of p: \(2p\).

    The multiplied piece and the added piece are different, so deal with them separately and keep the plus between them. The multiplied piece is \(3 \times a \times a = 3a^{2}\) and the added piece is \(b + b + b + b = 4b\), so the line simplifies to \(4b + 3a^{2}\).

  4. Either paper 1 mark

    Circle the expression that is equal to \(8m + 3 - 2m\)

    \(9m\) \qquad \(11 - 2m\) \qquad \(10m + 3\) \qquad \(6m + 3\)

    Show answer
    \(6m + 3\)

    Collect the m-terms and keep the number apart: it is not a like term, so it is never added to them.

    \(8m + 3 - 2m = 8m - 2m + 3\)

    Add the numbers in front of the letter:

    \(8 - 2 = 6\)

    So the expression to circle is \(6m + 3\).

  5. Either paper 2 marks

    Simplify \(9n^2 + 3n -n^2 - 6n\)

    Show answer
    \(8n^2 - 3n\)

    \(n^2\) and \(n\) are not like terms, so they can never be added together. Deal with them separately.

    \(n^2\) terms: \(9n^2 -n^2 = 8n^2\)

    \(n\) terms: \(3n - 6n = -3n\)

    So the expression simplifies to \(8n^2 - 3n\)

    • M1collects \(n^2\) terms \(8n^2\) or \(n\) terms \(-3n\)
    • A1fully simplified \(8n^2 - 3n\)
  6. Either paper 2 marks

    Simplify \(\frac{ 12k^{ 6 } }{ 2k^{ 3 } }\)

    Show answer
    \(6k^{ 3 }\)

    Divide the numbers, then subtract the indices.

    \(12 \div 2 = 6\)

    \(k^{ 6 } \div k^{ 3 } = k^{ 6 - 3 } = k^{ 3 }\)

    \(\frac{ 12k^{ 6 } }{ 2k^{ 3 } } = 6k^{ 3 }\)

    • M1divides the numbers \(12 \div 2 = 6\) or subtracts the indices \(6 - 3 = 3\)
    • A1answer \(6k^{ 3 }\)
  7. Either paper 4 marks

    Here is an expression.

    \(8s + 3t - 5s + 2t\)

    (a)

    Simplify the expression.

    (b)

    Work out the value of the expression when \(s = 2\) and \(t = -7\).

    Show answer
    (a) \(3s + 5t\)
    (b) \(-29\)

    \(s\) terms: \(8s - 5s = 3s\)

    \(t\) terms: \(3t + 2t = 5t\)

    So the expression simplifies to \(3s + 5t\)

    Substituting into the simplified form is quicker than into the original.

    \(3s = 3 \times 2 = 6\)

    \(5t = 5 \times (-7) = -35\)

    \(6 - 35 = -29\)

    • (a) M1collects like terms for one letter, \(3s\) or \(5t\)
    • (a) A1both correct: \(3s + 5t\)
    • (b) M1substitutes into the simplified expression: \(3 \times 2\) and \(5 \times (-7)\)
    • (b) A1value \(-29\)
  8. Either paper 2 marks

    When \(4p + 3q + kp + 7q\) is simplified, the result is \(3p + 10q\).

    Work out the value of \(k\).

    Show answer
    \(-1\)

    Only the \(p\) terms can tell you anything about \(k\), so collect those.

    \(4p + kp = 3p\)

    So \(4 + k = 3\), giving

    \(k = 3 - 4 = -1\)

    • M1collects the \(p\) terms, \(4 + k = 3\)
    • A1\(k = -1\)
  9. Either paper 4 marks
    Match each expression on the left with the simplified expression on the right. One of the simplified expressions is not needed.
    ExpressionSimplified expression
    \(9(n + 4) - 36 - 2n\)A \(25n^2\)
    \(12n + 4 - 4 - n\)B \(13n\)
    \(3n + 16n - 6n\)C \(25n\)
    \(5n \times 5n\)D \(7n\)
    E \(11n\)
    Write the letter of the matching simplified expression for each expression.
    Show answer
    Expression \(3n + 16n - 6n\) matches B; \(5n \times 5n\) matches A; \(9(n + 4) - 36 - 2n\) matches D; \(12n + 4 - 4 - n\) matches E. Option C is not needed.

    Simplify each expression, then find the option that is equal to it.

    Collecting like terms: \(3 + 16 - 6 = 13\), so \(3n + 16n - 6n = 13n\), which is option B.

    Multiplying: \(5 \times 5 = 25\) and \(n \times n = n^2\), so \(5n \times 5n = 25n^2\), which is option A. Option C, \(25n\), is the one that forgets the square.

    Expanding the bracket: \(9(n + 4) = 9n + 36\). The \(36\) then cancels, so \(9n - 2n = 7n\), which is option D.

    The \(4\) and \(-4\) cancel, leaving \(12n - n = 11n\), which is option E.

    • B1\(3n + 16n - 6n = 13n\), option B
    • B1\(5n \times 5n = 25n^2\), option A
    • B1\(9(n + 4) - 36 - 2n = 7n\), option D
    • B1\(12n + 4 - 4 - n = 11n\), option E
  10. Either paper 1 mark

    Zainab is asked to simplify \(8 \times 8n + 5n\)

    Here is her working.

    \(8 \times 8n + 5n = 88n + 5n = 93n\)

    Explain what is wrong with her working.

    Show answer
    The two numbers have been written side by side instead of being multiplied together. \(8 \times 8n = 64n\), not \(88n\).

    The mistake is in the first step, where \(8 \times 8n\) is rewritten as \(88n\). The two numbers have been written side by side instead of being multiplied together.

    Done correctly, \(8 \times 8n = 64n\).

    Adding the other term, \(64n + 5n = 69n\), not \(93n\).