GCSE Algebraic notation Practice Questions

Free GCSE Algebraic notation practice questions with full step-by-step worked solutions. Covers writing products, dropping the times sign, coefficients, powers. Practise exam-style problems and check your method.

writing productsdropping the times signcoefficientspowersindex notationalphabetical order
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
Write a×ba \times b using algebraic notation.
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Worked solution

  1. Recall the convention

    a×ba \times b

    In algebra we do not write the multiplication sign between letters.

  2. Write the letters together

    a×b=aba \times b = ab

    Placing the letters side by side means 'a times b'.

  3. State the answer

    ab

    So a × b is written ab.

Answer
ab
Question 2
1 markeasy
Write c÷dc \div d as a fraction.
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Worked solution

  1. Read the division

    c÷dc \div d

    Dividing c by d.

  2. Write as a fraction

    c÷d=cdc \div d = \frac{c}{d}

    c on top, d on the bottom.

  3. State the answer

    cd\frac{c}{d}

    So c ÷ d is c over d.

Answer
cd\frac{c}{d}
Question 3
2 marksintermediate
Write 12×b×h\frac{1}{2} \times b \times h using algebraic notation.
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Worked solution

  1. Read the product

    12×b×h\frac{1}{2} \times b \times h

    A half multiplied by b and h.

  2. Multiply the letters

    b×h=bhb \times h = bh

    Write the letters together.

  3. Multiply by a half

    12×bh=bh2\frac{1}{2} \times bh = \frac{bh}{2}

    Halving is the same as dividing by 2.

  4. Recognise the formula

    bh2\frac{bh}{2}

    This is the area of a triangle.

  5. Interpret

    bh2\frac{bh}{2}

    For example b=4b = 4, h=3h = 3 gives 12÷2=612 \div 2 = 6.

  6. State the answer

    bh2\frac{bh}{2}

    So the expression is bh over 2.

Answer
bh2\frac{bh}{2}
Question 4
3 markshard
Write p×p×q×q×qp \times p \times q \times q \times q using index notation.
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Worked solution

  1. Group the letters

    (p×p)×(q×q×q)(p \times p) \times (q \times q \times q)

    Two p's and three q's.

  2. Turn the p's into a power

    p×p=p2p \times p = p^2

    Two p's multiplied is p².

  3. Turn the q's into a power

    q×q×q=q3q \times q \times q = q^3

    Three q's multiplied is q³.

  4. Combine

    p2×q3p^2 \times q^3

    Bring the two powers together.

  5. Apply the product convention

    p2q3p^2 q^3

    No times sign needed.

  6. Check the order

    p2q3p^2 q^3

    p before q, alphabetical.

  7. State the coefficient

    coefficient=1\text{coefficient} = 1

    There is no number in front, so the coefficient is 1.

  8. Note we do not write the 1

    1×p2q3=p2q31 \times p^2 q^3 = p^2 q^3

    A coefficient of 1 is left out.

  9. Check with values

    p=1,q=1: 1×1=1p=1,q=1:\ 1 \times 1 = 1

    p2q3=1p²q³ = 1 when p=q=1p = q = 1. ✓

  10. State the answer

    p2q3p^2 q^3

    So it is written p²q³, with coefficient 1.

Answer
p2q3p^2 q^3
Question 5
6 markschallenging
A trapezium has parallel sides aa and bb and height hh. Write an expression for its area.
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Worked solution

  1. Recall the trapezium area rule

    average the parallel sides, times the height\text{average the parallel sides, times the height}

    Add the parallel sides, halve, then multiply by the height.

  2. Add the parallel sides

    a+ba + b

    Sum of the two parallel sides.

  3. Halve the sum

    a+b2\frac{a + b}{2}

    This is the average width.

  4. Multiply by the height

    a+b2×h\frac{a + b}{2} \times h

    Average width times height.

  5. Write neatly as one fraction

    (a+b)h2\frac{(a + b)h}{2}

    The height multiplies the whole numerator.

  6. Keep the sum bracketed

    (a+b)h(a + b)h

    The brackets show both sides are averaged.

  7. Warn about a common error

    (a+b)h2≠ah+b2\frac{(a+b)h}{2} \ne \frac{ah + b}{2}

    The height multiplies both a and b.

  8. Interpret

    a=3,b=5,h=4: 8×42=16a=3,b=5,h=4:\ \frac{8 \times 4}{2} = 16

    Area is 16 square units. ✓

  9. Check the average width

    3+52=4\frac{3+5}{2} = 4

    The average of the parallel sides is 4.

  10. Multiply out the check

    4×4=164 \times 4 = 16

    Average width 4 times height 4 is 16. ✓

  11. Note an equivalent form

    12(a+b)h\frac{1}{2}(a+b)h

    A common alternative way to write it.

  12. Check units idea

    (a+b)h2 is an area\frac{(a+b)h}{2} \text{ is an area}

    Measured in square units.

  13. Sense-check against a rectangle

    a=b: 2a⋅h2=aha=b:\ \frac{2a \cdot h}{2} = ah

    If both sides are equal it becomes a rectangle, area ah. ✓

  14. Reflect

    average width×height\text{average width} \times \text{height}

    The formula is a mean width times the height.

  15. State the answer

    (a+b)h2\frac{(a + b)h}{2}

    So the area is (a + b)h over 2.

Answer
(a+b)h2\frac{(a + b)h}{2}

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