GCSE Mathematics — Practice Paper 1 (non-calculator)

A non-calculator set covering number, algebra, ratio, geometry and probability. Method marks (M) and accuracy marks (A) are separated throughout, because knowing the difference is worth a grade.

How to use this: no calculator. If you reach for one you are practising a different exam. Where a question says "show your working", the working is most of the marks.

Section A

1. Work out $\dfrac{3}{4} + \dfrac{2}{5}$, giving your answer as a mixed number (2 marks)

Common denominator of 4 and 5 is 20.

 3     15          2      8
--- = ----        --- = ----
 4     20          5     20

15/20 + 8/20 = 23/20 = 1 3/20

Marks: M1 for a correct common denominator, A1 for $1\frac{3}{20}$.

Common mistake: adding numerators and denominators separately to get $5/9$. Fractions do not work that way.

2. Expand and simplify $(x + 3)(x - 5)$ (2 marks)

(x + 3)(x - 5)
= x² - 5x + 3x - 15
= x² - 2x - 15

Marks: M1 for four correct terms before simplifying, A1 for the simplified expression.

Common mistake: sign errors on the middle term. $-5x + 3x = -2x$, not $+2x$.

3. Write $\sqrt{72}$ in the form $a\sqrt{b}$, where $b$ is as small as possible (2 marks)

72 = 36 × 2          (36 is the largest square factor)

√72 = √36 × √2 = 6√2

Marks: M1 for identifying a square factor, A1 for $6\sqrt2$.

Common mistake: using $9 \times 8$ to get $3\sqrt8$ and stopping. $\sqrt8$ simplifies further, so the answer is not yet in the required form. Always take the largest square factor.

4. Solve $\dfrac{2x - 1}{3} = 5$ (2 marks)

Multiply both sides by 3:   2x - 1 = 15
Add 1:                      2x     = 16
Divide by 2:                x      = 8

Marks: M1 for multiplying through by 3, A1 for $x = 8$.

Check: $(2 \times 8 - 1)/3 = 15/3 = 5$ ✓

5. A recipe uses flour and sugar in the ratio $5 : 2$. If 350 g of flour is used, how much sugar is needed? (2 marks)

5 parts = 350 g
1 part  = 350 ÷ 5 = 70 g
2 parts = 2 × 70  = 140 g

Marks: M1 for finding one part, A1 for 140 g.

6. Solve the simultaneous equations (3 marks)

$$3x + y = 11 \qquad\qquad 2x - y = 4$$

Add the two equations (the y terms eliminate):

  3x + y = 11
+ 2x - y =  4
-----------------
  5x     = 15
  x      = 3

Substitute into the first:  3(3) + y = 11  ⟹  y = 2

Marks: M1 for a correct elimination or substitution, A1 for $x$, A1 for $y$.

Why adding works here: the $y$ coefficients are $+1$ and $-1$. When signs are opposite, add; when they match, subtract.

7. The probability that it rains on a given day is 0.3. Find the probability that it rains on exactly one of two consecutive days (3 marks)

P(rain) = 0.3      P(no rain) = 0.7

Exactly one  =  P(rain, no rain)  +  P(no rain, rain)
             =  (0.3 × 0.7)       +  (0.7 × 0.3)
             =  0.21              +  0.21
             =  0.42

Marks: M1 for one correct product, M1 for adding both orders, A1 for 0.42.

Common mistake: giving 0.21 — counting only one order. "Exactly one" does not specify which day, so both arrangements count.

8. A circle has radius 6 cm. Find its area, leaving your answer in terms of $\pi$ (2 marks)

A = π r²
  = π × 6²
  = 36π cm²

Marks: M1 for $\pi \times 6^2$, A1 for $36\pi$ with units.

Note: "in terms of $\pi$" means do not substitute 3.14. On a non-calculator paper, an exact answer is the required form.

9. Increase 80 by 15% (2 marks)

10% of 80 = 8
 5% of 80 = 4
15% of 80 = 12

80 + 12 = 92

Marks: M1 for a valid method, A1 for 92.

Faster route: multiply by 1.15 directly. Building from 10% and 5% is safer without a calculator.

10. Prove that the sum of two consecutive odd numbers is always divisible by 4 (4 marks)

Let the first odd number be   2n + 1        (n an integer)
The next odd number is        2n + 3

Sum = (2n + 1) + (2n + 3)
    = 4n + 4
    = 4(n + 1)

Since n is an integer, (n + 1) is an integer,
so the sum is 4 × an integer, and therefore divisible by 4.  ∎

Marks: M1 for a correct algebraic form of an odd number, M1 for the consecutive odd number, M1 for simplifying to $4n + 4$, A1 for factorising and stating the conclusion.

This is where proof questions are lost: stopping at $4n + 4$. You must factorise to $4(n+1)$ and say why that proves divisibility. Testing a few numbers is not a proof and scores 0.

Method marks versus accuracy marks

This is the single most useful thing to understand about GCSE Maths marking:

• M marks are for a correct method, awarded even if your arithmetic is wrong
• A marks are for the correct answer, and usually require the M marks first
• Follow-through (ft) means a later answer is marked as correct relative to your earlier mistake

The practical consequence: never erase your working, and never leave a question blank. A wrong answer with sound method routinely scores 2 of 3. A blank scores 0, and a bare answer with no working scores 1 at best.

Where to go next

• GCSE Biology Practice Paper 1 — for the data and graph questions
• SAT Math Practice Set 1 — if you are also sitting US admissions tests

Submit your working to StudyAI and it will identify the exact line where the method stopped earning marks.