Jump to: Sets & Venn Algebra Indices Fractions & BODMAS Word Problems Speed-Distance-Time Pie Charts Linear Equations Statistics Vectors Exam Checklist
ClassMaster · Last Mile · Mathematics

Formulas, methods &
examiner-backed fixes

Every high-yield formula and step-by-step method — drawn directly from the WAEC Chief Examiner's Reports for 2019, 2021, and 2023.

High-yield topics — appears every year
★ Sets & Venn Diagrams ★ Algebra (expand, simplify, factorize) ★ Fractions & BODMAS ★ Ratio & Proportion Laws of Indices Speed-Distance-Time Pie Charts Linear Equations Vectors Frequency Tables (mean, mode, median) Transformation / Rigid Motion Percentages & Sharing Probability
⚠
Always show your working

This is the single most repeated instruction in the Maths CE reports. Even if your final answer is wrong, showing clear working earns method marks. A correct answer with no working shown can still lose marks.

∗ Section 1 — Appears every year

Sets & Venn Diagrams

One of the most reliable topics in the paper. The errors are consistent — and easy to avoid once you know them.

Many candidates found n(M) and n(N) together instead of finding them separately in Venn diagram problems. Curly brackets were omitted when expressing sets. Entries in the intersection region were often counted twice.

WAEC Chief Examiner, 2021 & 2019
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Use this to find the total when you know the union and each set individually.
n(ε) = n(A only) + n(B only) + n(A ∩ B) + n(neither)
Total in universal set = all individual regions added together.
Worked Example — Venn Diagram with overlap Common exam question
In a class of 40 students, 25 study Maths, 20 study Science, and 10 study both. How many study neither?
1
Find the union (those who study at least one subject)
n(M ∪ S) = n(M) + n(S) − n(M ∩ S) = 25 + 20 − 10 = 35
2
Find those who study neither
Neither = n(ε) − n(M ∪ S) = 40 − 35 = 5
3
Fill in the Venn diagram regions
M only: 25 − 10 = 15  |  S only: 20 − 10 = 10  |  Both: 10  |  Neither: 5
Check: 15 + 10 + 10 + 5 = 40 ✓
Do
  • Always use curly brackets: {1, 2, 3} — never write sets without them
  • Find n(A only) and n(B only) separately — not together
  • Subtract the intersection when finding the union
  • Fill in the Venn diagram before answering questions about it
  • Check: all regions should add up to n(ε)
Don't
  • Don't count the intersection region twice
  • Don't write sets without curly brackets — it costs marks
  • Don't confuse A ∪ B (union = all) with A ∩ B (intersection = overlap)
  • Don't jump to the answer without drawing the diagram first
∗ Section 2 — Appears every year

Algebra: Expand, Simplify, Factorize

Three skills tested every year. Each has a specific method — and a specific error pattern the examiner flags.

In factorization, candidates factored out terms that were not common to all parts of the expression. For expansion, errors in BODMAS led to wrong signs. Always look for the highest common factor first.

WAEC Chief Examiner, 2023 & 2021
Expanding Brackets Sign errors are common
Expand and simplify: 3(2x + 5) − 2(x − 4)
1
Expand each bracket separately — multiply every term inside
3(2x + 5) = 6x + 15
−2(x − 4) = −2x + 8 ⚠ −2 × −4 = +8 (two negatives make a positive)
2
Collect like terms
6x − 2x + 15 + 8 = 4x + 23
Factorizing Expressions HCF first — always
Factorise completely: 6x² + 9x
1
Find the HCF of all terms
HCF of 6x² and 9x is 3x
2
Divide each term by the HCF and write in brackets
6x² ÷ 3x = 2x    9x ÷ 3x = 3
3
Write the final answer
3x(2x + 3)
Check by expanding: 3x × 2x + 3x × 3 = 6x² + 9x ✓
✗ Wrong factorization (common error)

6x² + 9x
= 3(2x² + 3x)

Error: factored out 3 but left the x inside — 3 is not the complete HCF.

✓ Correct factorization

6x² + 9x
= 3x(2x + 3)

HCF is 3x — take out the full common factor.

Section 3

Laws of Indices

The CE report names this as a consistent weakness. Memorise these six laws — they cover every type of indices question in BECE.

Candidates applied the laws of indices incorrectly — adding bases instead of exponents, or multiplying exponents when they should be added. The laws must be memorised and applied precisely.

WAEC Chief Examiner, 2021
Law Rule Example Wrong (common error) Correct
Multiplication am × an = am+n 23 × 24 47 27 = 128
Division am ÷ an = am−n 56 ÷ 52 53 54 = 625
Power of a power (am)n = amn (32)4 36 38
Zero exponent a0 = 1 70 0 1
Negative exponent a−n = 1/an 2−3 −8 1/8
Fractional exponent a1/n = √a 271/3 9 3 (cube root of 27)
💡
Key rule: the base must be the same

Laws of indices only work when the base is the same. You cannot apply am × an = am+n to expressions like 23 × 54 — the bases (2 and 5) are different.

∗ Section 4 — Appears every year

Fractions & BODMAS

Strong candidates can simplify mixed numbers and apply BODMAS correctly. Weak candidates skip the order of operations and get the wrong answer.

B → O → D → M → A → S
Brackets  •  Order (powers/roots)  •  Division  •  Multiplication  •  Addition  •  Subtraction. Always work in this order.
Mixed Numbers — Simplifying BODMAS errors common here
Evaluate: 2½ + 1⅓ × ¾
1
BODMAS: do multiplication first
1⅓ × ¾ = &frac43; × ¾ = &frac{12}{12} = 1
2
Then do the addition
2½ + 1 = 3½
Do
  • Convert mixed numbers to improper fractions before calculating
  • Always apply BODMAS — do multiplication/division before addition/subtraction
  • Find the LCM when adding or subtracting fractions with different denominators
  • Simplify your final answer to its lowest terms
  • Show every step — intermediate steps earn marks
Don't
  • Don't add or subtract fractions before doing any multiplication in the same expression
  • Don't add numerators and denominators separately (e.g. ½ + ⅓ ≠ ⅖)
  • Don't leave answers as improper fractions — convert back to mixed numbers
  • Don't skip the LCM step when denominators are different
Section 5 — High-loss area

Word Problems

The examiner flags this every single year. Candidates read the question, panic, and write numbers — without setting up an equation first. That costs marks.

Candidates answered a different question from what was asked. Many could not translate the word problem into a mathematical expression. Identify the unknown, write an equation, then solve — in that order.

WAEC Chief Examiner, 2023
Word Problem — 4-Step Method Use every time
A man is 4 times as old as his son. In 6 years, the man will be 3 times as old as his son. Find their present ages.
1
Define the unknown — let a variable represent what you don't know
Let son's age = x. Then man's age = 4x.
2
Write the equation from the condition given
In 6 years: 4x + 6 = 3(x + 6)
3
Solve
4x + 6 = 3x + 18
4x − 3x = 18 − 6
x = 12
4
Answer the question that was asked
Son's age = 12 years. Man's age = 4 × 12 = 48 years.
Check: In 6 years: 54 = 3 × 18 ✓
✗
Most common word problem error

Candidates solve for x but forget to use x to answer the actual question. If x = 12 is the son's age and the question asks for the father's age, write 4 × 12 = 48. Never stop at x.

Section 6 — Top weakness 2023

Speed, Distance & Time

The top-cited weakness in 2023. Multi-step journey questions (different speeds, stops, return trips) trip almost everyone. One triangle, three formulas.

Distance-time-speed problems were poorly answered — especially multi-step journeys where candidates had to deal with different speeds or stops. Units were frequently inconsistent.

WAEC Chief Examiner, 2023
Distance = Speed × Time
Cover D with your finger — S × T remains.
Speed = Distance ÷ Time
Cover S — D ÷ T remains.
Time = Distance ÷ Speed
Cover T — D ÷ S remains.
Multi-Step Journey 2023 exam type
A car travels 120 km at 60 km/h, then stops for 30 minutes, then travels a further 90 km at 45 km/h. Find the total time for the journey.
1
Time for first leg
Time = 120 ÷ 60 = 2 hours
2
Convert the stop to hours
30 min = 0.5 hours ⚠ Keep all times in the same unit — hours or minutes, not a mix of both.
3
Time for second leg
Time = 90 ÷ 45 = 2 hours
4
Total time
2 + 0.5 + 2 = 4.5 hours
⚠
Units must be consistent throughout

If speed is in km/h, time must be in hours and distance in km. If you have minutes, convert: minutes ÷ 60 = hours. Mixed units give wrong answers.

Section 7 — Top weakness 2023

Pie Charts

Candidates knew they needed a pie chart — but calculated the angles wrong, used the wrong total, or didn't show their working.

For pie charts, candidates did not show simplification steps and often used the wrong total. Always show your full working for every sector angle.

WAEC Chief Examiner, 2023
Sector angle = (Frequency ÷ Total) × 360°
Calculate each sector separately. All sector angles must add up to 360°. Use this as your check.
Pie Chart — Worked Example Show ALL working
Draw a pie chart for the following: Football: 15, Basketball: 10, Athletics: 5. Total: 30 students.
1
Find total (if not given)
Total = 15 + 10 + 5 = 30
2
Calculate each sector angle — show the full working
Football: (15 ÷ 30) × 360 = ½ × 360 = 180°
Basketball: (10 ÷ 30) × 360 = ⅓ × 360 = 120°
Athletics: (5 ÷ 30) × 360 = ⅙ × 360 = 60°
3
Check: angles must sum to 360°
180 + 120 + 60 = 360° ✓
4
Draw, label each sector, and give the chart a title
Use a protractor. Label each sector with its category and angle. Title the chart.
∗ Section 8 — Appears every year

Linear Equations

Including equations with fractions — where the LCM method is the key tool. Candidates who know this method score full marks.

Many candidates stopped mid-solution in linear equations without reaching the final answer. When fractions were involved, candidates did not multiply through by the LCM of the denominators.

WAEC Chief Examiner, 2021
Linear Equation with Fractions — LCM Method Most missed technique
Solve: x/3 + (x + 2)/2 = 5
1
Find the LCM of the denominators
Denominators are 3 and 2. LCM = 6
2
Multiply EVERY term on both sides by the LCM
6 × x/3 + 6 × (x+2)/2 = 6 × 5
2x + 3(x + 2) = 30
3
Expand and simplify
2x + 3x + 6 = 30
5x = 24
4
Solve and state the answer
x = 24/5 = 4.8
Section 9

Statistics: Mean, Median & Mode

All three measures are testable from a frequency table. Candidates consistently confuse median with mean — know the difference.

Mean = Σ(fx) ÷ Σf
Multiply each value by its frequency. Sum all fx. Divide by total frequency.
Median = middle value when data is ordered
From a frequency table: use cumulative frequencies to find the middle position.
Mode = value with highest frequency
Just look for the largest frequency in the table.
Mean from a Frequency Table Show the fx column
Score (x) Frequency (f) fx
133
2510
3721
4520
TotalΣf = 20Σfx = 54
1
Mean
Mean = 54 ÷ 20 = 2.7
2
Mode
Score 3 has the highest frequency (7). Mode = 3
3
Median
20 values, so median is between the 10th and 11th value.
Cumulative: scores 1+2 give 8 values; adding score 3 gives 15. So both 10th and 11th values fall in score 3.
Median = 3
Section 10

Vectors

Addition and magnitude are the most tested vector skills. Magnitude requires Pythagoras — always show the full square root working.

a + b = (a₁+b₁, a₂+b₂)
Add the x-components and y-components separately.
|a| = √(x² + y²)
Magnitude (size) of vector using Pythagoras. Always write the formula before substituting.
Vectors — Addition and Magnitude
Given a = (3, 4) and b = (1, −2), find a + b and |a + b|.
1
Add the vectors
a + b = (3+1, 4+(−2)) = (4, 2)
2
Find the magnitude
|a + b| = √(4² + 2²) = √(16 + 4) = √20 = 2√5
✅ Section 11

Exam-Day Checklist

Go through this the night before the Maths paper — and again when you sit down to start.

Before you write anything
  • Read each question at least twice before starting
  • Identify what is being asked — not what looks familiar
  • Write the formula or rule first, then substitute values
  • Show all working — even for questions that seem easy
Sets & Algebra
  • Sets written with curly brackets { }
  • Venn diagram regions filled in before answering
  • Intersection not counted twice in the union
  • Factorization: full HCF taken out, not partial
  • Expansion: checked signs, especially with negatives
Calculations
  • Pie chart: each angle = (f ÷ total) × 360, all sum to 360°
  • Speed problems: units consistent throughout (km/h & hours)
  • Indices: same base before applying laws
  • Mean: Σfx column shown, divided by Σf
  • Vectors: added component by component
Word Problems & Final Answers
  • Defined variable clearly (Let x = …)
  • Equation written before solving
  • Answered the question that was actually asked
  • Units included in every final answer
  • Checked answer makes sense in the context
✓
The examiner's most repeated instruction — every year

Show your working at every step. A wrong final answer with correct method earns method marks. A correct answer with no working shown risks earning nothing. Working is not optional in BECE Maths.

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