1. Place Value and Rounding
A district records 48,765 learners. Round this to the nearest thousand.
Show worked answer
The hundreds digit is 7, so increase the thousands digit by one. The estimate is 49,000 learners.
BECE Mathematics — Questions and Worked Answers
Original DAS BECE-style practice. Try each question before opening its worked solution. The complete NaCCA exemplar set is available in the Curriculum Reference and the source-page teaching lessons.
Number • Number and Numeration Systems
2. Factors, Multiples and Prime Numbers
Two bells ring every 12 and 18 minutes. They ring together at 8:00 a.m. When will they next ring together?
Show worked answer
12 = 2² × 3 and 18 = 2 × 3². LCM = 2² × 3² = 36 minutes. They next ring together at 8:36 a.m.
3. Number Bases
Express 101101 base two in base ten.
Show worked answer
Use powers of two: 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1 = 45.
4. Sets and Venn Diagrams
Of 40 learners, 25 like football and 18 like volleyball; 10 like both. How many like neither?
Show worked answer
Number liking at least one = 25 + 18 − 10 = 33. Neither = 40 − 33 = 7 learners.
5. Real Numbers and Rational Numbers
Which of −3, 0.75 and the square root of 2 is irrational?
Show worked answer
−3 = −3/1 and 0.75 = 3/4 are rational. The square root of 2 cannot be expressed as a ratio of integers, so it is irrational.
Number • Number Operations
6. Integers and Order of Operations
Evaluate −6 + 3 × (8 − 5).
Show worked answer
Calculate brackets first: 8 − 5 = 3. Multiply: 3 × 3 = 9. Add: −6 + 9 = 3.
7. Powers, Roots and Indices
Simplify 2⁵ × 2³ ÷ 2⁴.
Show worked answer
For the same base, add indices on multiplication and subtract on division. 2^(5+3−4) = 2⁴ = 16.
8. Standard Form and Estimation
Write 0.00072 in standard form.
Show worked answer
Move the decimal four places right to obtain 7.2. Compensate by multiplying by 10 to the power −4. Answer: 7.2 × 10^(−4).
9. Binary Operations
An operation is defined by a ★ b = 2a + b. Find 3 ★ 5.
Show worked answer
Substitute a=3 and b=5 into the given rule: 2(3)+5 = 6+5 = 11.
Number • Fractions, Decimals and Percentages
10. Fractions and Mixed Numbers
A tank is 3/4 full. After 1/6 of the whole tank is used, what fraction remains?
Show worked answer
Use denominator 12: 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
11. Decimal Operations
A rope is 7.2 m long. How many pieces of length 0.45 m can be cut?
Show worked answer
Number of pieces = 7.2 ÷ 0.45 = 720 ÷ 45 = 16 pieces.
12. Percentage Change and Reverse Percentages
A bag costs GH₵180 after a 10% discount. Find its original price.
Show worked answer
The sale price is 90% of the original. Original = 180 ÷ 0.90 = GH₵200.
13. Profit, Loss and Simple Interest
Find the simple interest on GH₵800 at 5% per annum for 3 years.
Show worked answer
I = PRT/100 = 800 × 5 × 3 / 100 = GH₵120. The total amount is GH₵920.
Number • Ratios and Proportion
14. Sharing in a Ratio
Share GH₵360 between Ama and Kojo in the ratio 2:3.
Show worked answer
Total parts = 2 + 3 = 5. One part = 360/5 = 72. Ama receives GH₵144; Kojo receives GH₵216.
15. Direct and Inverse Proportion
Six equally productive workers finish a job in 10 days. How long will 15 workers take?
Show worked answer
Work is constant: 6 × 10 = 60 worker-days. Time for 15 workers = 60/15 = 4 days.
16. Rates, Speed and Scale
A map scale is 1:50,000. Two villages are 6 cm apart on the map. Find the actual distance in kilometres.
Show worked answer
Actual distance = 6 × 50,000 = 300,000 cm. Since 100,000 cm = 1 km, the distance is 3 km.
Algebra • Patterns and Relations
17. Number Sequences
Find the next two terms of 5, 9, 13, 17, ...
Show worked answer
The common difference is 4. Add 4 to 17 to obtain 21, and add 4 again to obtain 25.
18. General Rules for Patterns
Find the 20th term of 3, 7, 11, 15, ...
Show worked answer
The first term is 3 and the difference is 4. T_n = 3 + 4(n−1) = 4n−1. T_20 = 80−1 = 79.
19. Mappings and Linear Graphs
For y = 2x + 1, find y when x = −2, 0 and 3.
Show worked answer
Substitute separately: y = −4+1 = −3; y = 0+1 = 1; y = 6+1 = 7. Plot (−2,−3), (0,1), (3,7).
20. Gradient and Intercepts
Find the gradient of the line through (1,3) and (5,11).
Show worked answer
Gradient = change in y / change in x = (11−3)/(5−1) = 8/4 = 2.
Algebra • Algebraic Expressions
21. Simplifying and Substitution
Simplify 3a + 2b − a + 5b, then evaluate when a=4 and b=−1.
Show worked answer
Collect like terms: 2a + 7b. Substitute: 2(4) + 7(−1) = 8−7 = 1.
22. Expansion and Factorisation
Factorise 6x² + 9x completely.
Show worked answer
The highest common factor is 3x. Divide each term by 3x to obtain 2x and 3. Answer: 3x(2x+3).
23. Algebraic Fractions and Formulae
Make h the subject of A = bh/2.
Show worked answer
Multiply both sides by 2: 2A = bh. Divide by b: h = 2A/b, where b is not zero.
Algebra • Variables and Equations
24. Linear Equations
Solve 3(x−2) = 2x + 5.
Show worked answer
Expand: 3x−6 = 2x+5. Subtract 2x: x−6 = 5. Add 6: x = 11. Check: both sides equal 27.
25. Word Problems and Simultaneous Equations
Two pens and one book cost GH₵13. One pen and one book cost GH₵9. Find each price.
Show worked answer
Let p be a pen and b a book: 2p+b=13, p+b=9. Subtract to get p=4. Substitute: b=9−4=5. Pen GH₵4; book GH₵5.
26. Linear Inequalities
Solve −2x + 3 < 11.
Show worked answer
Subtract 3: −2x < 8. Divide by −2 and reverse the inequality: x > −4. Use an open circle at −4 and shade to the right.
Geometry and Measurement • Shape and Space
27. Angles and Parallel Lines
Two angles on a straight line are 3x and 2x. Find the larger angle.
Show worked answer
Their sum is 180°. Thus 5x = 180°, giving x = 36°. The larger angle is 3×36° = 108°.
28. Triangles and Polygons
Find each interior angle of a regular hexagon.
Show worked answer
Sum of interior angles = (6−2)×180° = 720°. Each angle = 720°/6 = 120°.
29. Constructions and Loci
Describe how to construct the perpendicular bisector of a line segment AB.
Show worked answer
With the same compass radius greater than half AB, draw arcs above and below AB from A and B. Join the two arc intersections. The resulting line is perpendicular to AB and passes through its midpoint.
30. Solids, Nets and Symmetry
A triangular prism has 6 vertices and 9 edges. How many faces does it have?
Show worked answer
Use Euler’s relation V−E+F=2. Therefore F=2−6+9=5. These are two triangular and three rectangular faces.
Geometry and Measurement • Measurement
31. Perimeter and Area
Find the area of a trapezium with parallel sides 8 cm and 14 cm, and height 5 cm.
Show worked answer
A = (1/2)(a+b)h = (1/2)(8+14)×5 = 55 cm².
32. Circles and Sectors
Find the area of a 90° sector of radius 14 cm. Use pi = 22/7.
Show worked answer
Sector area = (90/360)×(22/7)×14² = (1/4)×616 = 154 cm².
33. Surface Area and Volume
Find the volume of a cylinder of radius 7 cm and height 10 cm. Use pi = 22/7.
Show worked answer
V = pi × r² × h = (22/7)×49×10 = 1,540 cm³.
34. Pythagoras and Right Triangles
A 10 m ladder reaches 8 m up a wall. How far is its foot from the wall?
Show worked answer
The ladder is the hypotenuse. x² + 8² = 10². Thus x²=36 and x=6 m. Use the positive root for a length.
35. Trigonometric Ratios
A right triangle has opposite side 6 cm, adjacent side 8 cm and hypotenuse 10 cm relative to angle theta. Find sin theta and tan theta.
Show worked answer
sin theta = opposite/hypotenuse = 6/10 = 3/5. tan theta = opposite/adjacent = 6/8 = 3/4.
Geometry and Measurement • Position and Transformation
36. Coordinates and Vectors
Find the vector from A(−2,3) to B(4,−1).
Show worked answer
Subtract coordinates of A from B: horizontal component = 4−(−2)=6; vertical component = −1−3=−4. Vector AB has components (6,−4).
37. Translations and Reflections
Reflect P(3,−2) in the y-axis, then translate by vector (1,4).
Show worked answer
Reflection in the y-axis changes the x sign: (−3,−2). Translation gives (−3+1,−2+4) = (−2,2).
38. Rotations and Enlargements
Rotate (2,1) through 90° anticlockwise about the origin.
Show worked answer
Use (x,y) → (−y,x). Therefore (2,1) maps to (−1,2). Distance from the origin remains the same.
39. Bearings and Directions
The bearing of B from A is 065°. Find the bearing of A from B.
Show worked answer
The reverse direction differs by 180°. 065°+180°=245°. The required bearing is 245°. Measure bearings clockwise from north.
40. Enlargements and Similar Figures
A triangle of area 12 cm² is enlarged by scale factor 3. Find the new area.
Show worked answer
Lengths are multiplied by 3, so areas are multiplied by 3² = 9. New area = 12×9 = 108 cm².
Handling Data • Data
41. Collecting and Organising Data
The scores are 2, 3, 2, 4, 3, 2, 5, 4. Make a frequency table.
Show worked answer
Count each score: 2 occurs 3 times; 3 occurs 2 times; 4 occurs 2 times; 5 occurs once. Total frequency = 3+2+2+1 = 8.
42. Bar Charts and Pie Charts
In a class of 40, 15 learners choose science club. Find its pie-chart sector angle.
Show worked answer
Fraction choosing science = 15/40. Sector angle = (15/40)×360° = 135°.
43. Mean, Median, Mode and Range
Find the mean, median, mode and range of 2, 4, 4, 5, 10.
Show worked answer
Sum = 25, so mean = 25/5 = 5. Middle value is 4, so median = 4. Most frequent value is 4, so mode = 4. Range = 10−2 = 8.
44. Interpreting Statistical Information
A frequency table gives values 1, 2, 3 with frequencies 2, 5, 3. Find the mean.
Show worked answer
Total frequency = 10. Sum of value × frequency = 1×2 + 2×5 + 3×3 = 21. Mean = 21/10 = 2.1.
45. Grouped Data and Histograms
Class intervals 0≤x<10 and 10≤x<20 have frequencies 3 and 7. Estimate the mean.
Show worked answer
Midpoints are 5 and 15. Estimated total = 3×5 + 7×15 = 120. Total frequency = 10. Estimated mean = 120/10 = 12.
Handling Data • Chance or Probability
46. Probability Scale and Single Events
A fair die is thrown. Find the probability of obtaining a prime number.
Show worked answer
Prime outcomes are 2, 3 and 5: 3 favourable outcomes out of 6. Probability = 3/6 = 1/2.
47. Complementary Events and Expected Frequency
A bag contains 3 red and 7 blue balls. Find the probability of not choosing red.
Show worked answer
Total balls = 10. P(red)=3/10. P(not red)=1−3/10=7/10, equal to the probability of blue.
48. Combined Outcomes and Sample Spaces
Two fair coins are tossed. Find the probability of exactly one head.
Show worked answer
The equally likely outcomes are HH, HT, TH, TT. Exactly one head occurs in HT and TH. Probability = 2/4 = 1/2.