1. Add the fractions
Calculate \(\frac{3}{4}+\frac{5}{8}\). Give your answer as a decimal.
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Use a common denominator of 8. \(\frac{3}{4}=\frac{6}{8}\). Then \(\frac{6}{8}+\frac{5}{8}=\frac{11}{8}=1.375\).
Answer: 1.375
Important: These are original CBE Nexus practice questions aligned to common international mathematics skills. They are not official examination questions. Follow the syllabus and calculator rules for your own course and paper.
Follow the arrows from left to right. The worked solutions below demonstrate the method for each question.
2. Find a percentage of an amount
A school has 250 learners. 18% participate in a mathematics club. How many learners is this?
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Convert 18% to a decimal: 0.18. Multiply: \(0.18\times250=45\).
Answer: 45 learners
3. Divide an amount in a given ratio
Divide 64 in the ratio 3 : 5. Calculate the smaller share.
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Total parts = 3 + 5 = 8. One part = 64 ÷ 8 = 8. Smaller share = 3 × 8 = 24.
Answer: 24
4. Calculate average speed
A car travels 150 km in 2.5 hours. Calculate its average speed in km/h.
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Average speed = distance ÷ time = 150 ÷ 2.5 = 60 km/h.
Answer: 60 km/h
5. Convert from standard form
Write \(6.72\times10^{-4}\) as an ordinary decimal.
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A power of \(10^{-4}\) moves the decimal point four places to the left: 6.72 → 0.000672.
Answer: 0.000672
6. Solve the equation
Solve \(4x+7=31\) for \(x\).
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Subtract 7 from both sides: 4x = 24. Divide both sides by 4: x = 6.
Answer: x = 6
7. Solve an equation with brackets
Solve \(3(2x-5)=21\).
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Expand: 6x − 15 = 21. Add 15: 6x = 36. Divide by 6: x = 6.
Answer: x = 6
8. Calculate the gradient
Find the gradient of the straight line passing through (2, 3) and (6, 11).
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Gradient = change in y ÷ change in x = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2.
Answer: 2
9. Find a term in a sequence
The nth term of a sequence is \(3n+2\). Calculate the 10th term.
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Substitute n = 10: 3(10) + 2 = 30 + 2 = 32.
Answer: 32
10. Calculate the hypotenuse
A right-angled triangle has shorter sides of 9 cm and 12 cm. Find the hypotenuse.
Show worked solution
By Pythagoras, \(c^2=9^2+12^2=81+144=225\). Therefore c = √225 = 15 cm.
Answer: 15 cm
11. Calculate the area of a circle
A circle has radius 7 cm. Calculate its area correct to 2 decimal places. Use π = 3.14159.
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Area = πr² = 3.14159 × 7² = 3.14159 × 49 = 153.938… cm².
Answer: 153.94 cm²
12. Find an angle using sine
In a right-angled triangle, the opposite side is 6 cm and the hypotenuse is 10 cm. Find the angle θ to 2 decimal places.
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sin θ = opposite ÷ hypotenuse = 6 ÷ 10 = 0.6. Therefore θ = sin⁻¹(0.6) = 36.87° (to 2 d.p.).
Answer: 36.87°
13. Solve a quadratic equation
Solve \(x^2-5x+6=0\). Enter the two roots separated by a comma.
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Factorise: (x − 2)(x − 3) = 0. Hence x − 2 = 0 or x − 3 = 0.
Answer: x = 2 or x = 3
14. Evaluate a function
Given \(f(x)=2x^2-3x+1\), calculate \(f(4)\).
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Substitute x = 4: f(4) = 2(4²) − 3(4) + 1 = 32 − 12 + 1 = 21.
Answer: 21
15. Calculate compound interest
KSh 1,000 is invested at 5% compound interest per year for 2 years. Calculate the final amount.
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Amount = P(1 + r)ⁿ = 1000(1.05)² = 1000 × 1.1025 = 1102.50.
Answer: KSh 1,102.50
16. Differentiate and evaluate
For \(y=x^3-4x^2+2x\), calculate \(\frac{dy}{dx}\) at x = 3.
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Differentiate term by term: dy/dx = 3x² − 8x + 2. At x = 3: 3(9) − 24 + 2 = 27 − 24 + 2 = 5.
Answer: 5
17. Evaluate a definite integral
Evaluate \(\int_0^2(6x^2-4)\,dx\).
Show worked solution
An antiderivative is 2x³ − 4x. Evaluate at the limits: [2(2³) − 4(2)] − [0] = 16 − 8 = 8.
Answer: 8
18. Calculate the mean
Find the arithmetic mean of 4, 6, 7, 9 and 14.
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Add the values: 4 + 6 + 7 + 9 + 14 = 40. Divide by the number of values, 5: 40 ÷ 5 = 8.
Answer: 8
19. Calculate a probability without replacement
A bag contains 5 red and 3 blue counters. Two counters are selected at random without replacement. Find the probability that both are red.
Show worked solution
Probability first red = 5/8. After one red is removed, probability second red = 4/7. Multiply: (5/8)(4/7) = 20/56 = 5/14 ≈ 0.35714.
Answer: 5/14 ≈ 0.35714
20. Find the magnitude of a vector
Calculate the magnitude of vector \(\begin{pmatrix}3\\4\end{pmatrix}\).
Show worked solution
Magnitude = √(3² + 4²) = √(9 + 16) = √25 = 5.
Answer: 5
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