15 reviewed questions with answers and explanations.
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P sold a bicycle to Q at a profit of 10%. Q then sold it to R for ₦209 at a loss of 5%. How much did the bicycle originally cost P?
A: ₦200
Q sold the bicycle for 95% of the amount Q paid. Therefore Q paid ₦209 ÷ 0.95 = ₦220. This was 110% of P’s original cost, so P paid ₦220 ÷ 1.10 = ₦200, option A. Check: ₦200 × 1.10 × 0.95 = ₦209.
AI-reviewed · calculation checkedIf the price of each orange were raised by ½ kobo, a customer spending ₦2.40 would be able to buy 16 fewer oranges. What is the price of an orange before the increase?
A: 2½ kobo
Convert the budget to 240 kobo. Let the original price be p kobo. Then 240/p − 240/(p + 0.5) = 16. Combining the fractions gives p(p + 0.5) = 7.5, or (p − 2.5)(p + 3) = 0. A price must be positive, so p = 2.5 kobo, option A. The customer could buy 96 oranges before the increase and 80 afterwards: 16 fewer.
AI-reviewed · calculation checkedBola chooses an integer at random from 1 through 300 inclusive, with each integer equally likely. What is the probability that it is divisible by 4?
B: 1/4
The multiples of 4 are 4, 8, 12, …, 300. There are 300 ÷ 4 = 75 such integers among 300 equally likely choices. The probability is 75/300 = 1/4, option B.
AI-reviewed · calculation checkedWithout logarithm tables, evaluate (log base 3 of 27 − log base 1/4 of 64) / (log base 3 of 1/81).
C: −3/2
Since 3³ = 27, the first logarithm is 3. Since (1/4)⁻³ = 64, the second is −3. Since 3⁻⁴ = 1/81, the denominator is −4. The fraction is therefore [3 − (−3)]/(−4) = 6/(−4) = −3/2, option C.
AI-reviewed · calculation checkedFactorise 6x² − 14x − 12.
C: 2(x - 3) (3x + 2)
First take out 2: 2(3x² − 7x − 6). Split the middle term as −9x + 2x, then group: 2[3x(x − 3) + 2(x − 3)] = 2(x − 3)(3x + 2), option C. Expanding gives 6x² − 14x − 12.
AI-reviewed · calculation checkedIn a racing competition, Musa covered 5x km in the first hour and (x + 10) km in the next hour. He finished behind Ngozi, who covered 118 km in those two hours. Which inequality describes the possible positive values of x?
E: 0 < x < 18
Musa’s total distance is 5x + (x + 10) = 6x + 10 km. Because it is less than 118 km, 6x + 10 < 118, giving x < 18. His first-hour distance is positive, so x > 0. Together, 0 < x < 18, option E.
AI-reviewed · calculation checkedTunde and Shola can finish a piece of work together in 18 days. Tunde can do it alone in x days, while Shola would take 15 days longer. Which equation is satisfied by x?
C: x² - 21x − 270 = 0
Their daily work rates are 1/x and 1/(x + 15). Together, 1/x + 1/(x + 15) = 1/18. Multiplying by 18x(x + 15) gives 18(2x + 15) = x(x + 15), hence x² − 21x − 270 = 0, option C. Its positive solution is x = 30; working with Shola’s 45-day rate gives 1/30 + 1/45 = 1/18.
AI-reviewed · calculation checkedWhich quadratic equation has roots 1 − √13 and 1 + √13?
E: x² − 2x − 12 = 0
The roots add to 2. Their product is (1 − √13)(1 + √13) = 1 − 13 = −12. A monic quadratic with these roots is x² − (sum)x + product = 0, so x² − 2x − 12 = 0, option E.
AI-reviewed · calculation checkedFind a factor which is common to all three binomial expressions 4a² − 9b², a³ + 27b³ and (4a + 6b)².
E: none
First, 4a² − 9b² = (2a − 3b)(2a + 3b), while (4a + 6b)² = 4(2a + 3b)². Their only possible shared nonconstant linear factor is 2a + 3b. But a³ + 27b³ = (a + 3b)(a² − 3ab + 9b²), which is not divisible by 2a + 3b. Therefore none of the listed factors divides all three expressions: option E.
AI-reviewed · calculation checkedA triangle has sides (x + 4) cm, x cm and (x − 4) cm. If the cosine of its largest angle is 1/5, find x.
A: 24 cm
The largest angle is opposite x + 4. By the cosine rule, 1/5 = [x² + (x − 4)² − (x + 4)²]/[2x(x − 4)] = (x − 16)/[2(x − 4)]. Thus 5x − 80 = 2x − 8, so 3x = 72 and x = 24, option A. The resulting sides 20, 24 and 28 form a valid triangle.
AI-reviewed · calculation checkedFind the integer values of x satisfying −3 < 2 − 5x < 12.
C: −1, 0
Subtract 2 throughout: −5 < −5x < 10. Dividing by −5 reverses both inequalities, giving 1 > x > −2, or −2 < x < 1. The integers strictly between these bounds are −1 and 0, option C. The endpoints −2 and 1 are excluded.
AI-reviewed · calculation checkedTwo independent fair six-sided dice are rolled. What is the probability that both show the same number?
E: 1/6
There are 6 × 6 = 36 equally likely ordered outcomes. Six are doubles: (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6). The probability is 6/36 = 1/6, option E.
AI-reviewed · calculation checkedIn triangle PST, Q lies on PS and R lies on PT. ∠TSP = ∠PRQ, QR = 8 cm, PR = 6 cm and ST = 12 cm. Find SP.
C: 9 cm
Triangles PRQ and PST share the angle at P, and ∠PRQ = ∠PST. They are similar, with PR corresponding to PS and RQ corresponding to ST. Therefore 6/SP = 8/12, so SP = 6 × 12 ÷ 8 = 9 cm.
AI-reviewed · calculation checkedIn triangle XYZ, W lies on YZ and XW is perpendicular to YZ. YZ = 10 cm, XZ = 5 cm and WZ = 4 cm. Calculate XY.
B: 3√5 cm
In right triangle XWZ, XW² = 5² − 4² = 9, so XW = 3 cm. Since YZ = 10 cm and WZ = 4 cm, YW = 6 cm. Thus XY² = 3² + 6² = 45 and XY = 3√5 cm.
AI-reviewed · calculation checkedDiameter range (cm) — Frequency 3.35–3.45 — 3 3.45–3.55 — 6 3.55–3.65 — 7 3.65–3.75 — 4
The grouped distribution describes the diameters of 20 copper spheres. Estimate their mean diameter.
C: 3.56 cm
Use the class midpoints 3.40, 3.50, 3.60 and 3.70 cm. Multiply each by its frequency: 3.40 × 3 = 10.20, 3.50 × 6 = 21.00, 3.60 × 7 = 25.20, and 3.70 × 4 = 14.80. Their sum is 71.20 cm. Divide by the total frequency, 20, to obtain an estimated mean of 3.56 cm, option C. It is an estimate because the individual diameters within each interval are not given.
AI-reviewed · calculation checked