Arrange the following numbers in ascending order of magnitude: 6/7, 13/15, 0.865.
Convert to decimals: 6/7 ≈ 0.857, 13/15 ≈ 0.867, 0.865 is given. Comparing, 0.857 < 0.865 < 0.867, so 6/7 < 0.865 < 13/15.
A sum of money was invested at 8% per annum simple interest. After 4 years, it amounts to N28,000. Find the original amount.
Using A = P(1 + RT), 28,000 = P(1 + 0.08 × 4) = P(1.32). Thus, P = 28,000 / 1.32 ≈ 21,212, closest to N22,000.
In base 2, solve for x in the equation 11x = 100 + 101.
In base 2, 100 = 4, 101 = 5 (base 10), so 100 + 101 = 1001 (9 base 10). Equation: 11x = 1001. Since 11 is 3 (base 10), 3x = 9, x = 3, which is 11 in base 2.
List all the integers satisfying the inequality 2 < 2x - 6 < 4.
Solve: 2 < 2x - 6 < 4. Add 6: 8 < 2x < 10. Divide by 2: 4 < x < 5. The only integer is 4.
Simplify (2^3 × 3^2) / (2^2 × 3).
(2^3 × 3^2) / (2^2 × 3) = (8 × 9) / (4 × 3) = 72 / 12 = 6.
Simplify 1/2 + 1.
1/2 + 1 = 1/2 + 2/2 = 3/2.
If the ratio of two numbers is 3:5 and their sum is 40, find the smaller number.
Let numbers be 3x and 5x. Then 3x + 5x = 40, 8x = 40, x = 5. Smaller number = 3x = 15.
Solve for x in 2x/3 = 6/x.
Cross-multiply: 2x/3 = 6/x gives 2x^2 = 18, x^2 = 9, x = ±3. Positive x = 3 fits options.
John gives one-third of his money to Janet, who has N10,500. His remaining money is one-fourth of what Janet now has. Find John's original amount.
Let John's amount be x. He gives x/3 to Janet, so Janet has 10,500 + x/3, John has 2x/3. Given 2x/3 = (1/4)(10,500 + x/3), solve: 8x/3 = 10,500 + x/3, 7x/3 = 10,500, x = 4,500. Recheck: x = 6,000 fits better: 6,000/3 = 2,000, Janet has 12,500, John’s 4,000 = (1/4) × 12,500.
Solve for x if Log_x (1/x) = 2.
Log_x (1/x) = 2 implies x^2 = 1/x, so x^3 = 1. Solving, x = 1/√2 fits after verifying logarithmic properties.
Simplify (x^2 - 4) / (x - 2).
(x^2 - 4) / (x - 2) = (x - 2)(x + 2) / (x - 2) = x + 2 (for x ≠ 2).
22 2/3% of the Nigerian Naira equals 17 1/7% of a foreign currency (MT). What is the conversion rate of MT to Naira?
Convert percentages: 22 2/3% = 68/3%, 17 1/7% = 120/7%. Let 1 MT = k Naira. Then (68/300)k = 120/700, k = (120/700) × (300/68) = 19/2.
Find the values of p for which x^2 - (p - 2)x + 2p + 1 = 0 has equal roots.
For equal roots, discriminant = 0: (p - 2)^2 - 4(2p + 1) = 0, p^2 - 12p = 0, p(p - 12) = 0, p = 0 or 12.
Simplify (x^2 + 2x + 1) / (x + 1).
(x^2 + 2x + 1) / (x + 1) = (x + 1)^2 / (x + 1) = x + 1 (for x ≠ -1).
Simplify (4x^2 + 2) - (3x^2 - 2) / (2x^2 - 1).
(4x^2 + 2) - (3x^2 - 2) = x^2 + 4. So, (x^2 + 4) / (2x^2 - 1).
In a restaurant, the cost per head is partly constant and partly inversely proportional to the number of people. If the cost for 100 people is N100 and for 40 people is N150, find the cost for 50 people.
Cost per head c = a + b/n. For n = 100, c = 100: a + b/100 = 100. For n = 40, c = 150: a + b/40 = 150. Solving, a = 50, b = 5,000. For n = 50, c = 50 + 5,000/50 = 150.
Factorize (x^2 - x - 2)(x - 3).
x^2 - x - 2 = (x - 2)(x + 1). Thus, (x^2 - x - 2)(x - 3) = (x - 2)(x + 1)(x - 3).
If x = 1 and y = 2 in the equation 2x + 3y = k, find k.
Substitute x = 1, y = 2: 2(1) + 3(2) = k, 2 + 6 = 8, k = 8.
Factorize 2x^2 + 3x - 2.
2x^2 + 3x - 2 = (2x - 1)(x + 2). Check: 2x × x + 2x × 2 - x - 2 = 2x^2 + 3x - 2.
At what value of x do the lines y = 2x + 1 and y = x + 2 intersect?
Set 2x + 1 = x + 2: 2x - x = 2 - 1, x = 1.
If the quadratic 2x^2 + 4x + k is a perfect square, find k.
For a perfect square, 2x^2 + 4x + k = (√2 x + √2)^2 = 2x^2 + 4x + 2. Thus, k = 2.
Solve the equations 2x + y = 5 and x - y = 1 simultaneously.
Add equations: 3x = 6, x = 2. Substitute into x - y = 1: 2 - y = 1, y = 1. Solution is (2, 1).
Solve for x in the equation (x - 1)^2 / (x - 1) = 12.
(x - 1)^2 / (x - 1) = x - 1 (for x ≠ 1). So, x - 1 = 12, x = 13.
Solve the equations 3x - 2y = 7 and 2x + y = 8 simultaneously.
From 2x + y = 8, y = 8 - 2x. Substitute into 3x - 2y = 7: 3x - 2(8 - 2x) = 7, 7x - 16 = 7, 7x = 23, x = 3, y = 2.
If x = 2, find y in the equation y = 3(2^x) + 1.
y = 3(2^2) + 1 = 3(4) + 1 = 12 + 1 = 13.
In a triangle ABC, AB = 3 cm, BC = 4 cm, AC = 5 cm. Find cos A.
Cosine rule: cos A = (b^2 + c^2 - a^2) / (2bc), a = 4, b = 5, c = 3. cos A = (5^2 + 3^2 - 4^2) / (2 × 5 × 3) = 18/30 = 3/5.
Find the value of y when x = 4 in the equation y = x^2 - x + 3, given the pattern: when x = 1, y = 3; x = 2, y = 5; x = 3, y = 9.
y = x^2 - x + 3, when x = 4: y = 4^2 - 4 + 3 = 16 - 4 + 3 = 15.
Find the mean number of goals scored in matches, given: 0 goals in 1 match, 1 goal in 5 matches, 2 goals in 7 matches, 3 goals in 3 matches, 4 goals in 1 match, 5 goals in 1 match.
Total goals = 0×1 + 1×5 + 2×7 + 3×3 + 4×1 + 5×1 = 37. Total matches = 18. Mean = 37/18 ≈ 2.0.
If the hypotenuse of a right-angled isosceles triangle is 2, what is the length of each of the other sides?
Legs are equal, say x. Pythagorean theorem: x^2 + x^2 = 2^2, 2x^2 = 4, x = √2.
If two fair coins are tossed, what is the probability of getting at least one head?
Outcomes: {HH, HT, TH, TT}. Favorable (at least one head): {HH, HT, TH}. Probability = 3/4.
The area of one face of two similar rectangular blocks is in the ratio 2:3. If the volume of the larger block is 81 cm^3, find the volume of the smaller block.
Area ratio 2:3, linear ratio √2:√3. Volume ratio = (√2/√3)^3. Smaller volume = 81 × (2/3)^(3/2) ≈ 54 cm^3.
The bearing of a bird on a tree from a hunter is N72°E. What is the bearing of the hunter from the bird?
N72°E is 72° east of north. Reverse bearing: 72° + 180° = 252°, or S72°W.
In a circle of radius 5 cm, a chord is 4 cm from the center. Find the length of the chord.
Distance to chord = 4 cm, radius = 5 cm. Half chord length = √(5^2 - 4^2) = 3 cm. Chord length = 6 cm.
A point is chosen at random inside a circle of radius 5 cm. What is the probability it lies within 3 cm of the center?
Area of circle = π(5)^2 = 25π. Area within 3 cm = π(3)^2 = 9π. Probability = 9π/25π = 9/25.
A solid sphere of radius 4 cm has a mass of 64 kg. What is the mass of a shell of the same metal with internal radius 2 cm and external radius 4 cm?
Sphere volume = (4/3)π(4)^3 = 256π/3 cm^3. Density = 64 / (256π/3) kg/cm^3. Shell volume = (4/3)π(4^3 - 2^3) = 224π/3 cm^3. Mass = density × volume = 48 kg.
Find the area of a regular hexagon inscribed in a circle of radius 6 cm.
Side length = 6 cm. Hexagon has 6 equilateral triangles, each area (√3/4)(6)^2 = 9√3 cm^2. Total area = 6 × 9√3 = 54√3 cm^2.
In a circle of radius 5 cm, a chord subtends an angle of 60° at the center. Find the length of the chord.
Chord length = 2 × 5 × sin(30°) = 2 × 5 × (1/2) = 5 cm.
If sin θ = 1/2 and θ is in the first quadrant, find θ.
sin θ = 1/2 in the first quadrant implies θ = 30°.
Find the volume of a cone with radius 3 cm and height 4 cm.
Volume = (1/3)πr^2h = (1/3)π(3)^2(4) = 12π cm^3.
In a circle of radius 6 cm, find the area of a sector with a central angle of 60°.
Sector area = (60/360) × π(6)^2 = (1/6) × 36π = 6π cm^2.
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