This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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1. (a) Simple Interest and Amount Step 1: Calculate the simple interest (SI) using the formula SI = (P × R × T)/(100). Given: Principal (P) = N720,000, Rate (R) = 5(1)/(2)\% = 5.5\%, Time (T) = 3 years. SI = (720000 × 5.5 × 3)/(100) SI = 7200 × 5.5 × 3 SI = 39600 × 3 SI = N118,800 Step 2: Calculate the total amount (A) at the end of 3 years using the formula A = P + SI. A = 720000 + 118800 A = N838,800 The simple interest is N118,800 and the total amount is N838,800. 1. (b) Ratio Problem Step 1: Determine the initial number of boys and girls. Total students = 75. Ratio of boys to girls = 2:3. Total ratio parts = 2+3 = 5. Number of boys = (2)/(5) × 75 = 30. Number of girls = (3)/(5) × 75 = 45. Step 2: Set up the equation for the new ratio. Let x be the number of boys added. The number of girls remains 45. New number of boys = 30+x. The new ratio of boys to girls is 7:9. (30+x)/(45) = (7)/(9) Step 3: Solve for x. Multiply both sides by 45: 30+x = (7)/(9) × 45 30+x = 7 × 5 30+x = 35 x = 35 - 30 x = 5 5 boys should be added to the class. 2. (a) Solve the Equation Step 1: Combine the terms on the right side of the equation. (3)/(a) = (2)/(a+1) + (3)/(3a-1) Find a common denominator for the right side, which is (a+1)(3a-1). (3)/(a) = (2(3a-1) + 3(a+1))/((a+1)(3a-1)) (3)/(a) = (6a-2 + 3a+3)/((a+1)(3a-1)) (3)/(a) = (9a+1)/((a+1)(3a-1)) Step 2: Cross-multiply and expand the equation. 3(a+1)(3a-1) = a(9a+1) 3(3a^2 - a + 3a - 1) = 9a^2 + a 3(3a^2 + 2a - 1) = 9a^2 + a 9a^2 + 6a - 3 = 9a^2 + a Step 3: Simplify and solve for a. Subtract 9a^2 from both sides: 6a - 3 = a Subtract a from both sides and add 3 to both sides: 6a - a = 3 5a = 3 a = (3)/(5) The solution is a = (3)/(5). 2. (b) Simplify Expression Step 1: Rewrite the division as multiplication by the reciprocal of the second fraction. (x-y)/(y+x) ÷ (y-x)/(x+y) = (x-y)/(y+x) × (x+y)/(y-x) Step 2: Recognize that y+x = x+y and y-x = -(x-y). Substitute these into the expression: = (x-y)/(x+y) × (x+y)/(-(x-y)) Step 3: Cancel common terms. = (1)/(1) × (1)/(-1) = -1 The simplified expression is -1. 3. (a) Shortest Distance from A to C Step 1: Draw a diagram and determine the angle ABC. A man travels 5 km due east from A to B. So, AB = 5 km. Then he travels 7 km on a bearing of 162^ from B to C. So, BC = 7 km. The bearing of 162^ is measured clockwise from the North line at B. Since AB is due East, the angle from the North line at B to the East direction is 90^. The angle from the East direction (extension of AB) to BC is 162^ - 90^ = 72^. The interior angle ABC in triangle ABC is 180^ - 72^ = 108^. Step 2: Use the Cosine Rule to find the distance AC. The Cosine Rule states c^2 = a^2 + b^2 - 2ab C. Here, AC^2 = AB^2 + BC^2 - 2(AB)(BC)( ABC). AC^2 = 5^2 + 7^2 - 2(5)(7)(108^) AC^2 = 25 + 49 - 70(-0.3090) AC^2 = 74 + 21.63 AC^2 = 95.63 AC = sqrt(95.63) AC ≈ 9.779 km Step 3: Correct to the nearest km. AC ≈ 10 km The shortest distance from A to C is 10 km. 3. (b) Bearing of C from A Step 1: Use the Sine Rule to find BAC. The Sine Rule states (a)/( A) = (b)/( B). Here, (BC)/(( BAC)) = (AC)/(( ABC)). (7)/(( BAC)) = (9.779)/((108^)) ( BAC) = (7 × (108^))/(9.779) ( BAC) = (7 × 0.9511)/(9.779) ( BAC) = (6.6577)/(9.779) ( BAC) ≈ 0.6808 BAC = (0.6808) BAC ≈ 42.9^ Step 2: Calculate the bearing of C from A. Since A to B is due East, the bearing of B from A is 090^. The angle BAC is measured clockwise from AB. The bearing of C from A is 90^ + BAC. Bearing = 90^ + 42.9^ = 132.9^. Step 3: Correct to the nearest degree. Bearing ≈ 133^. The bearing of C from A is 133^. 4. (a) Simplify Algebraic Expression Step 1: Factorize the denominators. (8)/(3r^2-48) - (4)/(3r^2-12r) 3r^2-48 = 3(r^2-16) = 3(r-4)(r+4) 3r^2-12r = 3r(r-4) Substitute the factored denominators back into the expression: = (8)/(3(r-4)(r+4)) - (4)/(3r(r-4)) Step 2: Find a common denominator and combine the fractions. The common denominator is 3r(r-4)(r+4). = (8r)/(3r(r-4)(r+4)) - (4(r+4))/(3r(r-4)(r+4)) = (8r - 4(r+4))/(3r(r-4)(r+4)) = (8r - 4r - 16)/(3r(r-4)(r+4)) = (4r - 16)/(3r(r-4)(r+4)) Step 3: Factor the numerator and simplify. = (4(r-4))/(3r(r-4)(r+4)) Cancel out the common factor (r-4): = (4)/(3r(r+4)) The simplified expression is (4)/(3r(r+4)). 4. (b) Evaluate Integral Step 1: Expand the integrand (3x^2+2)^2. (3x^2+2)^2 = (3x^2)^2 + 2(3x^2)(2) + 2^2 = 9x^4 + 12x^2 + 4 Step 2: Integrate each term with respect to x. (9x^4 + 12x^2 + 4) dx = 9x^4+14+1 + 12x^2+12+1 + 4x + C = (9x^5)/(5) + (12x^3)/(3) + 4x + C Step 3: Simplify the expression. = (9)/(5)x^5 + 4x^3 + 4x + C The evaluated integral is (9)/(5)x^5 + 4x^3 + 4x + C. Send me the next one 📸