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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Here are the solutions to the problems: RATIOS Section a) Step 1: Write the ratio as a fraction. (30)/(54) Step 2: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 6. (30 ÷ 6)/(54 ÷ 6) = (5)/(9) The ratio is 5:9. b) Step 1: Write the ratio as a fraction. (20)/(40) Step 2: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 20. (20 ÷ 20)/(40 ÷ 20) = (1)/(2) The ratio is 1:2. c) Step 1: Write the ratio as a fraction. (45)/(125) Step 2: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 5. (45 ÷ 5)/(125 ÷ 5) = (9)/(25) The ratio is 9:25. d) Step 1: Set up the proportion. (A)/(B) = (2)/(3) Step 2: Substitute the given value A = 30. (30)/(B) = (2)/(3) Step 3: Cross-multiply to solve for B. 30 × 3 = 2 × B 90 = 2B Step 4: Divide by 2. B = (90)/(2) B = 45 The value of B is 45. e) Step 1: Find the total number of parts in the ratio. 2 + 3 = 5 parts Step 2: Find the value of one part. sh. 1905 parts = sh. 38 per part Step 3: Calculate each share. First share: 2 × sh. 38 = sh. 76 Second share: 3 × sh. 38 = sh. 114 The shares are sh. 76 and sh. 114. f) Step 1: Find the total number of parts in the ratio. 2 + 3 + 5 = 10 parts Step 2: Find the value of one part. 1870 kg10 parts = 187 kg per part Step 3: Calculate each share. First share: 2 × 187 kg = 374 kg Second share: 3 × 187 kg = 561 kg Third share: 5 × 187 kg = 935 kg The shares are 374 kg, 561 kg, and 935 kg. g) Step 1: Set up the proportion. (L)/(S) = (3)/(5) Step 2: Substitute the given value L = 60. (60)/(S) = (3)/(5) Step 3: Cross-multiply to solve for S. 60 × 5 = 3 × S 300 = 3S Step 4: Divide by 3. S = (300)/(3) S = 100 The value of S is 100. h) Step 1: Set up the proportion. (K)/(H) = (7)/(9) Step 2: Substitute the given value H = 54. (K)/(54) = (7)/(9) Step 3: Multiply both sides by 54 to solve for K. K = (7)/(9) × 54 K = 7 × 6 K = 42 The value of K is 42. i) Step 1: Set up the proportion. (N)/(P) = (1)/(3) Step 2: Substitute the given value P = 42. (Assuming the question meant to find N, as P is given). (N)/(42) = (1)/(3) Step 3: Multiply both sides by 42 to solve for N. N = (1)/(3) × 42 N = 14 The value of N is 14. j) The question asks to find P, but P is not present in the given equation (x)/(y) = (2)/(13). This question is incomplete or contains a typo. If it meant to find the ratio (x)/(y), then it is already given. k) The question asks to find n, but the value of m is not given. This question is incomplete or contains a typo. l) Step 1: Write the equation. (1)/(17) Q = (5)/(34) Step 2: Multiply both sides by 17 to solve for Q. Q = (5)/(34) × 17 Q = (5 × 17)/(34) Q = (5 × 17)/(2 × 17) Step 3: Simplify the fraction. Q = (5)/(2) Q = 2.5 The value of Q is 2.5. m) Step 1: Write the ratio as a fraction. (25)/(75) Step 2: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 25. (25 ÷ 25)/(75 ÷ 25) = (1)/(3) The simplified ratio is 1:3. n) Step 1: Write the ratio as a fraction. (8)/(36) Step 2: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4. (8 ÷ 4)/(36 ÷ 4) = (2)/(9) The simplified ratio is 2:9. o) Step 1: Set up the proportion. (A)/(B) = (1)/(3) Step 2: Substitute the given value B = 240. (A)/(240) = (1)/(3) Step 3: Multiply both sides by 240 to solve for A. A = (1)/(3) × 240 A = 80 The value of A is 80. p) Step 1: Set up the proportion. (A)/(B) = (9)/(7) Step 2: Substitute the given value B = 175. (A)/(175) = (9)/(7) Step 3: Multiply both sides by 175 to solve for A. A = (9)/(7) × 175 A = 9 × 25 A = 225 The value of A is 225. q) Step 1: Set up the proportion. (P)/(T) = (9)/(11) Step 2: Substitute the given value P = 198. (198)/(T) = (9)/(11) Step 3: Cross-multiply to solve for T. 198 × 11 = 9 × T 2178 = 9T Step 4: Divide by 9. T = (2178)/(9) T = 242 The value of T is 242. r) Step 1: Write