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      Question

      Sum of first 20 odd natural numbers is (10m + 200),

      whereas sum of first 24 even natural numbers is (10n + 200). 'p' is the number of factors of (mn - 336). Ratio of milk and water in a mixture is (n + p):(m - p) respectively. When (n - p) litres of water is added, the ratio of milk to water becomes (m + n):3p. Find the final quantity of mixture.
      A 150 lit Correct Answer Incorrect Answer
      B 120 lit Correct Answer Incorrect Answer
      C 144 lit Correct Answer Incorrect Answer
      D 192 lit Correct Answer Incorrect Answer
      E 180 lit Correct Answer Incorrect Answer

      Solution

      Sum of first 'a' odd natural numbers = a²
      So, 10m + 200 = 20²
      Or, 10m + 200 = 400 → 10m = 200 → m = 20
      Sum of first 'b' even natural numbers = b × (b + 1)
      So, 10n + 200 = 24 × 25
      Or, 10n + 200 = 600 → 10n = 400 → n = 40
      Now, mn - 336 = 20 × 40 - 336 = 800 - 336 = 464
      464 = 2⁴ × 29
      Number of factors = (4 + 1) × (1 + 1) = 10
      So, p = 10
      Ratio of initial milk to water = (n + p):(m - p) = (40 + 10):(20 - 10) = 50:10 = 5:1
      Ratio of final milk to water = (m + n):(3p) = (20 + 40):(30) = 60:30 = 2:1
      Let initial milk and water be 5z and z litres.
      n - p = 40 - 10 = 30
      So, 5z:(z + 30) = 2:1 → 5z = 2z + 60 → 3z = 60 → z = 20

      Required mixture = 5z + z + 30 = 6z + 30 = 120 + 30 = 150 litres

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