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

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