Sum of first 33 odd natural numbers is (19m + 63), whereas sum of first 28 even natural numbers is (17n + 98). 'p' is the number of factors of (mn - 1288). 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): 4p. Find the final quantity of mixture.

Sum of first 'a' odd natural numbers = a² So, 19m + 63 = 33² Or, 19m + 63 = 1089 Or, 19m = 1026 Or, m = 54 Sum of first 'b' even natural numbers = b×(b+1) So, 17n + 98 = 28 × 29 Or, 17n + 98 = 812 Or, 17n = 714 Or, n = 42 Now, mn - 1288 = 54 × 42 - 1288 = 2268 - 1288 = 980 980 = 2² × 5 × 7² Number of factors = (2+1) × (1+1) × (2+1) So, p = 3 × 2 × 3 = 18 Ratio of initial quantity of milk to water = (42+18):(54-18) = 60:36 = 5:3 Ratio of final quantity of milk to water = (54+42):(4×18) = 96:72 = 4:3 Let the initial quantity of milk and water be '5z' and '3z' litres respectively. n - p = 42 - 18 = 24 So, 5z:(3z + 24) = 4:3 Or, 15z=12z+96 Or, 3z = 96 Or, z = 32 Required final quantity of mixture = 5z+3z+24 = 8z + 24 = 8 × 32 + 24 = 280 litres