Substitute x = 5k and y = 2k into the expression. (8(5k) + 9(2k)) : (8(5k) + 2(2k)) = (40k + 18k) : (40k + 4k) = 58k : 44k. Dividing by 2k gives 29 : 22.

Given that 15% of x = 20% of y. We can write this as 15x/100 = 20y/100, which simplifies to 15x = 20y. Dividing both sides by 5 gives 3x = 4y. Therefore, x/y = 4/3, or x : y = 4 : 3.

The compounded ratio of ratios (a:b), (c:d), and (e:f) is (a*c*e) : (b*d*f). Here, it is (2 * 4 * 6) : (3 * 5 * 7) = 48 : 105. Dividing both sides by 3 simplifies it to 16 : 35.

The sub-triplicate ratio of a : b is the ratio of their cube roots. For 64 : 125, the cube root of 64 is 4, and the cube root of 125 is 5. Therefore, the ratio is 4 : 5.

The triplicate ratio of a : b is the ratio of their cubes, a^3 : b^3. For 2 : 3, the triplicate ratio is 2^3 : 3^3, which equals 8 : 27.

The sub-duplicate ratio of two numbers a : b is the ratio of their square roots, sqrt(a) : sqrt(b). For 16 : 49, the sub-duplicate ratio is sqrt(16) : sqrt(49), which equals 4 : 7.

The duplicate ratio of two numbers a : b is the ratio of their squares, a^2 : b^2. For 3 : 5, the duplicate ratio is 3^2 : 5^2, which is 9 : 25.

These are standard properties of proportions. (a) is Componendo, (b) is Dividendo, and (c) is Componendo and Dividendo. If a/b = c/d, applying Componendo and Dividendo yields (a+b)/(a-b) = (c+d)/(c-d). All the given statements are correct transformations.

Initial quantities: Milk = (2/3) * 60 = 40 L, Water = (1/3) * 60 = 20 L. Let x liters of water be added. The new ratio of milk to water is 40 / (20 + x) = 1 / 2. Cross-multiplying gives 80 = 20 + x, so x = 60. 60 liters of water must be added.

Let the present ages be 7x and 3x. After 10 years, (7x + 10) / (3x + 10) = 2 / 1. Cross-multiplying gives 7x + 10 = 6x + 20, which simplifies to x = 10. The present age of the father is 7x = 7 * 10 = 70 years.