Question:

In what ratio three kinds of rice costing 1.45rs, 1.54rs and 1.70rs must be mixed so that the mixture can be sold at 1.65rs per kg.

Options:
a) 11:21:44
b) 20:11:44
c) 11:20:44
d) 44:20:11
e) None of these

Answer:

To solve this question, we will use the concept of weighted average.

Weighted average = (quantity of first item * price of first item + quantity of second item * price of second item + quantity of third item * price of third item) / (total quantity)

Let the quantities of first, second and third kind of rice be x, y and z respectively.

So, the weighted average of the mixture can be written as:

1.65 = (1.45x + 1.54y + 1.70z) / (x+y+z)

Simplifying this equation, we get:

1.65(x+y+z) = 1.45x + 1.54y + 1.70z

Multiplying by 100, we get:

165x + 165y + 165z = 145x + 154y + 170z

Simplifying further, we get:

20x = 9y + 5z

Now, we need to find the ratio of x:y:z. To do that, we will assume that x = 20k (where k is a constant). Substituting this value in the above equation, we get:

20k = 9y + 5z

Dividing by 5, we get:

4k = (9/5)y + z

We know that k, y and z are integers. To find the smallest possible values for k, y and z, we will assume k = 5 (as 20k needs to be divisible by 5). Substituting this value in the above equation, we get:

4(5) = (9/5)y + z

20 = 9y + 5z/5

20 = 9y + z

Now, we need to find two more equations in terms of y and z. To do that, we will assume that y = 11m (where m is a constant). Substituting this value in the above equation, we get:

20 = 9(11m) + z

z = 20 - 99m

Similarly, we will assume that z = 44n (where n is a constant). Substituting this value in the above equation, we get:

y = (20 - 44n)/11

Now, we have the values of x, y and z in terms of k, m and n. To find the ratio of x:y:z, we will assume k = 1, m = 1 and n = 2 (as these values will give us the smallest possible values for x, y and z).

Substituting these values, we get:

x = 20k = 20(1) = 20
y = 11m = 11(1) = 11
z = 44n = 44(2) = 88

Therefore, the ratio of x:y: