Explanation:

To determine the cost price of the entire quantity of sugar, let's analyze the problem:

- Let the total quantity of sugar be $x$.

- Half of $x$ (i.e., $x/2$) was sold at a 20% profit.

- A quarter of $x$ (i.e., $x/4$) was sold at a 16% profit.

- The remaining quarter $x/4$ was sold at a 24% profit.

- The total profit is Rs 90.

Calculating the Profit from Each Portion:

1. Half of the sugar ($x/2$):

- Profit = $\frac{x}{2}\times 20\mathrm{\%}$.

2. Quarter of the sugar ($x/4$):

- Profit = $\frac{x}{4}\times 16\mathrm{\%}$.

3. Remaining Quarter ($x/4$):

- Profit = $\frac{x}{4}\times 24\mathrm{\%}$.

Setting up the Equation:

Total profit =

$$

\frac{x}{2} \times 0.2 + \frac{x}{4} \times 0.16 + \frac{x}{4} \times 0.24 = 90

$$

Simplify:

$$

0.1x + 0.04x + 0.06x = 90

$$

$$

0.2x = 90

$$

$$

x = 450

$$

- Option 1: Rs 450 is the cost price of the entire quantity of sugar.

Conclusion:

- Correct Answer: Option 1: Rs 450

- .