Hi Deepanshi,
We have multiplied the ratios because we have been given the product of the prices, not the sum.

The ratio of smallest to medium was 2:5. We let the ratio be "k".
Now, we have prices of 2k and 5k.
Since there is no information about the largest cup, we assumed it as L only  because we have two equations and L is common in both so we can equate the both equations using L.

Now, in the question, we were given the product of the prices.

Therefore, $$2K\times\ 5K\times\ L=800$$
Similarly done for the other equation i.e. $$\left(2K+6\right)\times\ \left(5K+6\right)\times\ L=3200$$
Using L, we equate both.

I hope this clears your doubt.

Hi Chirag,

Here is the breakdown.
$$5x^2-7x-6=0$$
Now, we will factorise the equation: $$5x^2-10x+3x-6=0$$
$$5x\left(x^{ }-2\right)+3\left(x-2\right)=0$$
$$\left(5x+3\right)\left(x-2\right)=0$$
Now, putting both i,e, (5x+3) and (x-2) equal to 0.
We get x = $$-\frac{3}{5},2$$
Since this is the price, therefore it can't be negative. Hence, x = 2 is the only option.

I hope this clarifies your doubts.