Find the single discount rate percentage equivalent to the successive discounts of 10%, 20% and 20%.

The correct option is 42.4%.

To find the single discount rate percentage equivalent to successive discounts of 10%, 20%, and 20%, we can determine the final price after all three discounts are applied to an initial marked price.

Method 1: Step-by-Step Calculation (Assuming Initial Price = 100)

Let the initial Marked Price (MP) be 100.

Step 1: Apply the first discount of 10%
$\text{Price after 1st discount}=100\times \left(1-\frac{10}{100}\right)=100\times 0.90=90$

Step 2: Apply the second discount of 20% on 90
$\text{Price after 2nd discount}=90\times \left(1-\frac{20}{100}\right)=90\times 0.80=72$

Step 3: Apply the third discount of 20% on 72
$\text{Final Selling Price (SP)}=72\times \left(1-\frac{20}{100}\right)=72\times 0.80=57.6$

Step 4: Calculate the total single equivalent discount
$\text{Total Discount}=\text{Marked Price}-\text{Final Selling Price}$
$\text{Total Discount}=100-57.6=42.4$

Since the initial Marked Price was 100, the single equivalent discount rate percentage is 42.4%.

Method 2: Using Successive Discount Multipliers

The single equivalent selling price factor for successive discounts $d_1$, $d_2$, and $d_3$ is calculated as:
$\text{Selling Price Factor}=\left(1-d_1\right)\times \left(1-d_2\right)\times \left(1-d_3\right)$

Substituting $d_1=0.10$, $d_2=0.20$, and $d_3=0.20$:
$\text{Selling Price Factor}=\left(1-0.10\right)\times \left(1-0.20\right)\times \left(1-0.20\right)=0.90\times 0.80\times 0.80=0.576$

The single equivalent discount rate percentage is:
$\text{Equivalent Discount Rate}=\left(1-0.576\right)\times 100\%=0.424\times 100\%=42.4\%$