Understanding Successive Discounts

Successive discounts, also known as sequential discounts or chain discounts, are multiple discounts applied one after the other on the price of an item. Each discount is applied to the price remaining after the previous discount has been taken off, not the original price.

To find the total effect of successive discounts, you calculate the price after each discount is applied. The final price is then compared to the original price to find the total discount.

Calculating Total Discount Step-by-Step

Let's calculate the total discount for successive discounts of 20%, 10%, and 12.5%. We can assume an original price, for instance, $100, to make the calculation easy.

Step 1: Apply the first discount (20%)

The price after a 20% discount is \(100 \times (1 - 0.20) = 100 \times 0.80 = 80\).

The price is now $80.

Step 2: Apply the second discount (10%) to the new price

The price after a 10% discount on $80 is \(80 \times (1 - 0.10) = 80 \times 0.90 = 72\).

The price is now $72.

Step 3: Apply the third discount (12.5%) to the latest price

Note that 12.5% is equivalent to \(\frac{1}{8}\) as a fraction, or 0.125 as a decimal. The price after a 12.5% discount on $72 is \(72 \times (1 - 0.125) = 72 \times 0.875\).

Calculation: \(72 \times 0.875 = 72 \times \frac{7}{8} = 9 \times 7 = 63\).

The final price after all three discounts is $63.

Step 4: Calculate the total discount amount

The total discount is the original price minus the final price: \(100 - 63 = 37\).

The total discount amount is $37.

Step 5: Calculate the total discount percentage

The total discount percentage is the total discount amount divided by the original price, multiplied by 100.

\(\text{Total Discount Percentage} = \frac{\text{Total Discount}}{\text{Original Price}} \times 100\)

\(\text{Total Discount Percentage} = \frac{37}{100} \times 100 = 37\%\)

The total successive discount is 37%.

Successive Discount Calculation Revision Table

Original Price = $100

Final Price = $63

Total Discount = $100 - $63 = $37

Total Discount Percentage = \(\frac{37}{100} \times 100\% = 37\%\)

Additional Information on Equivalent Discounts

You can also find the single equivalent discount for successive discounts using a formula. For two successive discounts, say \(d_1\%\) and \(d_2\%\), the single equivalent discount \(D\) is given by:

\(D = d_1 + d_2 - \frac{d_1 \times d_2}{100}\)

For three or more discounts, you can first find the equivalent discount for the first two, and then use that equivalent discount with the third discount in the same formula, and so on. This formula gives the same result as the step-by-step method but requires careful calculation, especially with fractions or decimals.

For example, for 20% and 10% successive discounts:

\(D_{1,2} = 20 + 10 - \frac{20 \times 10}{100} = 30 - \frac{200}{100} = 30 - 2 = 28\%\)

Now, combine this 28% equivalent discount with the third discount of 12.5%:

\(D_{1,2,3} = 28 + 12.5 - \frac{28 \times 12.5}{100} = 40.5 - \frac{350}{100} = 40.5 - 3.5 = 37\%\)

Both methods confirm that the total discount applicable is 37%.