Understanding Successive Discounts and Equivalent Single Discount

Successive discounts are discounts applied one after another on the reduced price. When multiple discounts are applied in succession, they are equivalent to a single discount that provides the same final price reduction from the original price.

Let's consider the given problem where successive discounts of 5%, 10%, and p% are applied, and they are equivalent to a single discount of 31.6%.

Calculating the Price After Each Successive Discount

Assume the original price of an item is $100$.

1. After the first discount of 5%: The price becomes $100 \times (1 - \frac{5}{100}) = 100 \times (1 - 0.05) = 100 \times 0.95 = 95$.
2. After the second discount of 10%: This discount is applied to the reduced price of $95$. The price becomes $95 \times (1 - \frac{10}{100}) = 95 \times (1 - 0.10) = 95 \times 0.90$.

   Calculating $95 \times 0.90$: $95 \times \frac{9}{10} = \frac{855}{10} = 85.5$.

   So, after the second discount, the price is $85.5$.

3. After the third discount of p%: This discount is applied to the price of $85.5$. The price becomes $85.5 \times (1 - \frac{p}{100})$.

The final price after the three successive discounts is $85.5 \times (1 - \frac{p}{100})$.

Calculating the Final Price with the Equivalent Single Discount

The problem states that the successive discounts are equivalent to a single discount of 31.6% on the original price ($100$).

The final price with a single discount of 31.6% is $100 \times (1 - \frac{31.6}{100}) = 100 \times (1 - 0.316) = 100 \times 0.684 = 68.4$.

Equating the Final Prices and Solving for p

Since the successive discounts are equivalent to the single discount, the final prices must be equal:

Price after successive discounts = Price after single equivalent discount

\( 85.5 \times \left(1 - \frac{p}{100}\right) = 68.4 \)

Now, we need to solve this equation for p:

\( 1 - \frac{p}{100} = \frac{68.4}{85.5} \)

To simplify the fraction, we can remove the decimal points by multiplying the numerator and denominator by 10:

\( 1 - \frac{p}{100} = \frac{684}{855} \)

Let's simplify the fraction $\frac{684}{855}$. Both numbers are divisible by 9 (sum of digits $6+8+4=18$, $8+5+5=18$).

\( 684 \div 9 = 76 \)

\( 855 \div 9 = 95 \)

So, \( \frac{684}{855} = \frac{76}{95} \).

Now simplify $\frac{76}{95}$. Both numbers are divisible by 19 ($19 \times 4 = 76$, $19 \times 5 = 95$).

\( \frac{76}{95} = \frac{4}{5} \)

Convert the fraction to a decimal: \( \frac{4}{5} = 0.8 \).

So the equation becomes:

\( 1 - \frac{p}{100} = 0.8 \)

Now, isolate $\frac{p}{100}$:

\( \frac{p}{100} = 1 - 0.8 \)

\( \frac{p}{100} = 0.2 \)

Finally, solve for p:

\( p = 0.2 \times 100 \)

\( p = 20 \)

Thus, the value of p is 20.

Alternative Method: Formula for Equivalent Discount

If successive discounts are \( d_1\%, d_2\%, \) and \( d_3\% \), the equivalent single discount \( D\% \) is given by:

\( 1 - \frac{D}{100} = \left(1 - \frac{d_1}{100}\right) \times \left(1 - \frac{d_2}{100}\right) \times \left(1 - \frac{d_3}{100}\right) \)

In this problem, \( d_1 = 5\% \), \( d_2 = 10\% \), \( d_3 = p\% \), and \( D = 31.6\% \).

Substitute the values into the formula:

\( 1 - \frac{31.6}{100} = \left(1 - \frac{5}{100}\right) \times \left(1 - \frac{10}{100}\right) \times \left(1 - \frac{p}{100}\right) \)

\( 1 - 0.316 = (1 - 0.05) \times (1 - 0.10) \times \left(1 - \frac{p}{100}\right) \)

\( 0.684 = (0.95) \times (0.90) \times \left(1 - \frac{p}{100}\right) \)

Calculate the product of the known factors:

\( 0.95 \times 0.90 = 0.855 \)

So, \( 0.684 = 0.855 \times \left(1 - \frac{p}{100}\right) \)

Now, solve for \( \left(1 - \frac{p}{100}\right) \):

\( 1 - \frac{p}{100} = \frac{0.684}{0.855} \)

To simplify the fraction, multiply the numerator and denominator by 1000 to remove decimal points:

\( 1 - \frac{p}{100} = \frac{684}{855} \)

As calculated before, \( \frac{684}{855} = \frac{4}{5} = 0.8 \).

\( 1 - \frac{p}{100} = 0.8 \)

\( \frac{p}{100} = 1 - 0.8 = 0.2 \)

\( p = 0.2 \times 100 \)

\( p = 20 \)

Both methods yield the same result, confirming that the value of p is 20.

Summary of Calculation

Revision Table: Key Concepts

Additional Information: Applications of Discounts

Understanding discounts, whether single or successive, is crucial in various real-life scenarios and business calculations:

• Retail Pricing: Stores often use successive discounts (e.g., "30% off, plus an extra 10% for members") to attract customers.
• Financial Calculations: Concepts similar to successive discounts appear in calculations involving depreciation, where value decreases by a certain percentage each period.
• Sales Tax and Markups: While not discounts, these are related concepts involving percentage increases or decreases from a base price.

Remember that successive discounts are not simply added together. Applying a 10% discount and then a 20% discount is not the same as a single 30% discount. The second discount is always applied to the already reduced price.