Understanding the Discount Problem

The problem asks us to find the original amount of a bill given the difference between two ways of applying discounts: a single discount and two successive discounts. We are told this difference in the final discount amount is ₹56.

Calculating the Single Discount

A single discount of 27% means that the price is reduced by 27% of the original amount. If the original amount of the bill is represented by 'A', the discount amount is:

Discount 1 = 27% of A

Discount 1 Amount $= \frac{27}{100} \times A$

Calculating the Equivalent Single Discount for Successive Discounts

When two successive discounts are applied, the second discount is applied to the price *after* the first discount. We have successive discounts of 20% and 10%.

Let the original amount be A.

1. First Discount (20%): After a 20% discount, the price becomes (100% - 20%) = 80% of the original amount.
2. Price after first discount = 80% of A $= \frac{80}{100} \times A$
3. Second Discount (10%): The second discount of 10% is applied to this reduced price. So, the final price is (100% - 10%) = 90% of the price after the first discount.
4. Final Price after successive discounts = 90% of (80% of A) $= \frac{90}{100} \times (\frac{80}{100} \times A)$
5. Final Price after successive discounts $= \frac{90 \times 80}{100 \times 100} \times A = \frac{7200}{10000} \times A = \frac{72}{100} \times A$

The final price is 72% of the original amount. This means the total discount given by the two successive discounts is (100% - 72%) = 28% of the original amount.

Discount 2 = 28% of A

Discount 2 Amount $= \frac{28}{100} \times A$

Finding the Difference in Discount Amounts

The problem states that the difference between these two discount amounts is ₹56.

Difference in Discount Amount = Discount 2 Amount - Discount 1 Amount

$₹56 = \frac{28}{100} \times A - \frac{27}{100} \times A$

$₹56 = (\frac{28 - 27}{100}) \times A$

$₹56 = \frac{1}{100} \times A$

This means that 1% of the original amount (A) is equal to ₹56.

Calculating the Amount of the Bill

If 1% of the original amount is ₹56, then 100% of the original amount is 100 times ₹56.

$A = 56 \times 100$

$A = ₹5600$

The amount of the bill is ₹5600.

Verification

Let's check our answer:

• Single discount of 27% on ₹5600: $\frac{27}{100} \times 5600 = 27 \times 56 = ₹1512$
• Successive discounts of 20% and 10% on ₹5600:
  1. First discount (20%): $\frac{20}{100} \times 5600 = ₹1120$. Price after discount: $5600 - 1120 = ₹4480$.
  2. Second discount (10%): $\frac{10}{100} \times 4480 = ₹448$. Final Price: $4480 - 448 = ₹4032$.
• Total discount for successive discounts: $1120 + 448 = ₹1568$. Alternatively, Original price - Final price = $5600 - 4032 = ₹1568$.
• Difference in discounts: $₹1568 - ₹1512 = ₹56$.

The calculated difference matches the difference given in the question, confirming our amount is correct.

Revision Table: Discount Calculations

Additional Information: Successive Discounts Formula

For two successive discounts of $d_1$% and $d_2$%, the equivalent single discount percentage (D) can be calculated using the formula:

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

Let's apply this to our problem with $d_1 = 20$% and $d_2 = 10$%:

$D = 20 + 10 - \frac{20 \times 10}{100}$

$D = 30 - \frac{200}{100}$

$D = 30 - 2$

$D = 28$%

This confirms our calculation that two successive discounts of 20% and 10% are equivalent to a single discount of 28%. The difference between this and the 27% single discount is 1%, which is equal to ₹56 of the bill amount. Thus, 1% of the amount = ₹56, leading to the total amount being ₹5600.