Samacheer Kalvi Class 8 Maths Solutions Chapter 4 Life Mathematics Exercise 4.2

 

Question 1. Fill in the blanks:
(i) Loss or gain percentage is always calculated on the ______.
(ii) A mobile phone is sold for Rs 8400 at a gain of 20%. The cost price of the mobile phone is ______.
(iii) An article is sold for Rs 555 at a loss of \( 7\frac { 1 }{ 2 }\)%. The cost price of the article is ______.
(iv) A mixer grinder marked at Rs 4500 is sold for Rs 4140 after discount. The rate of discount is ______.
(v) The total bill amount of a shirt costing Rs 575 and a T-shirt costing Rs 325 with GST of 5% is ______.

Answer:
(i) Loss or gain percentage is always calculated on the Cost Price.
(ii) A mobile phone is sold for Rs 8400 at a gain of 20%. The cost price of the mobile phone is Rs 7000.
Hint:
Let the cost price of the mobile phone be \( x \).
The selling price (SP) is Rs 8400, and the gain is 20%.
Using the formula for selling price with gain:
\( SP = CP \times \left( \frac{100 + \text{gain}\%}{100} \right) \)
Substitute the given values:
\( 8400 = x \times \left( \frac{100 + 20}{100} \right) \)
\( 8400 = x \times \frac{120}{100} \)
Now, solve for \( x \):
\( x = \frac{8400 \times 100}{120} \)
\( x = \frac{840000}{120} \)
\( x = 7000 \)
So, the cost price is Rs 7000. It's important to remember that profit calculations always depend on the original cost.
(iii) An article is sold for Rs 555 at a loss of \( 7\frac { 1 }{ 2 }\)%. The cost price of the article is Rs 600.
Hint:
Given selling price (SP) is Rs 555, and loss is \( 7\frac { 1 }{ 2 }\)% \( = 7.5\% \).
Using the formula for selling price with loss:
\( SP = CP \times \left( \frac{100 - \text{loss}\%}{100} \right) \)
Substitute the given values:
\( 555 = CP \times \left( \frac{100 - 7.5}{100} \right) \)
\( 555 = CP \times \left( \frac{92.5}{100} \right) \)
\( CP = \frac{555 \times 100}{92.5} \)
\( CP = \frac{55500}{92.5} \)
\( CP = 600 \)
Therefore, the cost price of the article is Rs 600. Understanding how loss affects the selling price is key to these calculations.
(iv) A mixer grinder marked at Rs 4500 is sold for Rs 4140 after discount. The rate of discount is 8%.
Hint:
Marked price (MP) = Rs 4500
Selling price after discount = Rs 4140
First, find the discount amount:
Discount = Marked Price - Selling Price
Discount = Rs 4500 - Rs 4140 = Rs 360
Now, calculate the rate of discount using the formula:
\( \text{Rate of discount} = \left( \frac{\text{Discount}}{\text{Marked Price}} \right) \times 100 \)
\( \text{Rate of discount} = \left( \frac{360}{4500} \right) \times 100 \)
\( \text{Rate of discount} = 0.08 \times 100 \)
\( \text{Rate of discount} = 8\% \)
So, the discount rate is 8%. Discounts are usually calculated on the marked price of an item.
(v) The total bill amount of a shirt costing Rs 575 and a T-shirt costing Rs 325 with GST of 5% is Rs 945.
For the shirt:
Cost price (CP) = Rs 575
GST = 5%
Bill amount for shirt \( = CP \times \left( \frac{100 + \text{GST}\%}{100} \right) \)
Bill amount for shirt \( = 575 \times \left( \frac{100 + 5}{100} \right) \)
Bill amount for shirt \( = 575 \times \frac{105}{100} = 575 \times 1.05 = \text{Rs } 603.75 \)
For the T-shirt:
Cost price (CP) = Rs 325
GST = 5%
Bill amount for T-shirt \( = CP \times \left( \frac{100 + \text{GST}\%}{100} \right) \)
Bill amount for T-shirt \( = 325 \times \left( \frac{100 + 5}{100} \right) \)
Bill amount for T-shirt \( = 325 \times \frac{105}{100} = 325 \times 1.05 = \text{Rs } 341.25 \)
Total bill amount = Bill amount for shirt + Bill amount for T-shirt
Total bill amount = Rs 603.75 + Rs 341.25 = Rs 945.00
Adding GST increases the final price of the items.
In simple words: We find the final price for each item by adding 5% GST to its cost, then we add those two final prices together to get the total bill.

🎯 Exam Tip: When calculating percentages, always identify the base value correctly (e.g., cost price for profit/loss, marked price for discount, or original price for GST).

 

Question 2. If selling an article for Rs 820 causes 10% loss on the selling price, then find its cost price.
Answer: Given that the selling price (SP) is Rs 820 and the loss percentage is 10%.
We use the formula relating cost price (CP), selling price (SP), and loss percentage:
\( SP = CP \times \left( \frac{100 - \text{loss}\%}{100} \right) \)
Substitute the known values:
\( 820 = CP \times \left( \frac{100 - 10}{100} \right) \)
\( 820 = CP \times \left( \frac{90}{100} \right) \)
To find the cost price (CP), rearrange the equation:
\( CP = \frac{820 \times 100}{90} \)
\( CP = \frac{82000}{90} \)
\( CP = 911.11 \) (approximately)
The cost price of the article is approximately Rs 911.11. This calculation helps us understand the original value before any loss was incurred.
In simple words: If something is sold for Rs 820 and that means a 10% loss, we can work backwards to find out its original cost price by using a simple formula.

🎯 Exam Tip: Always make sure you understand if the percentage (profit/loss) is based on the cost price or selling price, as this changes the calculation. Usually, it's on the cost price unless stated otherwise.

 

Question 3. If the profit earned on selling an article for Rs 810 is the same as loss on selling it for Rs 530, then find the cost price of the article.
Answer: Let the cost price of the article be CP.
In Case 1: The article is sold for Rs 810, and there is a profit.
Profit = Selling price (SP) - Cost price (CP)
So, Profit (P) \( = 810 - CP \)
In Case 2: The article is sold for Rs 530, and there is a loss.
Loss = Cost price (CP) - Selling price (SP)
So, Loss (L) \( = CP - 530 \)
The problem states that the profit in Case 1 is the same as the loss in Case 2.
\( P = L \)
\( 810 - CP = CP - 530 \)
Now, we need to solve for CP. Gather the CP terms on one side and numbers on the other:
\( 810 + 530 = CP + CP \)
\( 1340 = 2CP \)
Divide by 2 to find CP:
\( CP = \frac{1340}{2} \)
\( CP = 670 \)
The cost price of the article is Rs 670. This type of problem shows how profit and loss relate to the cost price from different selling scenarios.
In simple words: If selling an item for Rs 810 makes the same profit as selling it for Rs 530 makes a loss, then the cost price of the item is exactly in the middle of these two selling prices.

🎯 Exam Tip: When profit equals loss for two different selling prices, the cost price is the average of those two selling prices. Always define your variables clearly before starting calculations.

 

Question 4. If the selling price of 10 rulers is the same as the cost price of 15 rulers, then find the profit percentage.
Answer: Let the cost price of one ruler be Rs \( x \).
According to the problem, the cost price of 15 rulers is \( 15x \).
The selling price of 10 rulers is equal to the cost price of 15 rulers.
So, selling price (SP) of 10 rulers \( = 15x \).
This means the selling price (SP) of one ruler \( = \frac{15x}{10} = 1.5x \).
Now, we can find the gain (profit) for one ruler:
Gain \( = \) SP of 1 ruler \( - \) CP of 1 ruler
Gain \( = 1.5x - x = 0.5x \)
To find the profit percentage, use the formula:
\( \text{Gain}\% = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100 \)
\( \text{Gain}\% = \left( \frac{0.5x}{x} \right) \times 100 \)
\( \text{Gain}\% = 0.5 \times 100 \)
\( \text{Gain}\% = 50\% \)
The profit percentage is 50%. This illustrates how a difference in quantity can lead to significant profit when selling prices are fixed.
In simple words: If you sell 10 rulers for the same amount it cost to buy 15 rulers, you are making a 50% profit on each ruler you sell.

🎯 Exam Tip: When quantities are involved in cost and selling price comparisons, always find the per-unit cost and selling price to calculate profit/loss percentages accurately.

 

Question 5. Some articles are bought at 2 for Rs 15 and sold at 3 for Rs 25. Find the gain percentage.
Answer: Let's find the cost price and selling price for one article.
Cost Price (CP):
2 articles are bought for Rs 15.
So, the cost price of one article \( = \frac{15}{2} = \text{Rs } 7.50 \).
Selling Price (SP):
3 articles are sold for Rs 25.
So, the selling price of one article \( = \frac{25}{3} = \text{Rs } 8.33 \) (approximately).
Now, calculate the gain for one article:
Gain \( = \) SP \( - \) CP
Gain \( = \frac{25}{3} - \frac{15}{2} \)
To subtract, find a common denominator, which is 6:
Gain \( = \frac{25 \times 2}{3 \times 2} - \frac{15 \times 3}{2 \times 3} = \frac{50}{6} - \frac{45}{6} = \frac{5}{6} \)
So, the gain on one article is Rs \( \frac{5}{6} \). This represents the small profit made on each item.
Now, find the gain percentage:
\( \text{Gain}\% = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100 \)
\( \text{Gain}\% = \left( \frac{5/6}{15/2} \right) \times 100 \)
\( \text{Gain}\% = \left( \frac{5}{6} \times \frac{2}{15} \right) \times 100 \)
\( \text{Gain}\% = \left( \frac{10}{90} \right) \times 100 \)
\( \text{Gain}\% = \left( \frac{1}{9} \right) \times 100 \)
\( \text{Gain}\% = \frac{100}{9} = 11\frac{1}{9}\% \)
The gain percentage is \( 11\frac{1}{9}\% \).
In simple words: We find the price of one item when bought and when sold. Then we calculate the profit per item and use that to find the overall percentage profit.

🎯 Exam Tip: For problems involving buying and selling different quantities, always find the unit price (cost and selling) first to simplify calculations and avoid errors.

 

Question 6. By selling a speaker for Rs 768, a man loses 20%. In order to gain 20%, how much should he sell the speaker?
Answer: First, we need to find the cost price (CP) of the speaker.
Given: Selling Price (SP) = Rs 768, Loss % = 20%.
Using the formula for SP with loss:
\( SP = CP \times \left( \frac{100 - \text{loss}\%}{100} \right) \)
\( 768 = CP \times \left( \frac{100 - 20}{100} \right) \)
\( 768 = CP \times \left( \frac{80}{100} \right) \)
Now, solve for CP:
\( CP = \frac{768 \times 100}{80} \)
\( CP = \frac{76800}{80} \)
\( CP = 960 \)
So, the cost price of the speaker is Rs 960.
Next, we want to find the selling price (SP) needed to gain 20%.
Given: CP = Rs 960, Gain % = 20%.
Using the formula for SP with gain:
\( SP = CP \times \left( \frac{100 + \text{gain}\%}{100} \right) \)
\( SP = 960 \times \left( \frac{100 + 20}{100} \right) \)
\( SP = 960 \times \left( \frac{120}{100} \right) \)
\( SP = 960 \times 1.20 \)
\( SP = 1152 \)
The man should sell the speaker for Rs 1152 to gain 20%. This two-step problem requires careful calculation of the cost price first.
In simple words: First, we find the real price the man paid for the speaker. Then, we use that real price to calculate how much he needs to sell it for to make a 20% profit.

🎯 Exam Tip: Break down two-part problems like this. First, find the unknown cost price using the given loss/profit. Second, use that cost price to calculate the new selling price for the desired profit/loss.

 

Question 7. Find the unknowns x, y and z.

Answer:
(i) For the Book:
Marked Price (MP) = Rs 225
Discount = 8%
The selling price (x) is calculated using the formula:
\( \text{Selling Price} = \text{MP} \times \left( \frac{100 - \text{Discount}\%}{100} \right) \)
\( x = 225 \times \left( \frac{100 - 8}{100} \right) \)
\( x = 225 \times \left( \frac{92}{100} \right) \)
\( x = 225 \times 0.92 \)
\( x = 207 \)
So, the selling price of the book is Rs 207.
(ii) For the LED TV:
Selling Price (SP) = Rs 11970
Discount = 5%
Marked Price = y
Using the same formula, but solving for MP:
\( SP = y \times \left( \frac{100 - \text{Discount}\%}{100} \right) \)
\( 11970 = y \times \left( \frac{100 - 5}{100} \right) \)
\( 11970 = y \times \left( \frac{95}{100} \right) \)
\( y = \frac{11970 \times 100}{95} \)
\( y = 126 \times 100 \)
\( y = 12600 \)
So, the marked price of the LED TV is Rs 12600.
(iii) For the Digital clock:
Marked Price (MP) = Rs 750
Selling Price (SP) = Rs 615
Discount = z
First, find the discount amount:
Discount Amount = MP - SP
Discount Amount \( = 750 - 615 = \text{Rs } 135 \)
Now, calculate the discount percentage (z):
\( \text{Discount}\% = \left( \frac{\text{Discount Amount}}{\text{MP}} \right) \times 100 \)
\( z = \left( \frac{135}{750} \right) \times 100 \)
\( z = 0.18 \times 100 \)
\( z = 18 \)
So, the discount for the digital clock is 18%. These calculations help us understand how discounts affect prices.
In simple words: For each item, we use the marked price, selling price, and discount percentage to find the missing value. We use formulas that connect these three numbers to solve for x, y, and z.

🎯 Exam Tip: Always remember that discount is calculated on the marked price. When finding a missing value, carefully apply the formula for selling price with discount, rearranging it as needed.

 

Question 8. Find the total bill amount for the data given below:

Answer: We need to calculate the bill amount for each item by first applying the discount to the marked price, and then adding the GST to the discounted price.
**Formula for Discounted Price (DP):** \( DP = \text{MP} \times \left( \frac{100 - \text{Discount}\%}{100} \right) \)
**Formula for Bill Amount (Total Price):** \( \text{Bill Amount} = \text{DP} \times \left( \frac{100 + \text{GST}\%}{100} \right) \)

(i) For the School bag:
Marked Price (MP) = Rs 500
Discount = 5%
GST = 12%
First, calculate the discounted price (DP):
\( DP = 500 \times \left( \frac{100 - 5}{100} \right) = 500 \times \left( \frac{95}{100} \right) = 500 \times 0.95 = \text{Rs } 475 \)
Next, calculate the bill amount after adding GST:
\( \text{Bill Amount} = 475 \times \left( \frac{100 + 12}{100} \right) = 475 \times \left( \frac{112}{100} \right) = 475 \times 1.12 = \text{Rs } 532 \)

(ii) For the Hair dryer:
Marked Price (MP) = Rs 2000
Discount = 10%
GST = 28%
First, calculate the discounted price (DP):
\( DP = 2000 \times \left( \frac{100 - 10}{100} \right) = 2000 \times \left( \frac{90}{100} \right) = 2000 \times 0.9 = \text{Rs } 1800 \)
Next, calculate the bill amount after adding GST:
\( \text{Bill Amount} = 1800 \times \left( \frac{100 + 28}{100} \right) = 1800 \times \left( \frac{128}{100} \right) = 1800 \times 1.28 = \text{Rs } 2304 \)

Total Bill Amount = Bill amount for School bag + Bill amount for Hair dryer
Total Bill Amount = Rs 532 + Rs 2304 = Rs 2836
This problem demonstrates how multiple percentage changes (discount and GST) are applied sequentially to find the final selling price.
In simple words: For each item, we first subtract the discount from its original price. Then, we add the GST to this new, discounted price to get the final cost. Finally, we add up the final costs of all items to get the total bill.

🎯 Exam Tip: When both discount and GST are applied, always calculate the discount first, then add the GST to the *discounted price*. Never add GST to the original marked price then apply discount, as that would give a different result.

 

Question 9. A branded Air-Conditioner (AC) has a marked price of Rs 38000. There are 2 options given for the customer. (i) Selling Price is the same 38000 but with attractive gifts worth Rs 3000 (or) (ii) Discount of 8% on the marked price but no free gifts. Which offer is better?
Answer: The marked price of the AC is Rs 38000.

**Option 1: Selling Price is Rs 38000 with attractive gifts worth Rs 3000.**
In this option, the customer pays the full marked price of Rs 38000. However, they receive free gifts worth Rs 3000. This means the customer effectively gains Rs 3000 in value through the gifts.
Net gain for the customer = Value of gifts = Rs 3000.

**Option 2: Discount of 8% on the marked price but no free gifts.**
Here, the customer gets a direct discount on the marked price.
Discount = 8% of Rs 38000
Discount \( = \frac{8}{100} \times 38000 = 8 \times 380 = \text{Rs } 3040 \)
The selling price after the discount would be Rs 38000 - Rs 3040 = Rs 34960.
In this option, the customer saves Rs 3040 directly from the price.

**Comparison:**
In Option 1, the customer's net gain (from gifts) is Rs 3000.
In Option 2, the customer's net saving (from discount) is Rs 3040.

Since Rs 3040 is greater than Rs 3000, Option 2 is better for the customer.
The second offer provides a slightly higher financial benefit. It is important to calculate the actual financial benefit for the customer in both scenarios to make the best choice.
In simple words: We compare two offers for an AC. One gives free gifts worth Rs 3000. The other gives an 8% discount. We calculate that the 8% discount saves Rs 3040, which is more than the gift value, so the discount offer is better.

🎯 Exam Tip: To compare offers, always convert all benefits (gifts, discounts, cashbacks) into a monetary value and choose the option that provides the highest net gain or saving for the customer.

 

Question 10. If a mattress is marked for Rs 7500 and is available at two successive discount of 10% and 20%, find the amount to be paid by the customer.
Answer: The marked price of the mattress is Rs 7500.
There are two successive discounts: \( d_1 = 10\% \) and \( d_2 = 20\% \).

First, calculate the price after the first discount of 10%:
\( \text{Price after discount } d_1 = \text{MP} \times \left( \frac{100 - d_1\%}{100} \right) \)
\( = 7500 \times \left( \frac{100 - 10}{100} \right) \)
\( = 7500 \times \left( \frac{90}{100} \right) \)
\( = 7500 \times 0.9 = \text{Rs } 6750 \)

Next, apply the second discount of 20% to this new price (Rs 6750):
\( \text{Price after second discount } d_2 = (\text{Price after } d_1) \times \left( \frac{100 - d_2\%}{100} \right) \)
\( = 6750 \times \left( \frac{100 - 20}{100} \right) \)
\( = 6750 \times \left( \frac{80}{100} \right) \)
\( = 6750 \times 0.8 = \text{Rs } 5400 \)

The amount to be paid by the customer is Rs 5400. Successive discounts are applied one after another, not added together.
In simple words: We start with the mattress's original price. First, we take off the 10% discount. Then, we take off the 20% discount from that new, lower price. The final price is what the customer pays.

🎯 Exam Tip: Remember that successive discounts are applied one after another, always on the *reduced price* from the previous discount, not on the original marked price for each discount.

 

Objective Type Questions

 

Question 11. A fruit vendor sells fruits for Rs 200 gaining Rs 40. His gain percentage is
(A) 20%
(B) 22%
(C) 25%
(D) 16
Answer: (C) 25%
The selling price (SP) is Rs 200.
The gain (profit) is Rs 40.
To find the cost price (CP), we subtract the gain from the selling price:
\( CP = SP - \text{Gain} \)
\( CP = 200 - 40 = \text{Rs } 160 \)
Now, calculate the gain percentage:
\( \text{Gain}\% = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100 \)
\( \text{Gain}\% = \left( \frac{40}{160} \right) \times 100 \)
\( \text{Gain}\% = \left( \frac{1}{4} \right) \times 100 \)
\( \text{Gain}\% = 25\% \)
In simple words: The vendor sold fruits for Rs 200 and made Rs 40 profit. This means the fruits cost him Rs 160. A Rs 40 profit on Rs 160 is a 25% gain.

🎯 Exam Tip: Always determine the cost price (CP) first when calculating profit or loss percentage. Profit/Loss percentages are typically based on the CP.

 

Question 12. By selling a flower pot for Rs 528, a woman gains 20%. At what price should she sell it to gain 25%?
Answer: First, find the cost price (CP) of the flower pot.
Given: Selling price (SP) = Rs 528, Gain % = 20%.
Using the formula for SP with gain:
\( SP = CP \times \left( \frac{100 + \text{gain}\%}{100} \right) \)
\( 528 = CP \times \left( \frac{100 + 20}{100} \right) \)
\( 528 = CP \times \left( \frac{120}{100} \right) \)
Solve for CP:
\( CP = \frac{528 \times 100}{120} \)
\( CP = \frac{52800}{120} \)
\( CP = 440 \)
The cost price of the flower pot is Rs 440.

Now, find the selling price (SP) required to gain 25%.
Given: CP = Rs 440, desired Gain % = 25%.
Using the formula for SP with gain:
\( SP = CP \times \left( \frac{100 + \text{gain}\%}{100} \right) \)
\( SP = 440 \times \left( \frac{100 + 25}{100} \right) \)
\( SP = 440 \times \left( \frac{125}{100} \right) \)
\( SP = 440 \times 1.25 \)
\( SP = 550 \)
She should sell the flower pot for Rs 550 to gain 25%. This problem shows a common two-step profit calculation.
In simple words: We first figure out the original price the woman paid for the flower pot, knowing she gained 20% by selling it for Rs 528. Then, we calculate a new selling price that would give her a 25% profit on that original price.

🎯 Exam Tip: In problems requiring a new selling price for a different profit margin, always calculate the original cost price first. This cost price acts as the base for all subsequent profit/loss calculations.

 

Question 13. A man buys an article for Rs 150 and makes overhead expenses which are 12% of the cost price. At what price must he sell it to gain 5%?
(A) Rs 180
(B) Rs 168
(C) Rs 176.40
(D) Rs 88.20
Answer: (C) Rs 176.40
The initial cost price of the article is Rs 150.
Overhead expenses are 12% of the cost price.
Overhead expenses \( = 12\% \text{ of } 150 = \frac{12}{100} \times 150 = 12 \times 1.5 = \text{Rs } 18 \).
The effective cost price (total cost for the man) is the initial cost plus overheads.
Effective cost price \( = 150 + 18 = \text{Rs } 168 \).
Now, the man wants to gain 5% on this effective cost price.
The desired selling price (SP) to gain 5% is:
\( SP = \text{Effective CP} \times \left( \frac{100 + \text{gain}\%}{100} \right) \)
\( SP = 168 \times \left( \frac{100 + 5}{100} \right) \)
\( SP = 168 \times \left( \frac{105}{100} \right) \)
\( SP = 168 \times 1.05 \)
\( SP = 176.40 \)
He must sell the article for Rs 176.40 to gain 5%. It's important to include all expenses when calculating the true cost of an item.
In simple words: First, we add the overhead costs to the original buying price to find the total money spent. Then, we calculate the selling price needed to make a 5% profit on this total cost.

🎯 Exam Tip: Always remember to add overhead expenses to the initial cost price to get the 'effective cost price' before calculating profit or loss percentages.

 

Question 14. What is the marked price of a hat which is bought for Rs 210 at 16% discount?
(A) Rs 243
(B) Rs 176
(C) Rs 230
(D) Rs 250
Answer: (D) Rs 250
Let the marked price (MP) of the hat be MP.
The hat is bought for Rs 210, which means this is the selling price (SP) after the discount.
The rate of discount is 16%.
Using the formula for selling price with discount:
\( SP = MP \times \left( \frac{100 - \text{Discount}\%}{100} \right) \)
Substitute the given values:
\( 210 = MP \times \left( \frac{100 - 16}{100} \right) \)
\( 210 = MP \times \left( \frac{84}{100} \right) \)
To find the marked price (MP), rearrange the equation:
\( MP = \frac{210 \times 100}{84} \)
\( MP = \frac{21000}{84} \)
\( MP = 250 \)
The marked price of the hat is Rs 250. This shows how to work backward from a discounted price to find the original marked price.
In simple words: The hat was bought for Rs 210 after a 16% discount. We need to find the original marked price before the discount was taken off.

🎯 Exam Tip: When given the selling price after a discount and asked for the marked price, set up the equation carefully and solve for the marked price (MP). Avoid applying the discount percentage directly to the selling price.

 

Question 15. What is the single discount equivalent to two successive discounts of 20% and 25%?
(A) 40%
(B) 45%
(C) 5%
(D) 22.5%
Answer: (A) 40%
Let the marked price (MP) be Rs 100 for easy calculation.

First discount \( d_1 = 20\% \):
Price after first discount \( = \text{MP} \times \left( \frac{100 - d_1}{100} \right) \)
\( = 100 \times \left( \frac{100 - 20}{100} \right) = 100 \times \frac{80}{100} = \text{Rs } 80 \)

Second discount \( d_2 = 25\% \):
This discount is applied to the price after the first discount (Rs 80).
Price after second discount \( = (\text{Price after } d_1) \times \left( \frac{100 - d_2}{100} \right) \)
\( = 80 \times \left( \frac{100 - 25}{100} \right) = 80 \times \frac{75}{100} = 80 \times 0.75 = \text{Rs } 60 \)

So, after two successive discounts, an item originally marked at Rs 100 is sold for Rs 60.
Now, find the single equivalent discount.
Total discount amount \( = \text{Original MP} - \text{Final SP} \)
Total discount amount \( = 100 - 60 = \text{Rs } 40 \)
The single equivalent discount percentage is:
\( \text{Equivalent Discount}\% = \left( \frac{\text{Total discount amount}}{\text{Original MP}} \right) \times 100 \)
\( = \left( \frac{40}{100} \right) \times 100 = 40\% \)
The single discount equivalent to successive discounts of 20% and 25% is 40%. This is often less than simply adding the percentages.
In simple words: We imagine an item costs Rs 100. We first take off 20%, then take off 25% from the new price. The total reduction from the original Rs 100 tells us the single discount that would give the same final price.

🎯 Exam Tip: For successive discounts, never just add the percentages. Always apply each discount sequentially to the *remaining amount*. A common shortcut is to calculate the final price factor: \((1 - d_1)(1 - d_2)\), and then subtract this from 1 to get the equivalent single discount percentage.

 

Question 1. Fill in the blanks:
(i) Loss or gain percentage is always calculated on the _____.
(ii) A mobile phone is sold for ₹ 8400 at a gain of 20%. The cost price of the mobile phone is _____.
(iii) An article is sold for ₹ 555 at a loss of 7\( \frac { 1 }{ 2 }% \). The cost price of the article is _____.
(iv) A mixer grinder marked at ₹ 4500 is sold for ₹ 4140 after discount. The rate of discount is _____.
(v) The total bill amount of a shirt costing ₹ 575 and a T-shirt costing ₹ 325 with GST of 5% is _______.
Answer:
(i) Loss or gain percentage is always calculated on the Cost Price. The cost price is the original price paid for an item before any profit or loss is considered.
(ii) Let the cost price of the mobile be \( x \).
Given that selling price (SP) \( = \) Rs 8400 and gain \( = \) 20%.
Using the formula:
SP \( = \) CP \( \times \frac { (100 + \text{gain} \%) }{ 100 } \)
Substitute the given values:
8400 \( = x \times \frac { (100 + 20) }{ 100 } \)
8400 \( = x \times \frac { 120 }{ 100 } \)
8400 \( = x \times \frac { 6 }{ 5 } \)
To find \( x \), rearrange the equation:
\( x = 8400 \times \frac { 5 }{ 6 } \)
\( x = 1400 \times 5 \)
\( x = \text{Rs } 7000 \)
So, the cost price of the mobile phone is Rs 7000.
(iii) Given selling price (SP) \( = \) Rs 555 and loss \( = 7\frac { 1 }{ 2 }% \).
First, convert the mixed fraction to an improper fraction: \( 7\frac { 1 }{ 2 }% = \frac { 15 }{ 2 }% = 7.5% \).
Using the formula:
SP \( = \) CP \( \times \frac { (100 - \text{loss} \%) }{ 100 } \)
Substitute the given values:
555 \( = \) CP \( \times \frac { (100 - 7.5) }{ 100 } \)
555 \( = \) CP \( \times \frac { 92.5 }{ 100 } \)
To find CP, rearrange the equation:
CP \( = 555 \times \frac { 100 }{ 92.5 } \)
CP \( = 555 \times \frac { 1000 }{ 925 } \)
CP \( = 555 \times \frac { 40 }{ 37 } \)
CP \( = 15 \times 40 \)
CP \( = \text{Rs } 600 \)
The cost price of the article is Rs 600.
(iv) Marked price \( = \) Rs 4500
Selling price \( = \) Rs 4140
Discount \( = \) Marked price \( - \) Selling price
Discount \( = \) 4500 \( - \) 4140 \( = \) Rs 360
Rate of discount \( = \frac { \text{Discount} }{ \text{Marked Price} } \times 100 \)
Rate of discount \( = \frac { 360 }{ 4500 } \times 100 \)
Rate of discount \( = \frac { 360 }{ 45 } \)
Rate of discount \( = \) 8%
The rate of discount is 8%.
(v) Cost price of shirt \( = \) Rs 575, GST \( = \) 5%
Bill amount for shirt \( = \) CP \( \times \frac { (100 + \text{GST} \%) }{ 100 } \)
\( = 575 \times \frac { (100 + 5) }{ 100 } = 575 \times \frac { 105 }{ 100 } = \text{Rs } 603.75 \)
Cost price of T-shirt \( = \) Rs 325, GST \( = \) 5%
Bill amount for T-shirt \( = \) CP \( \times \frac { (100 + \text{GST} \%) }{ 100 } \)
\( = 325 \times \frac { (100 + 5) }{ 100 } = 325 \times \frac { 105 }{ 100 } = \text{Rs } 341.25 \)
Total bill amount \( = \) Bill amount for shirt \( + \) Bill amount for T-shirt
Total bill amount \( = \) 603.75 \( + \) 341.25 \( = \text{Rs } 945 \)
The total bill amount is Rs 945.
In simple words: When calculating profit or loss, always use the original price (cost price). For discounts, find the difference between the marked price and selling price, then divide by the marked price to get the percentage. For GST, add the tax percentage to 100, divide by 100, and multiply by the cost price.

🎯 Exam Tip: Remember to always correctly identify whether the percentage is based on cost price or selling price, as this changes the calculation formula.

 

Question 2. If selling an article for ₹ 820 causes 10% loss on the selling price, then find its cost price.
Answer: Given that the selling price (SP) \( = \) Rs 820.
The loss percentage \( = \) 10% on the selling price. This means the loss amount is 10% of Rs 820.
Loss amount \( = 10\% \) of Rs 820 \( = \frac { 10 }{ 100 } \times 820 = \text{Rs } 82 \)
Cost Price (CP) \( = \) Selling Price (SP) \( + \) Loss
CP \( = 820 + 82 \)
CP \( = \text{Rs } 902 \)
The cost price of the article is Rs 902.
In simple words: If you sell something for Rs 820 and lose 10% of that selling price, your original cost was Rs 820 plus the Rs 82 you lost, making it Rs 902.

🎯 Exam Tip: Be careful if the loss or gain percentage is given on the selling price instead of the cost price, as the calculation method changes.

 

Question 3. If the profit earned on selling an article for ₹ 810 is the same as loss on selling it for ₹ 530, then find the cost price of the article.
Answer: Let the cost price of the article be CP.
Case 1: When the article is sold for Rs 810, there is a profit.
Profit \( = \) Selling Price (SP) \( - \) Cost Price (CP)
Profit \( = 810 - \text{CP} \)
Case 2: When the article is sold for Rs 530, there is a loss.
Loss \( = \) Cost Price (CP) \( - \) Selling Price (SP)
Loss \( = \text{CP} - 530 \)
Given that the profit is equal to the loss:
810 \( - \) CP \( = \) CP \( - \) 530
Now, bring all CP terms to one side and numbers to the other:
810 \( + \) 530 \( = \) CP \( + \) CP
1340 \( = \) 2CP
To find CP, divide the total by 2:
CP \( = \frac { 1340 }{ 2 } \)
CP \( = \text{Rs } 670 \)
The cost price of the article is Rs 670.
In simple words: When the money you gain from selling an item at a high price is the same as the money you lose when selling it at a low price, the cost price is exactly halfway between the two selling prices.

🎯 Exam Tip: When profit equals loss for different selling prices, the cost price is simply the average of those two selling prices. Always define variables clearly.

 

Question 4. If the selling price of 10 rulers is the same as the cost price of 15 rulers, then find the profit percentage.
Answer: Let the cost price of one ruler be Rs \( x \).
Then, the cost price of 15 rulers \( = 15x \).
Given that the selling price of 10 rulers is the same as the cost price of 15 rulers.
So, Selling Price (SP) of 10 rulers \( = 15x \).
This means the selling price of one ruler \( = \frac { 15x }{ 10 } = 1.5x \).
Now we can calculate the profit for one ruler:
Profit \( = \) SP of one ruler \( - \) CP of one ruler
Profit \( = 1.5x - x = 0.5x \)
To find the profit percentage:
Profit % \( = \frac { \text{Profit} }{ \text{CP} } \times 100 \)
Profit % \( = \frac { 0.5x }{ x } \times 100 \)
Profit % \( = 0.5 \times 100 \)
Profit % \( = 50\% \)
The profit percentage is 50%.
In simple words: If you sell 10 rulers for the same money it costs to buy 15, you are making a 50% profit. This is because you get the cost of 5 extra rulers for free.

🎯 Exam Tip: For problems where quantities are related (SP of X items = CP of Y items), always calculate the SP and CP for a single item to easily find the profit or loss.

 

Question 5. Some articles are bought at 2 for ₹ 15 and sold at 3 for ₹ 25. Find the gain percentage.
Answer: First, find the cost price (CP) and selling price (SP) of one article.
Given: 2 articles are bought for Rs 15.
So, Cost Price (CP) of 1 article \( = \frac { 15 }{ 2 } = \text{Rs } 7.50 \).
Given: 3 articles are sold for Rs 25.
So, Selling Price (SP) of 1 article \( = \frac { 25 }{ 3 } \approx \text{Rs } 8.33 \).
Now, calculate the gain (profit) for one article:
Gain \( = \) SP \( - \) CP
Gain \( = \frac { 25 }{ 3 } - \frac { 15 }{ 2 } \)
To subtract, find a common denominator, which is 6:
Gain \( = \frac { 25 \times 2 }{ 3 \times 2 } - \frac { 15 \times 3 }{ 2 \times 3 } \)
Gain \( = \frac { 50 }{ 6 } - \frac { 45 }{ 6 } \)
Gain \( = \frac { 50 - 45 }{ 6 } = \frac { 5 }{ 6 } \)
Now, calculate the gain percentage:
Gain % \( = \frac { \text{Gain} }{ \text{CP} } \times 100 \)
Gain % \( = \frac { \frac { 5 }{ 6 } }{ \frac { 15 }{ 2 } } \times 100 \)
Gain % \( = \frac { 5 }{ 6 } \times \frac { 2 }{ 15 } \times 100 \)
Gain % \( = \frac { 10 }{ 90 } \times 100 \)
Gain % \( = \frac { 1 }{ 9 } \times 100 \)
Gain % \( = \frac { 100 }{ 9 } = 11\frac { 1 }{ 9 }% \)
The gain percentage is \( 11\frac { 1 }{ 9 }% \).
In simple words: To compare different buying and selling rates, figure out the price for just one item. Then calculate how much profit you make on that single item and turn it into a percentage of its original cost.

🎯 Exam Tip: Always standardize the quantities (e.g., calculate for one article) before finding profit/loss and percentage to avoid errors.

 

Question 6. By selling a speaker for ₹ 768, a man loses 20%. In order to gain 20%, how much should he sell the speaker?
Answer: First, find the cost price (CP) of the speaker.
Given: Selling Price (SP) \( = \) Rs 768, Loss % \( = \) 20%.
Using the formula:
SP \( = \) CP \( \times \frac { (100 - \text{Loss} \%) }{ 100 } \)
768 \( = \) CP \( \times \frac { (100 - 20) }{ 100 } \)
768 \( = \) CP \( \times \frac { 80 }{ 100 } \)
768 \( = \) CP \( \times \frac { 4 }{ 5 } \)
To find CP:
CP \( = 768 \times \frac { 5 }{ 4 } \)
CP \( = 192 \times 5 \)
CP \( = \text{Rs } 960 \)
Now, calculate the selling price needed to gain 20%.
Desired Gain % \( = \) 20%.
Using the formula for new selling price:
New SP \( = \) CP \( \times \frac { (100 + \text{Gain} \%) }{ 100 } \)
New SP \( = 960 \times \frac { (100 + 20) }{ 100 } \)
New SP \( = 960 \times \frac { 120 }{ 100 } \)
New SP \( = 960 \times \frac { 6 }{ 5 } \)
New SP \( = 192 \times 6 \)
New SP \( = \text{Rs } 1152 \)
He should sell the speaker for Rs 1152 to gain 20%.
In simple words: First, use the selling price and loss percentage to find the original cost of the speaker. Once you know the cost, you can then calculate the new selling price required to make a specific profit.

🎯 Exam Tip: This is a two-step problem. Always find the cost price first when converting from a loss scenario to a gain scenario, or vice-versa.

 

Question 7. Find the unknowns x, y and z.

Answer:
(i) For the Book:
Marked Price (MP) \( = \) Rs 225
Discount \( = \) 8%
Selling Price (SP) \( = \) MP \( \times \frac { (100 - \text{Discount} \%) }{ 100 } \)
\( x = 225 \times \frac { (100 - 8) }{ 100 } \)
\( x = 225 \times \frac { 92 }{ 100 } \)
\( x = 225 \times 0.92 \)
\( x = \text{Rs } 207 \)
So, the selling price of the book is Rs 207.
(ii) For the LED TV:
Selling Price (SP) \( = \) Rs 11970
Discount \( = \) 5%
Let Marked Price \( = y \)
SP \( = \) MP \( \times \frac { (100 - \text{Discount} \%) }{ 100 } \)
11970 \( = y \times \frac { (100 - 5) }{ 100 } \)
11970 \( = y \times \frac { 95 }{ 100 } \)
To find \( y \), rearrange the equation:
\( y = 11970 \times \frac { 100 }{ 95 } \)
\( y = 126 \times 100 \)
\( y = \text{Rs } 12,600 \)
So, the marked price of the LED TV is Rs 12,600.
(iii) For the Digital clock:
Marked Price (MP) \( = \) Rs 750
Selling Price (SP) \( = \) Rs 615
Let Discount \( = z \% \)
SP \( = \) MP \( \times \frac { (100 - \text{Discount} \%) }{ 100 } \)
615 \( = 750 \times \frac { (100 - z) }{ 100 } \)
Rearrange to find \( (100 - z) \):
\( \frac { 615 \times 100 }{ 750 } = 100 - z \)
\( \frac { 61500 }{ 750 } = 100 - z \)
\( 82 = 100 - z \)
To find \( z \):
\( z = 100 - 82 \)
\( z = 18\% \)
So, the discount for the digital clock is 18%.
In simple words: To find the selling price, subtract the discount percentage from 100, then multiply by the marked price. To find the marked price, reverse this process by dividing the selling price by the percentage remaining after discount. To find the discount percentage, calculate the discount amount, divide it by the marked price, and multiply by 100.

🎯 Exam Tip: Always remember the relationship between Marked Price, Selling Price, and Discount. Selling Price = Marked Price - Discount Amount, and Discount Amount = (Discount % / 100) * Marked Price.

 

Question 8. Find the bill amount for the data given below:

Answer: The bill amount is calculated by first finding the discounted price, and then adding the GST to it.
(i) For School bag:
Marked Price (MP) \( = \) Rs 500
Discount \( = \) 5%
Discounted Price (DP) \( = \) MP \( \times \frac { (100 - \text{Discount} \%) }{ 100 } \)
DP \( = 500 \times \frac { (100 - 5) }{ 100 } \)
DP \( = 500 \times \frac { 95 }{ 100 } \)
DP \( = 5 \times 95 \)
DP \( = \text{Rs } 475 \)
Now, calculate the Bill amount including GST.
GST \( = \) 12%
Bill amount \( = \) DP \( \times \frac { (100 + \text{GST} \%) }{ 100 } \)
Bill amount \( = 475 \times \frac { (100 + 12) }{ 100 } \)
Bill amount \( = 475 \times \frac { 112 }{ 100 } \)
Bill amount \( = 475 \times 1.12 \)
Bill amount \( = \text{Rs } 532 \)
(ii) For Hair dryer:
Marked Price (MP) \( = \) Rs 2000
Discount \( = \) 10%
Discounted Price (DP) \( = \) MP \( \times \frac { (100 - \text{Discount} \%) }{ 100 } \)
DP \( = 2000 \times \frac { (100 - 10) }{ 100 } \)
DP \( = 2000 \times \frac { 90 }{ 100 } \)
DP \( = 20 \times 90 \)
DP \( = \text{Rs } 1800 \)
Now, calculate the Bill amount including GST.
GST \( = \) 28%
Bill amount \( = \) DP \( \times \frac { (100 + \text{GST} \%) }{ 100 } \)
Bill amount \( = 1800 \times \frac { (100 + 28) }{ 100 } \)
Bill amount \( = 1800 \times \frac { 128 }{ 100 } \)
Bill amount \( = 1800 \times 1.28 \)
Bill amount \( = \text{Rs } 2304 \)
Total bill amount for items in the table \( = \) Bill amount for School bag \( + \) Bill amount for Hair dryer
Total bill amount \( = 532 + 2304 = \text{Rs } 2836 \)
In simple words: First, calculate the price of each item after its discount. Then, add the GST percentage to this discounted price to find the final bill for each item. Finally, add all these final prices together to get the total bill.

🎯 Exam Tip: Remember to apply the discount first to the marked price, and then apply the GST to the *discounted* price, not the original marked price.

 

Question 9. A branded Air-Conditioner (AC) has a marked price of ₹ 38000. There are 2 options given for the customer. (i) Selling Price is the same ₹ 38000 but with attractive gifts worth ₹ 3000 (or) (ii) Discount of 8% on the marked price but no free gifts. Which offer is better?
Answer: Marked price of AC \( = \) Rs 38,000.
Option 1: Selling Price is Rs 38,000 with gifts worth Rs 3,000.
Net gain for customer from gifts \( = \text{Rs } 3,000 \). The customer pays the full marked price, but receives extra value.
Option 2: Discount of 8% on the marked price, with no gifts.
Discounted value \( = \) Marked Price \( \times \frac { (100 - \text{Discount} \%) }{ 100 } \)
Discounted value \( = 38000 \times \frac { (100 - 8) }{ 100 } \)
Discounted value \( = 38000 \times \frac { 92 }{ 100 } \)
Discounted value \( = 380 \times 92 \)
Discounted value \( = \text{Rs } 34,960 \)
Savings for the customer \( = \) Marked Price \( - \) Discounted value
Savings \( = 38000 - 34960 = \text{Rs } 3040 \)
Comparing the two options:
In Option 1, the customer gets a benefit of Rs 3,000 (in the form of gifts).
In Option 2, the customer saves Rs 3,040 (by paying less money).
Since Rs 3,040 is more than Rs 3,000, Option 2 is better for the customer as it provides greater savings.
In simple words: Compare the total benefit from each choice. One choice gives you gifts worth Rs 3000, while the other saves you Rs 3040 from the price. The option that saves you more money is usually the better deal.

🎯 Exam Tip: To compare offers, always calculate the exact monetary benefit (either in savings or value of gifts) for the customer in each scenario and choose the one that provides the highest benefit.

 

Question 10. If a mattress is marked for ₹ 7500 and is available at two successive discounts of 10% and 20%, find the amount to be paid by the customer.
Answer: Marked price of mattress \( = \) Rs 7500.
First discount (\( d_1 \)) \( = \) 10%.
Price after first discount \( = \) Marked Price \( \times \frac { (100 - d_1 \%) }{ 100 } \)
\( = 7500 \times \frac { (100 - 10) }{ 100 } \)
\( = 7500 \times \frac { 90 }{ 100 } \)
\( = 75 \times 90 \)
\( = \text{Rs } 6750 \)
This new price (Rs 6750) now becomes the base for the second discount.
Second discount (\( d_2 \)) \( = \) 20%.
Price after second discount \( = \) Price after first discount \( \times \frac { (100 - d_2 \%) }{ 100 } \)
\( = 6750 \times \frac { (100 - 20) }{ 100 } \)
\( = 6750 \times \frac { 80 }{ 100 } \)
\( = 675 \times 8 \)
\( = \text{Rs } 5400 \)
The amount to be paid by the customer is Rs 5400.
In simple words: When there are two discounts one after another, first calculate the price after the first discount. Then, take that new, lower price and apply the second discount to it. You don't add the discounts together before applying them.

🎯 Exam Tip: For successive discounts, always apply them one by one. The second discount is calculated on the price *after* the first discount has been applied.

 

Question 11. A fruit vendor sells fruits for ₹ 200 gaining ₹ 40. His gain percentage is
(A) 20%
(B) 22%
(C) 25%
(D) 16
Answer: (C) 25%
Given Selling Price (SP) \( = \) Rs 200.
Gain \( = \) Rs 40.
Cost Price (CP) \( = \) Selling Price \( - \) Gain
CP \( = 200 - 40 = \text{Rs } 160 \)
Gain Percentage \( = \frac { \text{Gain} }{ \text{CP} } \times 100 \)
Gain Percentage \( = \frac { 40 }{ 160 } \times 100 \)
Gain Percentage \( = \frac { 1 }{ 4 } \times 100 \)
Gain Percentage \( = 25\% \)
In simple words: First, find out the original cost by subtracting the gain from the selling price. Then, divide the gain by this original cost and multiply by 100 to get the gain percentage.

🎯 Exam Tip: Always calculate gain percentage based on the Cost Price (CP), unless specifically stated otherwise.

 

Question 12. By selling a flower pot for ₹ 528, a woman gains 20%. At what price should she sell it to gain 25%?
(B) ₹ 550
(C) ₹ 553
(D) ₹ 573
Answer: (B) ₹ 550
First, find the cost price (CP) of the flower pot.
Given: Selling Price (SP) \( = \) Rs 528, Gain % \( = \) 20%.
Using the formula:
SP \( = \) CP \( \times \frac { (100 + \text{Gain} \%) }{ 100 } \)
528 \( = \) CP \( \times \frac { (100 + 20) }{ 100 } \)
528 \( = \) CP \( \times \frac { 120 }{ 100 } \)
528 \( = \) CP \( \times \frac { 6 }{ 5 } \)
To find CP:
CP \( = 528 \times \frac { 5 }{ 6 } \)
CP \( = 88 \times 5 \)
CP \( = \text{Rs } 440 \)
Now, calculate the selling price needed to gain 25%.
Desired Gain % \( = \) 25%.
Using the formula for new selling price:
New SP \( = \) CP \( \times \frac { (100 + \text{Gain} \%) }{ 100 } \)
New SP \( = 440 \times \frac { (100 + 25) }{ 100 } \)
New SP \( = 440 \times \frac { 125 }{ 100 } \)
New SP \( = 440 \times \frac { 5 }{ 4 } \)
New SP \( = 110 \times 5 \)
New SP \( = \text{Rs } 550 \)
She should sell the flower pot for Rs 550 to gain 25%.
In simple words: To change the profit goal, first find the original cost of the item. Once you know the cost, you can then calculate the new selling price that will give you the desired higher profit.

🎯 Exam Tip: This type of question requires two steps: first calculate the cost price from the initial selling price and gain/loss, then calculate the new selling price for the desired gain/loss using that cost price.

 

Question 13. A man buys an article for ₹ 150 and makes overhead expenses which are 12% of the cost price. At what price must he sell it to gain 5%?
(A) ₹ 180
(B) ₹ 168
(C) ₹ 176.40
(D) ₹ 88.20
Answer: (C) ₹ 176.40
Cost price of article \( = \) Rs 150.
Overhead expenses \( = \) 12% of the cost price.
Overhead expenses \( = \frac { 12 }{ 100 } \times 150 = 12 \times 1.5 = \text{Rs } 18 \).
Effective cost price \( = \) Original cost price \( + \) Overhead expenses
Effective cost price \( = 150 + 18 = \text{Rs } 168 \).
Now, the man wants to gain 5% on this effective cost price.
Selling Price (SP) \( = \) Effective CP \( \times \frac { (100 + \text{Gain} \%) }{ 100 } \)
SP \( = 168 \times \frac { (100 + 5) }{ 100 } \)
SP \( = 168 \times \frac { 105 }{ 100 } \)
SP \( = 168 \times 1.05 \)
SP \( = \text{Rs } 176.40 \)
He must sell the article for Rs 176.40 to gain 5%.
In simple words: When you buy something, any extra money spent (like repair or transport) adds to its original cost, making a "total cost." Then, calculate your profit percentage based on this total cost.

🎯 Exam Tip: Remember to include all overhead expenses when calculating the true "cost price" before determining the selling price for profit.

 

Question 14. What is the marked price of a hat which is bought for ₹ 210 at 16% discount?
(A) ₹ 243
(B) ₹ 176
(C) ₹ 230
(D) ₹ 250
Answer: (D) ₹ 250
Let the marked price (MP) be MP.
Given: Discounted price (which is the selling price) \( = \) Rs 210.
Rate of discount \( = \) 16%.
Using the formula:
Discounted Price \( = \) MP \( \times \frac { (100 - \text{Discount} \%) }{ 100 } \)
210 \( = \) MP \( \times \frac { (100 - 16) }{ 100 } \)
210 \( = \) MP \( \times \frac { 84 }{ 100 } \)
To find MP, rearrange the equation:
MP \( = 210 \times \frac { 100 }{ 84 } \)
MP \( = \frac { 21000 }{ 84 } \)
MP \( = \text{Rs } 250 \)
The marked price of the hat is Rs 250.
In simple words: If you bought something after a discount, you can find its original marked price by reversing the discount calculation. Divide the price you paid by the percentage that was left after the discount (e.g., if 16% off, divide by 84%).

🎯 Exam Tip: When finding the marked price from a discounted price, divide the discounted price by (100 - discount percentage) and then multiply by 100.

 

Question 15. What single discount percentage is equivalent to two successive discounts of 20% and 25%?
(A) 40%
(B) 45%
(C) 5%
(D) 22.5%
Answer: (A) 40%
Let the marked price (MP) be Rs 100 for simplicity.
First discount (\( d_1 \)) \( = \) 20%.
Price after first discount \( = \) MP \( \times \frac { (100 - d_1 \%) }{ 100 } \)
\( = 100 \times \frac { (100 - 20) }{ 100 } \)
\( = 100 \times \frac { 80 }{ 100 } \)
\( = \text{Rs } 80 \)
Second discount (\( d_2 \)) \( = \) 25%.
Price after second discount \( = \) Price after first discount \( \times \frac { (100 - d_2 \%) }{ 100 } \)
\( = 80 \times \frac { (100 - 25) }{ 100 } \)
\( = 80 \times \frac { 75 }{ 100 } \)
\( = 80 \times \frac { 3 }{ 4 } \)
\( = 20 \times 3 \)
\( = \text{Rs } 60 \)
So, an item originally priced at Rs 100 is sold for Rs 60 after two successive discounts.
The total discount amount \( = \) Original MP \( - \) Final Selling Price
Total discount amount \( = 100 - 60 = \text{Rs } 40 \).
Since the original marked price was Rs 100, the single equivalent discount percentage is 40%.
In simple words: To find one discount that is the same as two discounts, imagine starting with a Rs 100 item. Apply the first discount, then apply the second discount to the new price. The final price subtracted from Rs 100 tells you the single total discount percentage.

🎯 Exam Tip: For successive discounts, use the formula: Equivalent Discount % \( = (d_1 + d_2 - \frac { d_1 d_2 }{ 100 }) \)%. This formula quickly combines two discounts into one effective rate.