Selina Concise Solutions for ICSE Class 8 Mathematics Chapter 8 Profit Loss and Discount

Exercise 8(A)

Question 1. Megha bought 10 note-books for Rs.40 and sold them at Rs.4.75 per note-book. Find, her gain percent.
Answer: Cost price of all 10 books is Rs. 40.
Each book is sold at Rs. 4.75.
So, the total money she made by selling is:
\( 10 \times \text{Rs. } 4.75 = \text{Rs. } 47.50 \)
Since she sold them for more than she spent, she made a profit.
Profit = \( \text{Rs. } 47.50 - \text{Rs. } 40 = \text{Rs. } 7.50 \)
Now, we calculate the gain percent:
\( \text{Gain \%} = \frac{\text{Gain}}{\text{C.P.}} \times 100 \)

\( \implies \text{Gain \%} = \frac{7.50}{40} \times 100 \)

\( \implies \text{Gain \%} = \frac{75}{4}\% = 18\frac{3}{4}\% \)
In simple words: First, find the total money she got from selling the books. Subtract the buying price from this to get her profit, then divide that profit by the original price and multiply by 100 to get the percentage.

Exam Tip: Remember to calculate the profit based on the total cost price of all items, not just a single item, to avoid arithmetic errors.

Question 2. A fruit-seller buys oranges at 4 for Rs.3 and sells them at 3 for Rs.4 Find his profit percent.
Answer: To keep the math simple, we can work with a common number of oranges. We find the L.C.M. of 4 and 3, which is 12.
Suppose the vendor purchases 12 oranges.
The buying price for 4 oranges is Rs. 3.
So, the total cost price of these 12 oranges is:
\( \text{C.P.} = \text{Rs. } \frac{3}{4} \times 12 = \text{Rs. } 9 \)
The selling rate is Rs. 4 for every 3 oranges.
So, the total selling price of these 12 oranges is:
\( \text{S.P.} = \text{Rs. } \frac{4}{3} \times 12 = \text{Rs. } 16 \)
Since the selling price is higher than the buying price, there is a gain:
Profit = \( \text{Rs. } 16 - \text{Rs. } 9 = \text{Rs. } 7 \)
To find the percentage profit:
\( \text{Profit \%} = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)

\( \implies \text{Profit \%} = \frac{7}{9} \times 100 \)

\( \implies \text{Profit \%} = \frac{700}{9}\% = 77\frac{7}{9}\% \)
In simple words: Assume the seller buys 12 oranges because 12 is easy to divide by both 4 and 3. Find the cost of 12 oranges and the selling price of 12 oranges, then calculate the profit percent using those values.

Exam Tip: When dealing with different buying and selling quantities, always find the L.C.M. of the quantities to work with simple whole numbers.

Question 3. A man buys a certain number of articles at 15 for Rs. 112.50 and sells them at 12 for Rs.108. Find ;
(i) his gain as percent;
(ii) the number of articles sold to make a profit of Rs.75.
Answer: To find the solution easily, let us use the L.C.M. of 15 and 12, which is 60. We will assume the man purchases 60 articles.
The buying cost for 15 articles is Rs. 112.50.
So, the cost price of 60 articles is:
\( \text{C.P.} = \text{Rs. } \frac{112.50}{15} \times 60 = \text{Rs. } 112.50 \times 4 = \text{Rs. } 450 \)
The selling price for 12 articles is Rs. 108.
So, the selling price of 60 articles is:
\( \text{S.P.} = \text{Rs. } \frac{108}{12} \times 60 = \text{Rs. } 108 \times 5 = \text{Rs. } 540 \)

(i) First, let's find the total profit earned on these 60 articles:
Profit = \( \text{S.P.} - \text{C.P.} = \text{Rs. } 540 - \text{Rs. } 450 = \text{Rs. } 90 \)
Now, let's find the profit percentage:
\( \text{Gain \%} = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)

\( \implies \text{Gain \%} = \frac{90}{450} \times 100 \)

\( \implies \text{Gain \%} = 20\% \)

(ii) Next, we use the unitary method to find the number of articles required to make Rs. 75 profit.
To earn a profit of Rs. 90, the number of articles he must sell = 60.
To earn a profit of Rs. 1, the number of articles he must sell is:
\( \frac{60}{90} \)
Thus, to make a profit of Rs. 75, the number of articles he needs to sell is:
\( \frac{60}{90} \times 75 = 50 \text{ articles} \)
In simple words: Find the cost and selling prices for a common quantity like 60 items. The profit from 60 items is Rs. 90, which is a 20% gain. Since 60 items give a profit of Rs. 90, you can calculate that selling 50 items will give a profit of Rs. 75.

Exam Tip: Always show each step of the unitary method clearly to secure full step-marks in section (ii).

Question 4. A boy buys an old bicycle for Rs. 162 and spends Rs. 18 on its repairs before selling the bicycles for Rs. 207. Find his gain or loss percent.
Answer: Purchase price of the bicycle = Rs. 162
Amount spent on its repair = Rs. 18
So, the actual cost price of the bicycle is:
\( \text{Total C.P.} = \text{Rs. } 162 + \text{Rs. } 18 = \text{Rs. } 180 \)
Selling price of the bicycle = Rs. 207
Since the selling price is higher than the total cost price, he makes a profit:
Profit = \( \text{S.P.} - \text{Total C.P.} \)

\( \implies \text{Profit} = \text{Rs. } 207 - \text{Rs. } 180 = \text{Rs. } 27 \)
Now, we calculate the profit percentage:
\( \text{Gain \%} = \frac{\text{Profit}}{\text{Total C.P.}} \times 100 \)

\( \implies \text{Gain \%} = \frac{27}{180} \times 100 \)

\( \implies \text{Gain \%} = 15\% \)
In simple words: Add the repair cost to the buying price to find the total money spent on the bicycle. Then, subtract this total from the selling price to find the profit, and convert it to a percentage.

Exam Tip: Remember to add any repair or extra costs to the buying price first to get the total cost price before calculating the profit or loss percentage.

Question 5. An article is bought from Jaipur for Rs. 4,800 and is sold in Delhi for Rs. 5,820. If Rs. 1,200 is spent on its transportations, etc. ; find he loss or the gain as percent.
Answer: Cost of the item in Jaipur = Rs. 4,800
Transportation and other extra expenses = Rs. 1,200
We calculate the real cost price by adding these values:
\( \text{Total C.P.} = \text{Rs. } 4,800 + \text{Rs. } 1,200 = \text{Rs. } 6,000 \)
Selling price of the item in Delhi = Rs. 5,820
Because the total cost price is higher than the selling price, it is a loss:
Loss = \( \text{Total C.P.} - \text{S.P.} \)

\( \implies \text{Loss} = \text{Rs. } 6,000 - \text{Rs. } 5,820 = \text{Rs. } 180 \)
Now, we calculate the loss percentage:
\( \text{Loss \%} = \frac{\text{Loss}}{\text{Total C.P.}} \times 100 \)

\( \implies \text{Loss \%} = \frac{180}{6,000} \times 100 \)

\( \implies \text{Loss \%} = 3\% \)
In simple words: Find the total money spent by adding the buying price and transport cost. Since this total is more than the selling price, there is a loss, which is then calculated as a percentage of the total money spent.

Exam Tip: Keep in mind that transport expenses are overhead charges and must always be added to the buying price to find the total cost price.

Question 6. Mohit sold a T.V. for Rs. 3,600 ; gaining one-sixth of its selling price. Find :
(i) the gain
(ii) the cost price of the T.V.
(iii) the gain percent.
Answer: Selling price of the T.V. = Rs. 3,600

(i) We are given that the profit is one-sixth of the selling price:
\( \text{Gain} = \frac{1}{6} \times \text{Rs. } 3,600 = \text{Rs. } 600 \)

(ii) To find the cost price, we subtract the gain from the selling price:
\( \text{Cost Price (C.P.)} = \text{S.P.} - \text{Gain} \)

\( \implies \text{C.P.} = \text{Rs. } 3,600 - \text{Rs. } 600 = \text{Rs. } 3,000 \)

(iii) Now, we calculate the gain percent:
\( \text{Gain \%} = \frac{\text{Gain}}{\text{C.P.}} \times 100 \)

\( \implies \text{Gain \%} = \frac{600}{3,000} \times 100 \)

\( \implies \text{Gain \%} = 20\% \)
In simple words: Find the profit first by taking one-sixth of the selling price. Subtract this profit from the selling price to find the cost price, then find the percentage profit on that cost price.

Exam Tip: Always remember that profit percentage is calculated on the cost price (C.P.), not on the selling price (S.P.), even if the initial gain was given as a fraction of S.P.

Question 7. By selling a certain number of goods for Rs. 5,500; a shopkeeper loses equal to one-tenth of their selling price. Find :
(i) the loss incured
(ii) the cost price of the goods
(iii) the loss as percent.
Answer: Selling price of the goods = Rs. 5,500

(i) We are given that the loss is equal to one-tenth of the selling price:
\( \text{Loss} = \frac{1}{10} \times \text{Rs. } 5,500 = \text{Rs. } 550 \)

(ii) Since there is a loss, the cost price is greater than the selling price:
\( \text{Cost Price (C.P.)} = \text{S.P.} + \text{Loss} \)

\( \implies \text{C.P.} = \text{Rs. } 5,500 + \text{Rs. } 550 = \text{Rs. } 6,050 \)

(iii) Now, we find the loss percentage:
\( \text{Loss \%} = \frac{\text{Loss}}{\text{C.P.}} \times 100 \)

\( \implies \text{Loss \%} = \frac{550}{6,050} \times 100 \)

\( \implies \text{Loss \%} = \frac{10}{110} \times 100 = \frac{100}{11}\% = 9\frac{1}{11}\% \)
In simple words: Calculate the loss first by finding one-tenth of the selling price. Add this loss to the selling price to find the cost price, then find the loss percentage using the cost price.

Exam Tip: Be careful to add the loss to the selling price to find the cost price, and always use the cost price in the denominator when calculating loss percent.

Question 8. The selling price of a sofa-set is 4/5 times of its cost price. Find the gain or the loss as percent.
Answer: Let us assume the cost price (C.P.) of the sofa-set is Rs. 100.
According to the question, the selling price (S.P.) is four-fifths of the cost price:
\( \text{S.P.} = \frac{4}{5} \times 100 = \text{Rs. } 80 \)
Since the selling price is lower than the cost price, it results in a loss:
Loss = \( \text{C.P.} - \text{S.P.} \)

\( \implies \text{Loss} = \text{Rs. } 100 - \text{Rs. } 80 = \text{Rs. } 20 \)
Now, we find the loss percentage:
\( \text{Loss \%} = \frac{\text{Loss}}{\text{C.P.}} \times 100 \)

\( \implies \text{Loss \%} = \frac{20}{100} \times 100 = 20\% \)
In simple words: If we assume the cost is Rs. 100, the selling price is four-fifths of that, which is Rs. 80. Since Rs. 80 is less than Rs. 100, there is a loss of Rs. 20, which is exactly a 20% loss.

Exam Tip: Assuming the base value (usually C.P.) as 100 is a smart shortcut that simplifies fractional relationships into easy percentages.

Question 9. The cost price of an article is 4/5 times of its selling price. Find the loss or the gain as percent.
Answer: Let us assume the selling price (S.P.) of the article is Rs. 100.
Since the cost price is four-fifths of the selling price, we get:
\( \text{C.P.} = \frac{4}{5} \times 100 = \text{Rs. } 80 \)
Since the selling price is higher than the cost price, there is a profit:
Profit = \( \text{S.P.} - \text{C.P.} \)

\( \implies \text{Profit} = \text{Rs. } 100 - \text{Rs. } 80 = \text{Rs. } 20 \)
Now, we calculate the gain percent on the cost price:
\( \text{Gain \%} = \frac{\text{Gain}}{\text{C.P.}} \times 100 \)

\( \implies \text{Gain \%} = \frac{20}{80} \times 100 \)

\( \implies \text{Gain \%} = 25\% \)
In simple words: If the selling price is Rs. 100, the buying cost is Rs. 80. This gives a profit of Rs. 20, which is a 25% gain when calculated against the buying price of Rs. 80.

Exam Tip: Be careful to base your final percentage calculation on the Cost Price (Rs. 80) and not on the assumed Selling Price (Rs. 100).

Question 10. A shopkeeper sells his goods at 80% of their cost price. Find the percent gain or loses ?
Answer: Let us assume the cost price (C.P.) of the goods is Rs. 100.
The selling price (S.P.) is 80% of the cost price:
\( \text{S.P.} = \frac{80}{100} \times \text{Rs. } 100 = \text{Rs. } 80 \)
Since the selling price is less than the cost price, the shopkeeper incurs a loss:
Loss = \( \text{C.P.} - \text{S.P.} \)

\( \implies \text{Loss} = \text{Rs. } 100 - \text{Rs. } 80 = \text{Rs. } 20 \)
Now, we calculate the loss percentage:
\( \text{Loss \%} = \frac{\text{Loss}}{\text{C.P.}} \times 100 \)

\( \implies \text{Loss \%} = \frac{20}{100} \times 100 = 20\% \)
In simple words: If the item costs Rs. 100 and sells for Rs. 80, the seller loses Rs. 20. This is exactly a 20% loss.

Exam Tip: Since the cost price is taken as 100, any actual value change directly represents the percentage change, making the final step straightforward.

Question 11. The cost price of an article is 90% of its selling price. What is the profit or the loss as percent ?
Answer: Let us assume the selling price (S.P.) of the article is Rs. 100.
Since the cost price (C.P.) is 90% of the selling price:
\( \text{C.P.} = \frac{90}{100} \times 100 = \text{Rs. } 90 \)
Here, the selling price is greater than the cost price, which means there is a gain:
Profit = \( \text{S.P.} - \text{C.P.} \)

\( \implies \text{Profit} = \text{Rs. } 100 - \text{Rs. } 90 = \text{Rs. } 10 \)
Now, we find the profit percentage on the cost price:
\( \text{Gain \%} = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)

\( \implies \text{Gain \%} = \frac{10}{90} \times 100 \)

\( \implies \text{Gain \%} = \frac{100}{9}\% = 11\frac{1}{9}\% \)
In simple words: If the selling price is Rs. 100, the cost price is Rs. 90. The profit of Rs. 10 is then divided by the cost of Rs. 90, which gives a gain of 11 and 1/9 percent.

Exam Tip: Do not divide by 100 at the end; remember that profit or loss percentage is always calculated over the Cost Price (Rs. 90 in this case).

Question 12. The cost price of an article is 30 percent less than its selling price. Find, the profit or loss as percent.
Answer: Let us assume the selling price (S.P.) of the article is Rs. 100.
Since the cost price (C.P.) is 30% lower than the selling price:
\( \text{C.P.} = \text{Rs. } 100 - 30 = \text{Rs. } 70 \)
Since the selling price is higher than the cost price, there is a profit:
Profit = \( \text{S.P.} - \text{C.P.} \)

\( \implies \text{Profit} = \text{Rs. } 100 - \text{Rs. } 70 = \text{Rs. } 30 \)
Now, we calculate the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)

\( \implies \text{Profit \%} = \frac{30}{70} \times 100 \)

\( \implies \text{Profit \%} = \frac{300}{7}\% = 42\frac{6}{7}\% \)
In simple words: If the selling price is Rs. 100, the cost is Rs. 70 because it is 30% less. This means you make a Rs. 30 profit on a Rs. 70 investment, which is 42 and 6/7 percent.

Exam Tip: Be careful with the fraction calculations. Keep your final fraction in the mixed number form \( 42\frac{6}{7}\% \) for clarity.

Question 13. A shop-keeper bought 300 eggs at 80 paisa each. 30 eggs were broken in transaction and then he sold the remaining eggs at one rupee each. Find, his gain or loss as percent.
Answer: Total eggs purchased = 300
Rate of purchase = 80 Paise per egg
Thus, the total cost price of all eggs is:
\( \text{C.P.} = 300 \times 80 \text{ Paise} = 24,000 \text{ Paise} = \text{Rs. } 240 \)
During transport, 30 eggs were broken.
So, the remaining good eggs are:
\( 300 - 30 = 270 \text{ eggs} \)
Selling rate of these eggs = Rs. 1 each
Therefore, the total selling price is:
\( \text{S.P.} = 270 \times \text{Rs. } 1 = \text{Rs. } 270 \)
Since the selling price is higher than the buying price, there is a profit:
Profit = \( \text{S.P.} - \text{C.P.} \)

\( \implies \text{Profit} = \text{Rs. } 270 - \text{Rs. } 240 = \text{Rs. } 30 \)
Now, we calculate the gain percent:
\( \text{Gain \%} = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)

\( \implies \text{Gain \%} = \frac{30}{240} \times 100 \)

\( \implies \text{Gain \%} = \frac{100}{8}\% = 12.5\% \)
In simple words: The shopkeeper spent Rs. 240 to buy 300 eggs. After 30 eggs broke, he sold the other 270 eggs for Rs. 270, making a Rs. 30 profit, which is a 12.5% gain.

Exam Tip: Be sure to convert paise into rupees correctly (\( \text{Rs. } 1 = 100 \text{ Paise} \)) before comparing the cost price and selling price.

Question 14. A man sold his bicycle for Rs.405 losing one-tenth of its cost price, find :
(i) its cc price;
(ii) the loss percent.
Answer:
(i) Let us assume the cost price (C.P.) of the bicycle is Rs. \( x \).
The loss is given as one-tenth of this cost price:
\( \text{Loss} = \text{Rs. } \frac{x}{10} \)
The selling price (S.P.) is equal to the cost price minus the loss:
\( \text{S.P.} = x - \frac{x}{10} = \frac{9x}{10} \)
We are given that the selling price is Rs. 405:
\( \frac{9x}{10} = 405 \)

\( \implies x = 405 \times \frac{10}{9} \)

\( \implies x = 45 \times 10 = 450 \)
Thus, the cost price of the bicycle is Rs. 450.

(ii) Now, we find the loss incurred:
\( \text{Loss} = \text{Rs. } \frac{450}{10} = \text{Rs. } 45 \)
Now, let's find the loss percentage:
\( \text{Loss \%} = \frac{\text{Loss}}{\text{C.P.}} \times 100 \)

\( \implies \text{Loss \%} = \frac{45}{450} \times 100 = 10\% \)
In simple words: (i) If he lost 1/10 of the cost price, he sold it for 9/10 of what he paid. Since 9/10 of the cost is Rs. 405, the total cost was Rs. 450. (ii) Losing 1/10 of the cost is exactly a 10% loss.

Exam Tip: "cc price" in the question is a typographical error for "cost price" (C.P.). Do not get confused by such typos in the question paper.

Question 15. A man sold a radio-set for Rs.250 and gained one-ninth of its cost price. Find ;
(i) its cost price;
(ii) the profit percent.
Answer:
(i) Let the cost price (C.P.) of the radio-set be Rs. \( x \).
The profit is one-ninth of the cost price:
\( \text{Gain} = \text{Rs. } \frac{x}{9} \)
The selling price (S.P.) is the cost price plus the profit:
\( \text{S.P.} = x + \frac{x}{9} = \frac{10x}{9} \)
We know the selling price is Rs. 250, so:
\( \frac{10x}{9} = 250 \)

\( \implies x = 250 \times \frac{9}{10} \)

\( \implies x = 225 \)
Thus, the cost price of the radio-set is Rs. 225.

(ii) Now, we calculate the profit amount:
\( \text{Profit} = \text{Rs. } \frac{225}{9} = \text{Rs. } 25 \)
Next, let's find the profit percentage:
\( \text{Profit \%} = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)

\( \implies \text{Profit \%} = \frac{25}{225} \times 100 = \frac{100}{9}\% = 11\frac{1}{9}\% \)
In simple words: (i) Since the profit is 1/9 of the cost, the selling price is 10/9 of the cost. Since 10/9 of the cost is Rs. 250, the cost price is Rs. 225. (ii) Gaining 1/9 of the cost is always equal to 11 and 1/9 percent.

Exam Tip: In algebraic methods, always express S.P. in terms of \( x \) first, then set it equal to the given numerical S.P. to solve for \( x \).

Exercise 8(B)

Question 1. Find the selling price, if:
(i) C.P. = Rs. 950 and profit = 8%
(ii) C.P. = Rs. 1,300 and loss = 13%
Answer:
(i) Given: Cost Price (C.P.) = Rs. 950, Profit = 8%
We can find the selling price (S.P.) using this formula:
\( \text{S.P.} = \text{C.P.} \times \frac{100 + \text{Profit \%}}{100} \)
\( \implies \text{S.P.} = \text{Rs. } 950 \times \frac{100 + 8}{100} \)

\( \implies \text{S.P.} = 950 \times \frac{108}{100} \)

\( \implies \text{S.P.} = 19 \times 54 = \text{Rs. } 1,026 \)

(ii) Given: Cost Price (C.P.) = Rs. 1,300, Loss = 13%
We find the selling price (S.P.) using this formula:
\( \text{S.P.} = \text{C.P.} \times \frac{100 - \text{Loss \%}}{100} \)
\( \implies \text{S.P.} = \text{Rs. } 1,300 \times \frac{100 - 13}{100} \)

\( \implies \text{S.P.} = 1,300 \times \frac{87}{100} \)

\( \implies \text{S.P.} = 13 \times 87 = \text{Rs. } 1,131 \)
In simple words: To find the selling price when there is a profit, add the profit percentage to 100, divide by 100, and multiply by the cost price. When there is a loss, subtract the loss percentage from 100 first, then do the same.

Exam Tip: Be careful with the basic calculations and make sure to correctly choose either addition for profit or subtraction for loss in the formula.

Question 2. Find the cost price, if :
(i) S.P. = Rs. 1,680 and profit = 12%
(ii) S.P. = Rs. 1,128 and loss = 6%
Answer:
(i) Given: Selling Price (S.P.) = Rs. 1,680, Profit = 12%
We can find the cost price (C.P.) using this formula:
\( \text{C.P.} = \frac{100}{100 + \text{Profit \%}} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{100 + 12} \times \text{Rs. } 1,680 \)

\( \implies \text{C.P.} = \frac{100}{112} \times 1,680 \)

\( \implies \text{C.P.} = 100 \times 15 = \text{Rs. } 1,500 \)

(ii) Given: Selling Price (S.P.) = Rs. 1,128, Loss = 6%
We can find the cost price (C.P.) using this formula:
\( \text{C.P.} = \frac{100}{100 - \text{Loss \%}} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{100 - 6} \times \text{Rs. } 1,128 \)

\( \implies \text{C.P.} = \frac{100}{94} \times 1,128 \)

\( \implies \text{C.P.} = 100 \times 12 = \text{Rs. } 1,200 \br />In simple words: To find the original cost price when there is a profit, multiply the selling price by 100 divided by (100 plus profit percentage). For a loss, divide by (100 minus loss percentage) instead.

Exam Tip: Be sure to write down the standard formula before putting in the values, as this helps you earn partial marks if you make a calculation error.

Question 3. By selling an article for Rs.900; a man gains 20%. Find his cost price and the gain.
Answer: Given: Selling Price (S.P.) = Rs. 900, Gain = 20%
First, we calculate the cost price (C.P.) of the article:
\( \text{C.P.} = \frac{100}{100 + \text{Gain \%}} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{100 + 20} \times \text{Rs. } 900 \)

\( \implies \text{C.P.} = \frac{100}{120} \times 900 = \text{Rs. } 750 \)
Next, we calculate the total gain earned:
\( \text{Gain} = \text{S.P.} - \text{C.P.} \)

\( \implies \text{Gain} = \text{Rs. } 900 - \text{Rs. } 750 = \text{Rs. } 150 \)
In simple words: Use the formula to find that the item originally cost Rs. 750. Subtract this cost from the selling price of Rs. 900 to find that the profit made is Rs. 150.

Exam Tip: Always double check that the sum of your calculated cost price and the gain equals the given selling price to verify your answer.

Question 4. By selling an article for Rs.704; a person loses 12%. Find his cost price and the loss
Answer: Given: Selling Price (S.P.) = Rs. 704, Loss = 12%
First, we calculate the cost price (C.P.) of the article:
\( \text{C.P.} = \frac{100}{100 - \text{Loss \%}} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{100 - 12} \times \text{Rs. } 704 \)

\( \implies \text{C.P.} = \frac{100}{88} \times 704 = 100 \times 8 = \text{Rs. } 800 \)
Next, we calculate the total loss incurred:
\( \text{Loss} = \text{C.P.} - \text{S.P.} \)

\( \implies \text{Loss} = \text{Rs. } 800 - \text{Rs. } 704 = \text{Rs. } 96 \)
In simple words: The item sold at a loss of 12%, which means it was sold for 88% of its cost. Using this, we find the cost price is Rs. 800, and the loss amount is Rs. 96.

Exam Tip: Since it is a loss, verify that the calculated cost price (Rs. 800) is indeed higher than the given selling price (Rs. 704).

Question 5. Find the selling price, if :
(i) C.P. = Rs.352; overheads = Rs.28 and profit = 20
(ii) C.P. = Rs.576; overheads = Rs.44 and loss = 16%
Answer:
(i) Given: Cost Price (C.P.) = Rs. 352, Overheads = Rs. 28, Profit = 20%
First, we calculate the total cost price by adding overhead expenses:
\( \text{Total C.P.} = \text{Rs. } 352 + \text{Rs. } 28 = \text{Rs. } 380 \)
Now, we find the selling price:
\( \text{S.P.} = \text{Total C.P.} \times \frac{100 + \text{Profit \%}}{100} \)
\( \implies \text{S.P.} = \text{Rs. } 380 \times \frac{100 + 20}{100} \)

\( \implies \text{S.P.} = 380 \times \frac{120}{100} \)

\( \implies \text{S.P.} = 38 \times 12 = \text{Rs. } 456 \)

(ii) Given: Cost Price (C.P.) = Rs. 576, Overheads = Rs. 44, Loss = 16%
First, we calculate the total cost price:
\( \text{Total C.P.} = \text{Rs. } 576 + \text{Rs. } 44 = \text{Rs. } 620 \)
Now, we calculate the selling price:
\( \text{S.P.} = \text{Total C.P.} \times \frac{100 - \text{Loss \%}}{100} \)
\( \implies \text{S.P.} = \text{Rs. } 620 \times \frac{100 - 16}{100} \)

\( \implies \text{S.P.} = 620 \times \frac{84}{100} \)

\( \implies \text{S.P.} = \frac{5,208}{10} = \text{Rs. } 520.80 \br />In simple words: Add overhead costs to the basic buying price to find the total money spent first. Then, apply the standard percentage formulas to find the selling price for profit or loss.

Exam Tip: Never calculate profit or loss percentage on the raw C.P. when overheads are present; always add overheads to C.P. first.

Question 6. If John sells his bicycle for Rs. 637, he will suffer a loss of 9%. For how much should it be sold, if he desires a profit of 5% ?
Answer: In the first case, S.P. = Rs. 637, and Loss = 9%
We calculate the cost price (C.P.) first:
\( \text{C.P.} = \frac{100}{100 - \text{Loss \%}} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{100 - 9} \times \text{Rs. } 637 \)

\( \implies \text{C.P.} = \frac{100}{91} \times 637 = 100 \times 7 = \text{Rs. } 700 \)
Now, we want to find the new selling price for a desired profit of 5%:
\( \text{New S.P.} = \frac{100 + \text{Profit \%}}{100} \times \text{C.P.} \)
\( \implies \text{New S.P.} = \frac{100 + 5}{100} \times \text{Rs. } 700 \)

\( \implies \text{New S.P.} = \frac{105}{100} \times 700 = 105 \times 7 = \text{Rs. } 735 \br />In simple words: First, find the buying price of the bicycle by using the 9% loss info, which gives Rs. 700. Then, calculate how much it should sell for to make a 5% profit on Rs. 700, which is Rs. 735.

Exam Tip: Solve this type of double-step question in two distinct stages: find C.P. first, then use it as the base to calculate the new S.P.

Question 7. A man sells a radio-set for Rs.605 and gains 10%. At what price should he sell another radio of the same kind, in order to gain 16% ?
Answer: In the first scenario, S.P. = Rs. 605, and Gain = 10%
Let's calculate the cost price (C.P.) first:
\( \text{C.P.} = \frac{100}{100 + \text{Gain \%}} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{100 + 10} \times \text{Rs. } 605 \)

\( \implies \text{C.P.} = \frac{100}{110} \times 605 = 10 \times 55 = \text{Rs. } 550 \)
In the second scenario, the desired gain is 16% on the same cost price (Rs. 550):
\( \text{New S.P.} = \frac{100 + \text{Gain \%}}{100} \times \text{C.P.} \)
\( \implies \text{New S.P.} = \frac{100 + 16}{100} \times \text{Rs. } 550 \)

\( \implies \text{New S.P.} = \frac{116}{100} \times 550 = 58 \times 11 = \text{Rs. } 638 \br />In simple words: Use the first sale to find that the radio costs Rs. 550. Then calculate the new price needed to make a 16% profit on that Rs. 550 cost, which gives Rs. 638.

Exam Tip: Be sure to write the final statement clearly concluding that the radio should be sold for Rs. 638 to earn the desired profit.

Question 8. By selling a sofa-set for Rs.2,500; the shopkeeper loses 20%. Find his loss percent or profit percent ; if he sells the same sofa-set for Rs.3150.
Answer: First, we use the given loss of 20% to find the cost price (C.P.) of the sofa-set:
\( \text{C.P.} = \frac{100}{100 - \text{Loss \%}} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{100 - 20} \times \text{Rs. } 2,500 \)

\( \implies \text{C.P.} = \frac{100}{80} \times 2,500 = \frac{5}{4} \times 2,500 = \text{Rs. } 3,125 \)
Now, in the second case, the new selling price (S.P.) is Rs. 3,150.
Since this new S.P. is higher than the C.P., there is a profit:
Profit = \( \text{S.P.} - \text{C.P.} \)

\( \implies \text{Profit} = \text{Rs. } 3,150 - \text{Rs. } 3,125 = \text{Rs. } 25 \)
Now, we calculate the profit percentage:
\( \text{Gain \%} = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)
\( \implies \text{Gain \%} = \frac{25}{3,125} \times 100 \)

\( \implies \text{Gain \%} = \frac{100}{125}\% = \frac{4}{5}\% = 0.8\% \)
In simple words: The shopkeeper bought the sofa-set for Rs. 3,125. If he sells it for Rs. 3,150, he makes a Rs. 25 profit, which is equal to 0.8 percent profit.

Exam Tip: When simplifying fractions in the final step, always reduce to the simplest terms first before converting into decimal percentage like 0.8%.

Question 9. Mr. Sinha sold two tape-recorders for Rs.990 each; gaining 10% on one and losing 10% on the other. Find his total loss or gain as percent on the whole transaction.
Answer:
For the first device:
S.P. = Rs. 990
Gain = 10%
C.P. = \( \frac{100}{100 + \text{Gain}\%} \times \text{S.P.} \)
\( \implies \) C.P. = \( \frac{100}{100 + 10} \times \text{Rs. } 990 \)
\( \implies \) C.P. = \( \frac{100}{110} \times \text{Rs. } 990 \)
\( \implies \) C.P. = \( \text{Rs. } 100 \times 9 = \text{Rs. } 900 \)

For the second device:
S.P. = Rs. 990
Loss = 10%
C.P. = \( \frac{100}{100 - \text{Loss}\%} \times \text{S.P.} \)
\( \implies \) C.P. = \( \frac{100}{100 - 10} \times \text{Rs. } 990 \)
\( \implies \) C.P. = \( \frac{100}{90} \times \text{Rs. } 990 \)
\( \implies \) C.P. = \( \text{Rs. } 100 \times 11 = \text{Rs. } 1100 \)

Combined cost price for both devices:
C.P. = \( \text{Rs. } 900 + \text{Rs. } 1100 = \text{Rs. } 2000 \)

Combined selling price for both devices:
S.P. = \( \text{Rs. } 990 + \text{Rs. } 990 = \text{Rs. } 1980 \)

Since the combined cost price is higher than the combined selling price, there is a loss on the entire transaction:
Loss = C.P. - S.P.
\( \implies \) Loss = \( \text{Rs. } 2000 - \text{Rs. } 1980 = \text{Rs. } 20 \)

Loss percent on the entire sale:
Loss% = \( \frac{\text{Loss}}{\text{C.P.}} \times 100 \)
\( \implies \) Loss% = \( \frac{20}{2000} \times 100 \)
\( \implies \) Loss% = \( 1\% \)
In simple words: When you sell two items for the same price with equal gain and loss percentages, you always end up with an overall loss of 1%.

Exam Tip: Remember that in such transactions where two items are sold at the same price with equal gain and loss percents, there is always a loss. You can use the shortcut formula: \( \text{Loss}\% = \left( \frac{\text{Common Loss or Gain}\%}{10} \right)^2 \).

Question 10. A tape-recorder is sold for Rs. 2,760 at a gain of 15% and a C.D. player is sold for Rs. 3,240 at a loss of 10% Find :
(i) the C.P. of the tape-recorder
(ii) the C.P. of the C.D. player.
(iii) the total C.P. of both.
(iv) the total S.P. of both
(v) the gain % or the loss % on the whole
Answer:
(i) For the tape-recorder:
S.P. = Rs. 2,760
Gain = 15%
C.P. = \( \frac{100}{100 + \text{Gain}\%} \times \text{S.P.} \)
\( \implies \) C.P. = \( \frac{100}{115} \times \text{Rs. } 2760 \)
\( \implies \) C.P. = \( \frac{20}{23} \times \text{Rs. } 2760 \)
\( \implies \) C.P. = \( 20 \times 120 = \text{Rs. } 2400 \)

(ii) For the C.D. player:
S.P. = Rs. 3,240
Loss = 10%
C.P. = \( \frac{100}{100 - \text{Loss}\%} \times \text{S.P.} \)
\( \implies \) C.P. = \( \frac{100}{100 - 10} \times \text{Rs. } 3240 \)
\( \implies \) C.P. = \( \frac{100}{90} \times \text{Rs. } 3240 \)
\( \implies \) C.P. = \( 100 \times 36 = \text{Rs. } 3600 \)

(iii) Total cost price of both items:
Total C.P. = \( \text{Rs. } 2400 + \text{Rs. } 3600 = \text{Rs. } 6000 \)

(iv) Total selling price of both items:
Total S.P. = \( \text{Rs. } 2760 + \text{Rs. } 3240 = \text{Rs. } 6000 \)

(v) Comparing total S.P. and total C.P.:
Since S.P. = C.P. = Rs. 6000, there is no gain and no loss on the entire transaction.
In simple words: First find the buying price of each item. Then add them up to find the total buying price and compare it with the total selling price.

Exam Tip: Be careful with the calculations. Write down each step clearly. Showing that the total cost price is exactly equal to the total selling price makes it easy to conclude there is no profit or loss.

Question 11. Rajesh sold his scooter to Rahim at 8% loss and Rahim, in turn, sold the same scooter to Prem at 5% gain. If Prem paid Rs. 14,490 for the scooter ; find :
(i) the S.P. and the C.P. of the scooter for Rahim
(ii) the S.P. and the C.P. of the scooter for Rajesh
Answer:
Let us assume Rajesh bought the scooter for \( \text{Rs. } 100x \)
Since Rajesh sold it to Rahim at an 8% loss:
S.P. for Rajesh = \( \frac{100 - 8}{100} \times 100x = 92x \)
This is Rahim's buying price:
C.P. for Rahim = \( 92x \)

Rahim sold the vehicle to Prem at a 5% profit:
S.P. for Rahim = \( \frac{100 + 5}{100} \times 92x \)
\( \implies \) S.P. for Rahim = \( \frac{105}{100} \times 92x \)
\( \implies \) S.P. for Rahim = \( \frac{21 \times 92x}{20} = \frac{21 \times 46x}{10} = \frac{966x}{10} \)

This is the purchase price for Prem, which is Rs. 14,490:
\( \implies \frac{966x}{10} = 14490 \)
\( \implies x = \frac{14490 \times 10}{966} \)
\( \implies x = \frac{14490}{483} \times 5 = 30 \times 5 = 150 \)

Now we can find the required prices:
(i) For Rahim:
C.P. of the scooter = \( 92x = 92 \times 150 = \text{Rs. } 13800 \)
S.P. of the scooter = \( \frac{966 \times 150}{10} = 966 \times 15 = \text{Rs. } 14490 \)

(ii) For Rajesh:
C.P. of the scooter = \( 100x = 100 \times 150 = \text{Rs. } 15000 \)
S.P. of the scooter = \( 92x = 92 \times 150 = \text{Rs. } 13800 \)
In simple words: Start by assuming Rajesh's cost is 100x. Calculate how the price changes with each sale to find Prem's cost in terms of x, then solve for x.

Exam Tip: Using a variable like 100x makes calculations with percentages much simpler than using a single variable x. Always label who each price belongs to clearly.

Question 12. John sold an article to Peter at 20% profit and Peter sold it to Mohan at 5% loss. If Mohan paid Rs.912 for the article; find how much did John pay for it ?
Answer:
Let us work backward from Mohan's purchase price.
Mohan paid Rs. 912 for the article, which means:
S.P. of the article for Peter = Rs. 912
Loss incurred by Peter = 5%

Cost price of the article for Peter:
C.P. = \( \frac{100}{100 - \text{Loss}\%} \times \text{S.P.} \)
\( \implies \) C.P. = \( \frac{100}{100 - 5} \times \text{Rs. } 912 \)
\( \implies \) C.P. = \( \frac{100}{95} \times \text{Rs. } 912 \)
\( \implies \) C.P. = \( 20 \times 48 = \text{Rs. } 960 \)

Since John sold this article to Peter, Peter's cost price is John's selling price:
S.P. of the article for John = Rs. 960
Profit gained by John = 20%

Cost price of the article for John:
C.P. = \( \frac{100}{100 + \text{Profit}\%} \times \text{S.P.} \)
\( \implies \) C.P. = \( \frac{100}{100 + 20} \times \text{Rs. } 960 \)
\( \implies \) C.P. = \( \frac{100}{120} \times \text{Rs. } 960 \)
\( \implies \) C.P. = \( \text{Rs. } 100 \times 8 = \text{Rs. } 800 \)

Thus, John paid Rs. 800 for the article.
In simple words: Since we know the final price Mohan paid, we can work backward step-by-step to find out what each person spent before him.

Exam Tip: Working backward is a very reliable method for chain transaction problems. Be sure to use the correct formulas for cost price based on profit or loss.

Exercise 8(C)

Question 1. A stationer buys pens at 5 for Rs.28 and sells them at a profit of 25 %. How much should a customer pay; if he buys
(i) only one pen ;
(ii) three pens ?
Answer:
Cost price of 5 pens = Rs. 28
Cost price of a single pen = \( \frac{\text{Rs. } 28}{5} = \text{Rs. } 5.60 \)
Profit percentage = 25%

Selling price of 1 pen:
S.P. = \( \frac{100 + \text{Profit}\%}{100} \times \text{C.P. of 1 pen} \)
\( \implies \) S.P. = \( \frac{100 + 25}{100} \times \text{Rs. } 5.60 \)
\( \implies \) S.P. = \( \frac{125}{100} \times \text{Rs. } 5.60 \)
\( \implies \) S.P. = \( 1.25 \times \text{Rs. } 5.60 = \text{Rs. } 7 \)

Therefore, a customer has to pay:
(i) For only one pen = Rs. 7
(ii) For three pens, the customer pays = \( 3 \times \text{Rs. } 7 = \text{Rs. } 21 \)
In simple words: First find the cost of a single pen. Then add the 25% profit to find the selling price of one pen, and multiply to find the price for three pens.

Exam Tip: Finding the cost of a single unit (the unitary method) is usually the safest way to avoid mistakes in such multi-unit profit and loss questions.

Question 2. A fruit-seller sells 4 oranges for Rs. 3, gaining 50%. Find :
(i) C.P. of 4 oranges,
(ii) C.P. of one orange.
(iii) S.P. of one orange.
(iv) profit made by selling one orange.
(v) number of oranges brought and sold in order to gain Rs. 24.
Answer:
Selling price of 4 oranges is Rs. 3
S.P. of 1 orange = \( \text{Rs. } \frac{3}{4} = \text{Rs. } 0.75 \)
Gain percentage = 50%

Cost price for one orange is:
C.P. = \( \frac{100}{100 + \text{Gain}\%} \times \text{S.P. of 1 orange} \)
\( \implies \) C.P. = \( \frac{100}{100 + 50} \times \text{Rs. } \frac{3}{4} \)
\( \implies \) C.P. = \( \frac{100}{150} \times \text{Rs. } \frac{3}{4} \)
\( \implies \) C.P. = \( \frac{2}{3} \times \text{Rs. } \frac{3}{4} = \text{Re. } \frac{1}{2} = \text{Rs. } 0.50 \)

Now, let us calculate each required value:
(i) Cost price of 4 oranges:
C.P. of 4 oranges = \( 4 \times \text{Re. } \frac{1}{2} = \text{Rs. } 2 \)

(ii) Cost price of one orange:
C.P. of 1 orange = Rs. 0.50

(iii) Selling price of one orange:
S.P. of 1 orange = \( \text{Rs. } \frac{3}{4} = \text{Rs. } 0.75 \)

(iv) Profit earned from selling one orange:
Profit = S.P. - C.P.
\( \implies \) Profit = \( \text{Rs. } 0.75 - \text{Rs. } 0.50 = \text{Rs. } 0.25 \) (or \( \text{Re. } \frac{1}{4} \))

(v) Total oranges to buy and sell to gain Rs. 24:
Since a profit of \( \text{Re. } \frac{1}{4} \) is made on 1 orange,
Number of oranges needed to gain Rs. 24 = \( \frac{24}{\frac{1}{4}} = 24 \times 4 = 96 \)
In simple words: Work out the cost and selling price of just one orange first. After that, it is easy to answer all the other parts of the question.

Exam Tip: Keeping values in fraction form (like \( \frac{1}{2} \) and \( \frac{1}{4} \)) during calculations can make division steps much simpler to handle than using decimals.

Question 3. A man sells 12 articles for Rs. 80 gaining \( 33\frac{1}{3}\% \). Find the number of articles bought by the man for Rs. 90.
Answer:
Selling price of 12 articles = Rs. 80
Gain percentage = \( 33\frac{1}{3}\% = \frac{100}{3}\% \)

Cost price of these 12 articles:
C.P. = \( \frac{100}{100 + \text{Gain}\%} \times \text{S.P.} \)
\( \implies \) C.P. = \( \frac{100}{100 + \frac{100}{3}} \times \text{Rs. } 80 \)
\( \implies \) C.P. = \( \frac{100}{\frac{300 + 100}{3}} \times \text{Rs. } 80 \)
\( \implies \) C.P. = \( \frac{100 \times 3}{400} \times \text{Rs. } 80 \)
\( \implies \) C.P. = \( \frac{300}{400} \times \text{Rs. } 80 = \frac{3}{4} \times \text{Rs. } 80 = \text{Rs. } 60 \)

So, 12 articles were bought for Rs. 60.
To buy articles with Rs. 90:
Number of articles = \( \frac{12}{60} \times 90 \)
\( \implies \) Number of articles = \( \frac{1}{5} \times 90 = 18 \)
In simple words: First find how much the 12 articles actually cost. Then use that rate to see how many articles can be bought with Rs. 90.

Exam Tip: Be comfortable converting mixed fraction percentages like \( 33\frac{1}{3}\% \) to improper fractions (\( \frac{100}{3}\% \)) before placing them into the cost price formula.

Question 4. The cost price of 20 articles is same as the selling price of 16 articles. Find the gain percent.
Answer:
We are given that buying 20 articles costs the same as selling 16 articles.
Assume each article costs Re. 1
Then, the cost price of 20 articles = Rs. 20
This means 16 articles cost Rs. 16
S.P. of 16 articles = Rs. 20

Profit made on selling 16 articles:
Gain = S.P. - C.P.
\( \implies \) Gain = \( \text{Rs. } 20 - \text{Rs. } 16 = \text{Rs. } 4 \)

Gain percentage:
Gain% = \( \frac{\text{Gain}}{\text{C.P.}} \times 100 \)
\( \implies \) Gain% = \( \frac{4}{16} \times 100 \)
\( \implies \) Gain% = \( \frac{1}{4} \times 100 = 25\% \)
In simple words: Imagine each article costs Rs. 1 to buy. This helps you easily find the profit percentage by comparing costs and selling prices for the same number of items.

Exam Tip: When dealing with relations between cost and selling prices of different quantities, assuming the cost of a single item to be Re. 1 is an excellent way to simplify the problem.

Question 5. The selling price of 15 articles is equal to the cost price of 12 articles. Find the gain or loss as percent.
Answer:
Let the cost price of 1 article = Re. 1
Then, the cost price of 15 articles = Rs. 15
We are given that:
S.P. of 15 articles = C.P. of 12 articles = Rs. 12
Since S.P. of 15 articles (Rs. 12) is less than the C.P. of 15 articles (Rs. 15), there is a loss.

Loss incurred:
Loss = C.P. - S.P.
\( \implies \) Loss = \( \text{Rs. } 15 - \text{Rs. } 12 = \text{Rs. } 3 \)

Loss percentage:
Loss% = \( \frac{\text{Loss}}{\text{C.P.}} \times 100 \)
\( \implies \) Loss% = \( \frac{3}{15} \times 100 \)
\( \implies \) Loss% = \( \frac{1}{5} \times 100 = 20\% \)
In simple words: If you sell 15 items for the price of 12, you are losing money. By setting a simple cost of Re. 1 per item, you can quickly find that the loss is 20%.

Exam Tip: Always make sure you compare the S.P. and C.P. for the exact same number of items (in this case, 15 articles) before calculating profit or loss percentage.

Question 6. By selling 8 pens, Shyam loses equal to the cost price of 2 pens. Find his loss percent.
Answer:
Let the cost price of 1 pen = Re. 1
Then, the cost price of 8 pens = Rs. 8
Shyam's loss is equal to the cost price of 2 pens:
Loss = Rs. 2

Loss percentage is calculated as:
Loss% = \( \frac{\text{Loss}}{\text{C.P. of 8 pens}} \times 100 \)
\( \implies \) Loss% = \( \frac{2}{8} \times 100 \)
\( \implies \) Loss% = \( \frac{1}{4} \times 100 = 25\% \)
In simple words: If you buy 8 pens and lose the cost of 2 of them, your loss is 2 out of 8, which works out to one-fourth or 25%.

Exam Tip: Be sure not to mix up "equal to the cost price of 2 pens" with "equal to the selling price". Since it is based on cost price, the calculation is very straightforward.

Question 7. A shop-keeper bought rice worth Rs.4,500. He sold one-third of it at 10% profit. If he desires a profit of 12% on the whole ; find :
(i) the selling price of the rest of the rice ;
(ii) the percentage profit on the rest of the rice.
Answer:
The shopkeeper purchased rice for Rs. 4,500
Total profit desired on the whole amount = 12%

Total target selling price for all the rice:
Total S.P. = \( \frac{100 + \text{Profit}\%}{100} \times \text{Total C.P.} \)
\( \implies \) Total S.P. = \( \frac{100 + 12}{100} \times \text{Rs. } 4500 \)
\( \implies \) Total S.P. = \( \frac{112}{100} \times \text{Rs. } 4500 \)
\( \implies \) Total S.P. = \( 112 \times 45 = \text{Rs. } 5040 \)

Now, let us look at the first portion sold:
C.P. of \( \frac{1}{3} \) of the rice = \( \frac{1}{3} \times \text{Rs. } 4500 = \text{Rs. } 1500 \)
Profit on this portion = 10%
S.P. of this portion = \( \frac{100 + 10}{100} \times \text{Rs. } 1500 \)
\( \implies \) S.P. of this portion = \( \frac{110}{100} \times \text{Rs. } 1500 = 110 \times 15 = \text{Rs. } 1650 \)

Let us find the values for the remaining rice:
The cost price of the remaining portion = \( \text{Rs. } 4500 - \text{Rs. } 1500 = \text{Rs. } 3000 \)
(i) S.P. required for the rest of the rice:
S.P. of remaining rice = \( \text{Rs. } 5040 - \text{Rs. } 1650 = \text{Rs. } 3390 \)

(ii) Percentage profit on the remaining rice:
Profit made on the remaining rice = \( \text{Rs. } 3390 - \text{Rs. } 3000 = \text{Rs. } 390 \)
Profit percentage on remaining rice = \( \frac{\text{Profit}}{\text{C.P. of remaining rice}} \times 100 \)
\( \implies \) Profit percentage = \( \frac{390}{3000} \times 100 \)
\( \implies \) Profit percentage = \( \frac{390}{30} = 13\% \)
In simple words: First find the total money the seller wants to collect. Then subtract the money already made from selling the first portion to find what is needed from the rest.

Exam Tip: Ensure you keep track of the different parts of the transaction separately: the total, the first portion, and the remaining portion.

Question 8. Mohan bought a certain number of note-books for Rs.600. He sold \( \frac{1}{4} \) of them at 5 percent loss. At what price should he sell the remaining note-books so as to gain 10% on the whole ?
Answer:
Total cost price of the note-books = Rs. 600
Desired gain percentage on the whole = 10%

Required total selling price to achieve this gain:
Total S.P. = \( \frac{100 + \text{Gain}\%}{100} \times \text{Total C.P.} \)
\( \implies \) Total S.P. = \( \frac{100 + 10}{100} \times \text{Rs. } 600 \)
\( \implies \) Total S.P. = \( \frac{110}{100} \times \text{Rs. } 600 = 110 \times 6 = \text{Rs. } 660 \)

Now, let us calculate the details for the first portion sold:
C.P. of \( \frac{1}{4} \) of the note-books = \( \frac{1}{4} \times \text{Rs. } 600 = \text{Rs. } 150 \)
Loss on this portion = 5%
S.P. of this portion = \( \frac{100 - \text{Loss}\%}{100} \times \text{C.P. of this portion} \)
\( \implies \) S.P. of this portion = \( \frac{100 - 5}{100} \times \text{Rs. } 150 \)
\( \implies \) S.P. of this portion = \( \frac{95}{100} \times \text{Rs. } 150 = 0.95 \times 150 = \text{Rs. } 142.50 \)

Now, let us calculate the values for the remaining portion:
C.P. of the remaining note-books = \( \text{Rs. } 600 - \text{Rs. } 150 = \text{Rs. } 450 \)
Required S.P. of the remaining note-books = \( \text{Total S.P.} - \text{S.P. of the first portion} \)
\( \implies \) Required S.P. = \( \text{Rs. } 660 - \text{Rs. } 142.50 = \text{Rs. } 517.50 \)
In simple words: Find the total target money to earn a 10% profit overall. Subtract the money already made from selling the first quarter to find how much the remaining notebooks must sell for.

Exam Tip: Double-check your decimal subtraction when subtracting \( \text{Rs. } 142.50 \) from \( \text{Rs. } 660 \). Arithmetic slips here can cost valuable marks.

Question 9. Raju sells a watch at 5% profit. Had he sold it for Rs.24 more ; he would have gained 11%. Find the cost price of the watch.
Answer:
Let us assume the cost price of the watch = Rs. 100
First Scenario:
Profit = 5%
S.P. = \( \text{Rs. } (100 + 5) = \text{Rs. } 105 \)

Second Scenario:
Profit = 11%
S.P. = \( \text{Rs. } (100 + 11) = \text{Rs. } 111 \)

The difference between the two selling prices:
Difference = \( \text{Rs. } 111 - \text{Rs. } 105 = \text{Rs. } 6 \)

We can find the actual cost price using the unitary method:
If the difference in S.P. is Rs. 6, then the C.P. = Rs. 100
If the difference in S.P. is Rs. 24, then the C.P. = \( \frac{100}{6} \times \text{Rs. } 24 \)
\( \implies \) C.P. = \( 100 \times 4 = \text{Rs. } 400 \)
In simple words: Imagine the watch costs Rs. 100. Compare the two selling prices at different profit rates, and use that difference to scale up to the real price difference of Rs. 24.

Exam Tip: This type of problem can also be solved using simple algebra: let C.P. be \( x \), then \( 11\% \text{ of } x - 5\% \text{ of } x = \text{Rs. } 24 \). Both methods are perfectly valid and will get you full marks.

Question 10. A man sold a bicycle at 5% profit. If the cost had been 30% less and the selling price Rs.63 less, he would have made a profit of 30%. What is the cost price of the bicycle ?
Answer:
Let us assume the cost price of the bicycle = Rs. 100
First Case:
Profit = 5%
S.P. = \( \text{Rs. } (100 + 5) = \text{Rs. } 105 \)

Second Case:
The cost is 30% less:
New C.P. = \( \text{Rs. } 100 - \left( \frac{30}{100} \times 100 \right) = \text{Rs. } 70 \)
Profit on this new cost = 30%
New S.P. = \( \frac{100 + \text{Profit}\%}{100} \times \text{New C.P.} \)
\( \implies \) New S.P. = \( \frac{100 + 30}{100} \times \text{Rs. } 70 \)
\( \implies \) New S.P. = \( \frac{130}{100} \times \text{Rs. } 70 = 13 \times 7 = \text{Rs. } 91 \)

The difference between the two selling prices is:
Difference = \( \text{Rs. } 105 - \text{Rs. } 91 = \text{Rs. } 14 \)

Using the unitary method to find the actual cost price:
If the difference in selling prices is Rs. 14, then the C.P. = Rs. 100
If the difference is Re. 1, then the C.P. = \( \text{Rs. } \frac{100}{14} \)
If the difference is Rs. 63, then the C.P. = \( \frac{100}{14} \times 63 \)
\( \implies \) C.P. = \( \frac{100 \times 9}{2} = 50 \times 9 = \text{Rs. } 450 \)
In simple words: Assume the bicycle costs Rs. 100. Work out the first and second selling prices based on the changes, find their difference, and scale it to find the real cost price.

Exam Tip: Be careful when calculating the new selling price. The 30% profit in the second case must be calculated on the new cost price of Rs. 70, not the original Rs. 100.

Question 11. Renu sold an article at a loss of 8 percent. Had she bought it at 10% less and sold for Rs.36 more; she would have gained 20%. Find the cost price of the article.
Answer:
Let us assume the cost price of the article = Rs. 100
First Case:
Loss = 8%
S.P. = \( \text{Rs. } (100 - 8) = \text{Rs. } 92 \)

Second Case:
The cost is 10% less:
New C.P. = \( \text{Rs. } (100 - 10) = \text{Rs. } 90 \)
Profit on this new cost = 20%
New S.P. = \( \frac{100 + \text{Profit}\%}{100} \times \text{New C.P.} \)
\( \implies \) New S.P. = \( \frac{100 + 20}{100} \times \text{Rs. } 90 \)
\( \implies \) New S.P. = \( \frac{120}{100} \times \text{Rs. } 90 = 12 \times 9 = \text{Rs. } 108 \)

The difference between the two selling prices is:
Difference = \( \text{Rs. } 108 - \text{Rs. } 92 = \text{Rs. } 16 \)

Using the unitary method to find the actual cost price:
If the difference in selling prices is Rs. 16, then the C.P. = Rs. 100
If the difference is Re. 1, then the C.P. = \( \text{Rs. } \frac{100}{16} \)
If the difference is Rs. 36, then the C.P. = \( \frac{100}{16} \times 36 \)
\( \implies \) C.P. = \( 25 \times 9 = \text{Rs. } 225 \)
In simple words: Treat the cost price as Rs. 100. Find the two selling prices, check the difference between them, and use that to work out the actual cost price.

Exam Tip: Keep your calculations neat and organized. Labeling each case clearly as "Case 1" and "Case 2" ensures you do not mix up the values.

Exercise 8(D)

Question 1. An article is marked for Rs. 1,300 and is sold for Rs. 1,144 ; find the discount percent.
Answer:
The list price is Rs. 1,300, and it is sold for Rs. 1,144:
Marked price (M.P.) = Rs. 1,300
Selling price (S.P.) = Rs. 1,144

Discount amount = M.P. - S.P.
\( \implies \) Discount = \( \text{Rs. } 1300 - \text{Rs. } 1144 = \text{Rs. } 156 \)

Discount percentage:
Discount% = \( \frac{\text{Discount}}{\text{M.P.}} \times 100 \)
\( \implies \) Discount% = \( \frac{156}{1300} \times 100 \)
\( \implies \) Discount% = \( \frac{156}{13} = 12\% \)
In simple words: Subtract the selling price from the printed price to find the discount amount. Then divide this discount by the original printed price to get the percentage.

Exam Tip: Remember that discount percentage is always calculated based on the marked price (M.P.) of the article, not the selling price (S.P.).

Question 2. The marked price of a dinning table is Rs. 23,600 and is available at a discount of 8%. Find its selling price.
Answer:
Given details:
Marked price = Rs. 23,600
Discount rate = 8%

We can find the selling price (S.P.) using the formula:
\( \text{S.P.} = \frac{\text{Marked Price} \times (100 - \text{Discount}\% sample)}{100} \)

Let us put the values in the formula:
\( \text{S.P.} = \frac{23600 \times (100 - 8)}{100} \)
\( \implies \text{S.P.} = 236 \times 92 \)
\( \implies \text{S.P.} = \text{Rs. } 21,712 \)

Thus, the selling price is Rs. 21,712.
In simple words: The table starts at Rs. 23,600. After taking off 8%, the final price you pay is Rs. 21,712.

Exam Tip: Always subtract the discount percent from 100 before multiplying with the marked price to quickly get the selling price.

Question 3. A wrist-watch is available at a discount of 9%. If the list-price of the watch is Rs. 1,400 ; find the discount given and the selling price of the watch.
Answer:
Given details:
List price of the watch = Rs. 1,400
Discount rate = 9%

First, let us calculate the discount amount:
\( \text{Discount} = \frac{9}{100} \times 1400 \)
\( \implies \text{Discount} = 9 \times 14 \)
\( \implies \text{Discount} = \text{Rs. } 126 \)

Now, we find the selling price (S.P.):
\( \text{S.P.} = \text{List price} - \text{Discount} \)
\( \implies \text{S.P.} = \text{Rs. } (1400 - 126) \)
\( \implies \text{S.P.} = \text{Rs. } 1274 \)

So, the discount is Rs. 126 and the selling price is Rs. 1,274.
In simple words: First, find 9% of Rs. 1,400, which is Rs. 126. Then, subtract this discount from the original price to get Rs. 1,274.

Exam Tip: Be sure to calculate and state both values separately when the question asks for both the discount and the selling price.

Question 4. A shopkeeper sells an article for Rs. 248.50 after allowing a discount of 10%. Find the list price of the article.
Answer:
Given details:
Selling price (S.P.) = Rs. 248.50
Discount rate = 10%

Let us assume the list price (M.P.) is Rs. 100.
Then, the discount amount is:
\( \text{Discount} = \frac{10}{100} \times 100 = \text{Rs. } 10 \)

The selling price would be:
\( \text{S.P.} = 100 - 10 = \text{Rs. } 90 \)

Now, we can use the unitary method:
When the S.P. equals Rs. 90, the corresponding M.P. is Rs. 100.
For an S.P. of Rs. 1, the M.P. becomes \( \frac{100}{90} \).
Therefore, for an S.P. of Rs. 248.50, the list price is:
\( \text{M.P.} = \frac{100}{90} \times 248.50 \)
\( \implies \text{M.P.} = \frac{24850}{90} \)
\( \implies \text{M.P.} \approx \text{Rs. } 276.11 \)

Thus, the list price of the article is approximately Rs. 276.11.
In simple words: Since the selling price is 90% of the list price, dividing the selling price by 0.9 gives the original price.

Exam Tip: When the list price is unknown, assuming it to be 100 makes setting up the unitary method or direct ratio easy and clear.

Question 5. A shop-keeper buys an article for Rs.450. He marks it at 20% above the cost price. Find :
(i) the marked price of the article.
(ii) the selling price, if he sells the articles at 10 percent discount.
(iii) the percentage discount given by him, if he sells the article for Rs.496.80
Answer:
Given cost price (C.P.) = Rs. 450

(i) To find the marked price (M.P.):
The article is marked 20% higher than C.P.
\( \text{M.P.} = \frac{100 + 20}{100} \times 450 \)
\( \implies \text{M.P.} = \frac{120}{100} \times 450 \)
\( \implies \text{M.P.} = 12 \times 45 = \text{Rs. } 540 \)
So, the marked price is Rs. 540.

(ii) To find the selling price (S.P.) at a 10% discount:
\( \text{Discount} = \frac{10}{100} \times \text{M.P.} \)
\( \implies \text{Discount} = \frac{10}{100} \times 540 = \text{Rs. } 54 \)
\( \text{S.P.} = \text{M.P.} - \text{Discount} \)
\( \implies \text{S.P.} = 540 - 54 = \text{Rs. } 486 \)
So, the selling price is Rs. 486.

(iii) To find the discount percentage if S.P. is Rs. 496.80:
Here, S.P. = Rs. 496.80 and M.P. = Rs. 540.
\( \text{Discount} = \text{M.P.} - \text{S.P.} \)
\( \implies \text{Discount} = 540 - 496.80 = \text{Rs. } 43.20 \)
\( \text{Discount}\% = \frac{\text{Discount}}{\text{M.P.}} \times 100 \)
\( \implies \text{Discount}\% = \frac{43.20}{540} \times 100 = \frac{4320}{540}\% = 8\% \)
So, the discount given is 8%.
In simple words: First, add 20% to the cost price to get Rs. 540. Then, subtract a 10% discount to find Rs. 486. Finally, if sold at Rs. 496.80, the price drop of Rs. 43.20 is equal to an 8% discount.

Exam Tip: Remember that discount is always calculated on the marked price (M.P.), whereas profit is calculated on the cost price (C.P.). Keep these bases clear!

Question 6. The list price of an article is Rs.800 and is available at a discount of 15 percent. Find :
(i) selling price of the article ;
(ii) cost price of the article, if a profit of 13 \frac{1}{3} % is made on selling it.
Answer:
Given list price = Rs. 800
Discount rate = 15%

(i) To find the selling price (S.P.):
\( \text{Discount amount} = \frac{15}{100} \times 800 = 15 \times 8 = \text{Rs. } 120 \)
\( \text{S.P.} = \text{List price} - \text{Discount} \)
\( \implies \text{S.P.} = 800 - 120 = \text{Rs. } 680 \)
Thus, the selling price is Rs. 680.

(ii) To find the cost price (C.P.) if profit is \( 13\frac{1}{3}\% \):
Here, Profit = \( 13\frac{1}{3}\% = \frac{40}{3}\% \)
We use the formula:
\( \text{C.P.} = \frac{100}{100 + \text{Profit}\%} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{100 + \frac{40}{3}} \times 680 \)
\( \implies \text{C.P.} = \frac{100}{\frac{340}{3}} \times 680 \)
\( \implies \text{C.P.} = \frac{3 \times 100 \times 680}{340} \)
\( \implies \text{C.P.} = 3 \times 100 \times 2 = \text{Rs. } 600 \)
So, the cost price of the article is Rs. 600.
In simple words: First, take 15% off Rs. 800 to get a selling price of Rs. 680. Then, use the profit rate to work backward and find that the original cost was Rs. 600.

Exam Tip: Convert mixed fraction percentages like \( 13\frac{1}{3}\% \) to an improper fraction \( \frac{40}{3}\% \) to make calculation steps simpler and avoid decimal errors.

Question 7. An article is marked at Rs. 2,250. By selling it at a discount of 12%, the dealer makes a profit of 10%. Find :
(i) the selling price of the article.
(ii) the cost price of the article for the dealer.
Answer:
(i) To find the selling price (S.P.):
Given Marked Price (M.P.) = Rs. 2,250
Discount rate = 12%
\( \text{S.P.} = \frac{\text{M.P.} \times (100 - \text{Discount}\%)}{100} \)
\( \implies \text{S.P.} = \frac{2250 \times (100 - 12)}{100} \)
\( \implies \text{S.P.} = \frac{2250 \times 88}{100} \)
\( \implies \text{S.P.} = 45 \times 44 = \text{Rs. } 1980 \)
Thus, the selling price is Rs. 1,980.

(ii) To find the cost price (C.P.) for the dealer:
Given S.P. = Rs. 1,980 and Profit = 10%
\( \text{C.P.} = \frac{100}{100 + \text{Profit}\%} \times \text{S.P.} \)
\( \implies \text{C.P.} = \frac{100}{110} \times 1980 \)
\( \implies \text{C.P.} = 100 \times 18 = \text{Rs. } 1800 \)
Thus, the cost price is Rs. 1,800.
In simple words: The article is priced at Rs. 2,250. A 12% discount drops the price to Rs. 1,980. Since this price includes a 10% profit, the original cost was Rs. 1,800.

Exam Tip: Work step-by-step: find the selling price first from the marked price and discount, then find the cost price using the profit percent and the selling price.

Question 8. By selling an article at 20% discount, a shopkeeper gains 25%. If the selling price of the article is Rs. 1,440 ; find :
(i) the marked price of the article.
(ii) the cost price of the article.
Answer:
Given Selling Price (S.P.) = Rs. 1,440

(i) To find the marked price (M.P.):
Let the marked price be \( 100x \).
Since discount = 20%, we have:
\( \text{S.P.} = \frac{100x \times (100 - 20)}{100} = 80x \)
We are given that S.P. = Rs. 1,440.

So,
\( 80x = 1440 \)
\( \implies x = \frac{1440}{80} \)
\( \implies x = 18 \)

Therefore, Marked Price = \( 100 \times 18 = \text{Rs. } 1800 \).

(ii) To find the cost price (C.P.):
We know S.P. = Rs. 1,440 and profit = 25%.
\( \text{C.P.} = \frac{100 \times \text{S.P.}}{100 + \text{Profit}\%} \)
\( \implies \text{C.P.} = \frac{100 \times 1440}{100 + 25} \)
\( \implies \text{C.P.} = \frac{100 \times 1440}{125} \)
\( \implies \text{C.P.} = \frac{4}{5} \times 1440 \)
\( \implies \text{C.P.} = 4 \times 288 = \text{Rs. } 1152 \)
So, the cost price of the article is Rs. 1,152.
In simple words: The selling price of Rs. 1,440 is 80% of the marked price, making it Rs. 1,800. At the same time, Rs. 1,440 is 125% of the cost price, making it Rs. 1,152.

Exam Tip: Be careful not to confuse the bases: marked price is the base for discount, while cost price is the base for profit.

Question 9. A shop-keeper marks his goods at 30 percent above the cost price and then gives a discount of 10 percent. Find his gain percent.
Answer:
Let the cost price (C.P.) of the goods be Rs. 100.
Since the marked price (M.P.) is 30% higher, we have:
\( \text{M.P.} = 100 + 30 = \text{Rs. } 130 \)

Now, the discount given is 10% on M.P.:
\( \text{Discount} = \frac{10}{100} \times 130 = \text{Rs. } 13 \)

Therefore, the selling price (S.P.) is:
\( \text{S.P.} = \text{M.P.} - \text{Discount} \)
\( \implies \text{S.P.} = 130 - 13 = \text{Rs. } 117 \)

Now we find the gain:
\( \text{Gain} = \text{S.P.} - \text{C.P.} \)
\( \implies \text{Gain} = 117 - 100 = \text{Rs. } 17 \)

The gain percentage is calculated as:
\( \text{Gain}\% = \frac{\text{Gain}}{\text{C.P.}} \times 100 \)
\( \implies \text{Gain}\% = \frac{17}{100} \times 100 = 17\% \)
So, the shopkeeper's gain is 17%.
In simple words: If an item costs Rs. 100, the seller prices it at Rs. 130. Offering a 10% discount cuts Rs. 13, selling it for Rs. 117 and making a Rs. 17 (or 17%) profit.

Exam Tip: Assuming the cost price to be Rs. 100 makes calculating profit percent very simple, since the profit amount directly equals the profit percentage.

Question 10. A ready-made garments shop in Delhi, allows 20 percent discount on its garments and still makes a profit of 20 percent. Find the marked price of a dress which is bought by the shop-keeper for Rs.400.
Answer:
Given cost price (C.P.) of the dress = Rs. 400
Profit rate = 20%

First, find the profit amount:
\( \text{Profit} = \frac{20}{100} \times 400 = \text{Rs. } 80 \)

Now, calculate the selling price (S.P.):
\( \text{S.P.} = \text{C.P.} + \text{Profit} \)
\( \implies \text{S.P.} = 400 + 80 = \text{Rs. } 480 \)

Let the marked price (M.P.) of the dress be Rs. 100.
With a 20% discount, the S.P. would be:
\( \text{S.P.} = \text{M.P.} - \text{Discount} = 100 - 20 = \text{Rs. } 80 \)

Now, using the unitary method:
If the S.P. is Rs. 80, the M.P. is Rs. 100.
If the S.P. is Rs. 1, the M.P. is Rs. \( \frac{100}{80} \).
Since the actual S.P. is Rs. 480, the actual M.P. is:
\( \text{M.P.} = \frac{100}{80} \times 480 \)
\( \implies \text{M.P.} = 100 \times 6 = \text{Rs. } 600 \)
Thus, the marked price of the dress is Rs. 600.
In simple words: The shopkeeper buys the dress for Rs. 400 and sells it for Rs. 480 to earn a 20% profit. Since there is a 20% discount on the tag, Rs. 480 is 80% of the tag price, meaning the tag price is Rs. 600.

Exam Tip: When both profit percent and discount percent are given, find the selling price first from the cost price, then work backward to get the marked price.

Question 11. At 12% discount, the selling price of a pen is Rs. 13.20. Find its marked price. Also, find the new selling price of the pen, if it is sold at 5% discount.
Answer:
Given initial selling price (S.P.) = Rs. 13.20
Initial discount rate = 12%

Let the marked price (M.P.) be \( 100x \).
\( \text{S.P.} = \frac{100x \times (100 - 12)}{100} = 88x \)
We are given:
\( 88x = 13.20 \)
\( \implies x = \frac{13.20}{88} \)
\( \implies x = \frac{1320}{88 \times 100} \)
\( \implies x = \frac{30}{200} = \frac{3}{20} \)

Thus, the marked price is:
\( \text{M.P.} = 100x = 100 \times \frac{3}{20} = \text{Rs. } 15 \)

Now, find the new S.P. with a 5% discount:
\( \text{New S.P.} = \frac{\text{M.P.} \times (100 - 5)}{100} \)
\( \implies \text{New S.P.} = \frac{15 \times 95}{100} \)
\( \implies \text{New S.P.} = \frac{15 \times 19}{20} \)
\( \implies \text{New S.P.} = \frac{3 \times 19}{4} = \frac{57}{4} = \text{Rs. } 14.25 \)
So, the marked price is Rs. 15 and the new selling price is Rs. 14.25.
In simple words: The original tag price of the pen is Rs. 15. If we only give a 5% discount instead of 12%, the price drops to Rs. 14.25.

Exam Tip: Simplify the fraction values early in your calculations to make finding the decimal values easier in the final step.

Question 12. The cost price of an article is Rs. 2,400 and it is marked at 25% above the cost price. Find the profit and the profit percent, if the article is sold at 15% discount.
Answer:
Given cost price (C.P.) = Rs. 2,400
The article is marked 25% higher than C.P.

First, find the marked price (M.P.):
\( \text{M.P.} = \frac{2400 \times (100 + 25)}{100} \)
\( \implies \text{M.P.} = 24 \times 125 = \text{Rs. } 3000 \)

Next, calculate the selling price (S.P.) with a 15% discount:
\( \text{S.P.} = \frac{\text{M.P.} \times (100 - 15)}{100} \)
\( \implies \text{S.P.} = \frac{3000 \times 85}{100} \)
\( \implies \text{S.P.} = 30 \times 85 = \text{Rs. } 2550 \)

Now, calculate the profit:
\( \text{Profit} = \text{S.P.} - \text{C.P.} \)
\( \implies \text{Profit} = 2550 - 2400 = \text{Rs. } 150 \)

Finally, calculate the profit percentage:
\( \text{Profit}\% = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)
\( \implies \text{Profit}\% = \frac{150}{2400} \times 100 \)
\( \implies \text{Profit}\% = \frac{50}{8} = \frac{25}{4} = 6\frac{1}{4}\% \)
So, the profit is Rs. 150 and the profit percentage is \( 6\frac{1}{4}\% \).
In simple words: The shopkeeper lists the item at Rs. 3,000 and sells it for Rs. 2,550 after a 15% discount. This leaves a profit of Rs. 150, which is \( 6\frac{1}{4}\% \) of the original cost.

Exam Tip: Double-check that you divide by the cost price (C.P. = Rs. 2,400) and not the marked price when calculating the final profit percentage.

Question 13. Thirty articles are bought at Rs. 450 each. If one-third of these articles be sold at 6% loss; at what price must each of the remaining articles be sold in order to make a profit of 10% on the whole?
Answer:
Given details:
Total number of articles = 30
Cost price (C.P.) of 1 article = Rs. 450

Total C.P. of all 30 articles is:
\( \text{Total C.P.} = 450 \times 30 = \text{Rs. } 13500 \)

Number of articles sold at a loss = \( \frac{1}{3} \times 30 = 10 \text{ articles} \).
C.P. of these 10 articles = \( 450 \times 10 = \text{Rs. } 4500 \).
Loss on these 10 articles = 6%.
Selling price (S.P.) of these 10 articles is:
\( \text{S.P. of 10 articles} = \frac{\text{C.P.} \times (100 - \text{Loss}\%)}{100} \)
\( \implies \text{S.P. of 10 articles} = \frac{4500 \times (100 - 6)}{100} \)
\( \implies \text{S.P. of 10 articles} = 45 \times 94 = \text{Rs. } 4230 \)

We want a total profit of 10% on the entire transaction.
Required total S.P. for all 30 articles is:
\( \text{Total S.P.} = \frac{\text{Total C.P.} \times (100 + \text{Profit}\%)}{100} \)
\( \implies \text{Total S.P.} = \frac{13500 \times (100 + 10)}{100} \)
\( \implies \text{Total S.P.} = 135 \times 110 = \text{Rs. } 14850 \)

The remaining number of articles is:
\( 30 - 10 = 20 \text{ articles} \).
S.P. needed for these 20 articles is:
\( \text{Required S.P. of 20 articles} = \text{Total S.P.} - \text{S.P. of 10 articles} \)
\( \implies \text{Required S.P. of 20 articles} = 14850 - 4230 = \text{Rs. } 10620 \)

Therefore, the S.P. for each remaining article must be:
\( \text{S.P. of 1 article} = \frac{10620}{20} = \text{Rs. } 531 \)
So, each remaining article must be sold for Rs. 531.
In simple words: The total cost for 30 items is Rs. 13,500, and we want to get Rs. 14,850 back overall. Since we sold the first 10 items at a loss for Rs. 4,230, we must sell the remaining 20 items for Rs. 10,620 total, which is Rs. 531 each.

Exam Tip: Be careful with multi-step word problems. Keep track of the total cost and total desired revenue separately to stay on track.

Question 14. The cost price of an article is 25% below the marked price. If the article is available at 15% discount and its cost price is Rs. 2,400; find:
(i) Its marked price
(ii) its selling price
(iii) the profit percent.
Answer:
Let the marked price (M.P.) of the article be Rs. 100.
Since the cost price (C.P.) is 25% below the M.P., we have:
\( \text{C.P.} = \frac{100 \times (100 - 25)}{100} = \text{Rs. } 75 \)
With a 15% discount, the selling price (S.P.) would be:
\( \text{S.P.} = 100 - 15 = \text{Rs. } 85 \)

Given that the actual cost price is Rs. 2,400.

(i) To find the marked price (M.P.):
\( \text{Actual M.P.} = 2400 \times \frac{100}{75} \)
\( \implies \text{Actual M.P.} = 32 \times 100 = \text{Rs. } 3200 \)
So, the marked price is Rs. 3,200.

(ii) To find the selling price (S.P.):
\( \text{Actual S.P.} = \frac{3200 \times 85}{100} \)
\( \implies \text{Actual S.P.} = 32 \times 85 = \text{Rs. } 2720 \)
So, the selling price is Rs. 2,720.

(iii) To find the profit percent:
\( \text{Profit} = \text{Actual S.P.} - \text{Actual C.P.} \)
\( \implies \text{Profit} = 2720 - 2400 = \text{Rs. } 320 \)
Now, calculate the profit percentage:
\( \text{Profit}\% = \frac{\text{Profit}}{\text{C.P.}} \times 100 \)
\( \implies \text{Profit}\% = \frac{320 \times 100}{2400} \)
\( \implies \text{Profit}\% = \frac{40}{3}\% = 13\frac{1}{3}\% \)
So, the profit is \( 13\frac{1}{3}\% \).
In simple words: If the cost is 75% of the marked price, the actual marked price is Rs. 3,200. With a 15% discount, the items sell for Rs. 2,720, making a profit of Rs. 320, which is \( 13\frac{1}{3}\% \).

Exam Tip: Set up ratios using a baseline value of 100 first. This lets you find actual values easily once the real C.P. is known.

Question 15. Find a single discount (as percent) equivalent to following successive discounts:
(i) 20% and 12%
(ii) 10%, 20% and 20%
(iii) 20%, 10% and 5%
Answer:
Let the marked price (M.P.) be Rs. 100.

(i) Successive discounts of 20% and 12%:
\( \text{S.P.} = \frac{100 \times (100 - 20) \times (100 - 12)}{100 \times 100} \)
\( \implies \text{S.P.} = \frac{100 \times 80 \times 88}{10000} \)
\( \implies \text{S.P.} = \frac{352}{5} \)
Total discount on Rs. 100 is:
\( 100 - \frac{352}{5} = \frac{500 - 352}{5} = \frac{148}{5} = 29\frac{3}{5}\% \text{ or } 29.6\% \)

(ii) Successive discounts of 10%, 20% and 20%:
\( \text{S.P.} = \frac{100 \times (100 - 10) \times (100 - 20) \times (100 - 20)}{100 \times 100 \times 100} \)
\( \implies \text{S.P.} = \frac{100 \times 90 \times 80 \times 80}{1000000} \)
\( \implies \text{S.P.} = \frac{576}{10} \)
Total discount on Rs. 100 is:
\( 100 - \frac{576}{10} = \frac{1000 - 576}{10} = \frac{424}{10} = 42.4\% \)

(iii) Successive discounts of 20%, 10% and 5%:
\( \text{S.P.} = \frac{100 \times (100 - 20) \times (100 - 10) \times (100 - 5)}{100 \times 100 \times 100} \)
\( \implies \text{S.P.} = \frac{100 \times 80 \times 90 \times 95}{1000000} \)
\( \implies \text{S.P.} = \frac{342}{5} \)
Total discount on Rs. 100 is:
\( 100 - \frac{342}{5} = \frac{500 - 342}{5} = \frac{158}{5} = 31.6\% \)
In simple words: To find a single equivalent discount, find the final selling price when starting with Rs. 100. The total discount is Rs. 100 minus that final selling price.

Exam Tip: Never just add successive discounts together (like 20% + 12% = 32%). The second discount is applied to an already reduced price, making the total discount less than the sum.

Question 16. Find the single discount (as percent) equivalent to successive discounts of:
(i) 80% and 80%
(ii) 60% and 60%
(iii) 60% and 80%
Answer:
Let the marked price (M.P.) be Rs. 100.

(i) Successive discounts of 80% and 80%:
\( \text{S.P.} = \frac{100 \times (100 - 80) \times (100 - 80)}{100 \times 100} \)
\( \implies \text{S.P.} = \frac{100 \times 20 \times 20}{10000} = \text{Rs. } 4 \)
Total discount on Rs. 100 is:
\( 100 - 4 = 96\% \)

(ii) Successive discounts of 60% and 60%:
\( \text{S.P.} = \frac{100 \times (100 - 60) \times (100 - 60)}{100 \times 100} \)
\( \implies \text{S.P.} = \frac{100 \times 40 \times 40}{10000} = \text{Rs. } 16 \)
Total discount on Rs. 100 is:
\( 100 - 16 = 84\% \)

(iii) Successive discounts of 60% and 80%:
\( \text{S.P.} = \frac{100 \times (100 - 60) \times (100 - 80)}{100 \times 100} \)
\( \implies \text{S.P.} = \frac{100 \times 40 \times 20}{10000} = \text{Rs. } 8 \)
Total discount on Rs. 100 is:
\( 100 - 8 = 92\% \)
In simple words: When you start with Rs. 100 and apply the discounts one after another, the final price is what remains. The discount is the difference between Rs. 100 and that final price.

Exam Tip: Be sure to write the final answer with a percent symbol, as the question asks for the equivalent discount percent.

Exercise 8(E)

Question 1. Rajat purchases a wrist-watch costing Rs. 540. The rate of Sales Tax is 8%. Find the total amount paid by Rajat for the watch.
Answer:
Given details:
Cost of watch = Rs. 540
Sales Tax rate = 8%

First, find the sales tax amount:
\( \text{Sales Tax} = 540 \times \frac{8}{100} = \text{Rs. } 43.20 \)

Next, calculate the total price:
\( \text{Total Amount} = 540 + 43.20 = \text{Rs. } 583.20 \)
So, Rajat paid Rs. 583.20 in total.
In simple words: The watch is Rs. 540. An 8% tax adds Rs. 43.20 to the bill. So, the total amount paid is Rs. 583.20.

Exam Tip: Always add the sales tax amount to the cost price of the item to get the final bill amount.

Question 2. Ramesh paid Rs. 345.60 as Sales Tax on a purchase of Rs. 3,840. Find the rate of Sales Tax.
Answer:
Given details:
Cost of purchase = Rs. 3,840
Sales tax paid = Rs. 345.60

We can find the rate of Sales Tax using the formula:
\( \text{Percent of Sales Tax} = \frac{\text{Sales Tax}}{\text{Cost of purchase}} \times 100 \)
\( \implies \text{Percent of Sales Tax} = \frac{345.60 \times 100}{3840} \)
\( \implies \text{Percent of Sales Tax} = \frac{34560}{3840} = 9\% \)
So, the rate of Sales Tax is 9%.
In simple words: The tax rate is found by dividing the tax paid by the purchase cost, then multiplying by 100 to get a percent. Here, that is 9%.

Exam Tip: Keep your calculation simple by cancelling the zeros first before performing division.

Question 3. The price of a washing machine, inclusive of sales tax is Rs. 13,530/-. If the Sales Tax is 10%, find its basic (cost) price.
Answer:
Given details:
Selling price (inclusive of tax) = Rs. 13,530
Sales Tax rate = 10%

We can calculate the basic cost price using the formula:
\( \text{Cost Price} = \frac{\text{Selling Price} \times 100}{100 + \text{Rate of Sales Tax}} \)
\( \implies \text{Cost Price} = \frac{13530 \times 100}{100 + 10} \)
\( \implies \text{Cost Price} = \frac{13530 \times 100}{110} \)
\( \implies \text{Cost Price} = \frac{1353000}{110} \)
\( \implies \text{Cost Price} = \text{Rs. } 12300 \)
So, the basic price of the washing machine is Rs. 12,300.
In simple words: The total price of Rs. 13,530 includes the 10% tax. Removing this tax leaves the basic machine price at Rs. 12,300.

Exam Tip: Remember that "inclusive of sales tax" means the selling price is equal to 110% (100% + tax%) of the basic cost price.

Question 4. Sarita purchases biscuits costing Rs. 158 on which the rate of Sales Tax is 6%. She also purchases some cosmetic goods costing Rs. 354 on which rate of Sales Tax is 9%. Find the total amount to be paid by Sarita.
Answer:
The price of the biscuits is Rs. 158.
Sales tax on biscuits at 6% is calculated as:
\( \text{Sales Tax} = 158 \times \frac{6}{100} = \text{Rs. } 9.48 \)
Therefore, the total cost for the biscuits is:
\( 158 + 9.48 = \text{Rs. } 167.48 \)
The price of the cosmetics is Rs. 354.
Sales tax on cosmetics at 9% is calculated as:
\( \text{Sales Tax} = 354 \times \frac{9}{100} = \text{Rs. } 31.86 \)
Thus, the total cost for the cosmetics is:
\( 354 + 31.86 = \text{Rs. } 385.86 \)
To find the complete bill amount, we combine both costs:
\( 167.48 + 385.86 = \text{Rs. } 553.34 \)
Hence, the total amount payable by Sarita is Rs. 553.34.
In simple words: First, calculate the sales tax for each item separately and add it to their original prices. Finally, add the two total amounts together to find the final price.

Exam Tip: Be careful with decimal additions and always calculate sales tax on the base cost before adding it to get the final selling price.

Question 5. The price of a T.V. set inclusive of sales tax of 9% is Rs. 13,407. Find its marked price. If Sales Tax is increased to 13%, how much more does the customer has to pay for the T.V. ?
Answer:
Let the marked price of the television be \( x \).
With a sales tax of 9%, the final price becomes:
\( x + \frac{9}{100} \times x = \frac{109x}{100} \)
According to the given information:
\( \frac{109x}{100} = 13,407 \)
Solving for \( x \):
\( x = \frac{13,407 \times 100}{109} \)
\( x = 12,300 \)
So, the marked price is Rs. 12,300.
If the sales tax rises to 13%, the tax amount on the marked price is:
\( 12,300 \times \frac{13}{100} = \text{Rs. } 1,599 \)
The revised selling price is:
\( 12,300 + 1,599 = \text{Rs. } 13,899 \)
The extra amount the customer needs to pay is the difference between the new and old prices:
\( 13,899 - 13,407 = \text{Rs. } 492 \)
Thus, the customer has to pay Rs. 492 more.
In simple words: Use the initial tax rate and final price to find the original marked price of the television. Then, find the new price with the 13% tax rate to see how much extra it costs.

Exam Tip: Setting up an algebraic equation like \( x + \frac{9x}{100} \) helps avoid errors when working backward from a tax-inclusive price.

Question 6. The price of an article is Rs. 8,250 which includes Sales Tax at 10%. Find how much more or less does a customer pay for the article, if the Sales Tax on the article: (i) increases to 15% (ii) decreases to 6% (iii) increases by 2% (iv) decreases by 3%
Answer:
Let the list price of the article be \( x \).
Given that the price including 10% sales tax is Rs. 8,250, we have:
\( x + \frac{10}{100} \times x = 8,250 \)
\( \frac{11x}{10} = 8,250 \)
\( x = \frac{8,250 \times 10}{11} = 7,500 \)
So, the base list price is Rs. 7,500.

(i) When the sales tax rate increases to 15%:
The new selling price is:
\( 7,500 \times \frac{100 + 15}{100} = 7,500 \times \frac{115}{100} = \text{Rs. } 8,625 \)
The customer has to pay more by:
\( 8,625 - 8,250 = \text{Rs. } 375 \)

(ii) When the sales tax rate decreases to 6%:
The new selling price is:
\( 7,500 \times \frac{100 + 6}{100} = 7,500 \times \frac{106}{100} = \text{Rs. } 7,950 \)
The customer has to pay less by:
\( 8,250 - 7,950 = \text{Rs. } 300 \)

(iii) When the sales tax rate increases by 2%:
The new rate of tax is \( 10\% + 2\% = 12\% \).
The new selling price is:
\( 7,500 \times \frac{100 + 12}{100} = 7,500 \times \frac{112}{100} = \text{Rs. } 8,400 \)
The customer has to pay more by:
\( 8,400 - 8,250 = \text{Rs. } 150 \)

(iv) When the sales tax rate decreases by 3%:
The new rate of tax is \( 10\% - 3\% = 7\% \).
The new selling price is:
\( 7,500 \times \frac{100 + 7}{100} = 7,500 \times \frac{107}{100} = \text{Rs. } 8,025 \)
The customer has to pay less by:
\( 8,250 - 8,025 = \text{Rs. } 225 \)
In simple words: Find the basic price of the item first. After that, calculate the new total price for each of the four tax situations and find how much they differ from the starting price.

Exam Tip: Pay close attention to the phrasing: "increases/decreases to" means the rate becomes that value, whereas "increases/decreases by" means you must add or subtract that percentage from the initial rate.

Question 7. A bicycle is available for Rs. 1,664 including Sales Tax. If the list price of the bicycle is Rs. 1,600, find : (i) the rate of Sales Tax (ii) the price a customer will pay for the bicycle if the Sales Tax is increased by 6%.
Answer:
(i) The sales price of the bicycle is Rs. 1,664, and its list price is Rs. 1,600.
The sales tax amount paid is:
\( 1,664 - 1,600 = \text{Rs. } 64 \)
The tax rate is calculated as:
\( \text{Rate of Sales Tax} = \frac{64}{1,600} \times 100 = 4\% \)

(ii) If the sales tax is raised by 6%, the new tax rate is:
\( 4\% + 6\% = 10\% \)
The new tax amount is:
\( 1,600 \times \frac{10}{100} = \text{Rs. } 160 \)
The final amount to be paid by the customer is:
\( 1,600 + 160 = \text{Rs. } 1,760 \)
In simple words: Find the tax amount by subtracting the base price from the final price, then find the percentage it represents. For the second part, add 6% to this tax rate and calculate the new total price.

Exam Tip: Ensure you calculate the percentage of tax relative to the original list price (Rs. 1,600), not the tax-inclusive price.

Question 8. When the rate of sale-tax is decreased from 9% to 6% for a coloured T.V. ; Mrs Geeta will save Rs. 780 in buying this T.V. Find the list price of the T.V.
Answer:
The initial sales tax rate is 9%, which is reduced to 6%.
The reduction in the tax rate is:
\( 9\% - 6\% = 3\% \)
This 3% drop in tax leads to savings of Rs. 780 for the customer.
Let the list price of the television be \( T \).
\( T \times \frac{3}{100} = 780 \)
Solving for \( T \):
\( T = \frac{780 \times 100}{3} \)
\( T = 26,000 \)
Therefore, the list price of the television is Rs. 26,000.
In simple words: The difference in the tax rate is 3%, which corresponds to a savings of Rs. 780. By setting up a simple ratio, we can find the 100% value, which is the list price.

Exam Tip: Identifying the percentage difference directly simplifies calculations and saves time compared to setting up two separate equations.

Question 9. A shopkeeper sells an article for Rs. 21,384 including 10% sales-tax. However, the actual rate of sales-tax is 8%. Find the extra profit made by the dealer.
Answer:
The sale price of the article including a 10% tax is Rs. 21,384.
Let the actual price excluding tax be \( P \).
\( P \times \frac{100 + 10}{100} = 21,384 \)
\( \frac{110P}{100} = 21,384 \)
\( P = \frac{21,384 \times 100}{110} = 19,440 \)
So, the actual price is Rs. 19,440.
If the tax rate is actually 8%, the correct selling price should be:
\( 19,440 \times \frac{100 + 8}{100} = 19,440 \times \frac{108}{100} = \text{Rs. } 20,995.20 \)
The extra margin pocketed by the dealer is the difference between the collected price and this correct price:
\( 21,384 - 20,995.20 = \text{Rs. } 388.80 \)
Thus, the extra profit made by the dealer is Rs. 388.80.
In simple words: First, calculate the true value of the product before any tax was added. Then, find what the total cost would be with the correct 8% tax rate. The dealer's extra profit is the difference between what was charged and this correct total.

Exam Tip: Remember to calculate the 8% tax on the actual price (Rs. 19,440), not on the tax-inclusive price of Rs. 21,384.

Exercise 8(F)

[In this exercise, all the prices are excluding tax/VAT unless specified]

Question 1. A shopkeeper buys an article for Rs. 8,000 and sells it for Rs. 10,000. If the rate of tax under VAT is 10%, find : (i) tax paid by the shopkeeper (ii) tax charged by the shopkeeper (iii) VAT paid by the shopkeeper
Answer:
The purchase cost for the shopkeeper is Rs. 8,000, and the selling price is Rs. 10,000. The tax rate is 10%.

(i) The tax paid by the shopkeeper upon purchasing the article is:
\( 8,000 \times \frac{10}{100} = \text{Rs. } 800 \)

(ii) The tax collected by the shopkeeper when selling the article is:
\( 10,000 \times \frac{10}{100} = \text{Rs. } 1,000 \)

(iii) The net VAT payable by the shopkeeper is the difference between the tax collected and the tax paid:
\( 1,000 - 800 = \text{Rs. } 200 \)
In simple words: The shopkeeper pays tax when buying the item and collects tax when selling it. The difference between these two tax amounts is the VAT that must be paid to the government.

Exam Tip: VAT is always the net tax liability, which is Output Tax (tax charged) minus Input Tax (tax paid).

Question 2. A trader buys some goods for Rs. 12,000 and sells them for Rs. 15,000. If the rate of tax under VAT is 12%, find the VAT paid by the trader?
Answer:
The purchase price of the goods is Rs. 12,000 and the selling price is Rs. 15,000.
The tax charged on the purchase (Input Tax) at 12% is:
\( 12,000 \times \frac{12}{100} = \text{Rs. } 1,440 \)
The tax collected on the sale (Output Tax) at 12% is:
\( 15,000 \times \frac{12}{100} = \text{Rs. } 1,800 \)
The net VAT to be paid by the trader is:
\( 1,800 - 1,440 = \text{Rs. } 360 \)
In simple words: Find the tax on the purchase price and the tax on the selling price. Subtract the purchase tax from the selling tax to get the VAT.

Exam Tip: Make sure to clearly show both the Input Tax and Output Tax calculations before subtracting them to find the VAT.

Question 3. The marked price of an article is Rs. 7,000 and is available at 20% discount. Manoj buys this article and then sold it at its marked price. If the rate of tax at each state is 10%, find the VAT paid by Manoj.
Answer:
The list price of the article is Rs. 7,000 with a discount of 20%.
The cost price for Manoj before tax is calculated as:
\( 7,000 - \left(7,000 \times \frac{20}{100}\right) = 7,000 - 1,400 = \text{Rs. } 5,600 \)
The tax paid by Manoj (Input Tax) at 10% is:
\( 5,600 \times \frac{10}{100} = \text{Rs. } 560 \)
Manoj sells the article at its list price of Rs. 7,000.
The tax collected by Manoj (Output Tax) at 10% is:
\( 7,000 \times \frac{10}{100} = \text{Rs. } 700 \)
The net VAT paid by Manoj is:
\( 700 - 560 = \text{Rs. } 140 \)
In simple words: Manoj buys the item at a discount, so his purchase tax is lower. He sells it at the full original price, collecting higher tax. The difference between these two tax amounts is the VAT.

Exam Tip: First calculate the discounted price before applying the input tax rate, as tax is levied on the actual transaction value.

Question 4. A buys some goods for Rs. 4,000 and sold them to B for Rs. 5,000. B sold these goods to C for Rs. 6,000. If the rate of tax (under VAT) at each stage is 5%, find : (i) VAT paid by A (ii) VAT paid by B
Answer:
The purchase and sale details for A, B, and C are as follows, with a tax rate of 5% at each level:
- Purchase cost for A = Rs. 4,000
- Sale price from A to B (which is B's purchase cost) = Rs. 5,000
- Sale price from B to C = Rs. 6,000

The tax details are:
- Tax charged by A = \( 4,000 \times \frac{5}{100} = \text{Rs. } 200 \)
- Tax charged by B = \( 5,000 \times \frac{5}{100} = \text{Rs. } 250 \)
- Tax charged by C = \( 6,000 \times \frac{5}{100} = \text{Rs. } 300 \)

(i) The VAT paid by A is:
\( 250 - 200 = \text{Rs. } 50 \)

(ii) The VAT paid by B is:
\( 300 - 250 = \text{Rs. } 50 \)
In simple words: Both A and B add value to the goods when they resell them. Since the increase in value is Rs. 1,000 at each stage, they each pay a 5% tax on this added value, which is Rs. 50.

Exam Tip: An alternative quick way to find VAT at each stage is to calculate the tax rate directly on the profit (value added) at that stage.

Question 5. A buys an article for Rs. 8,000 and sold it to B at 20% profit. If the rate of tax under VAT is 8%, find : (i) tax paid by A (ii) tax charged by A (iii) VAT paid by A
Answer:
The purchase price for A is Rs. 8,000, and the VAT rate is 8%.

(i) The tax paid by A when purchasing the article is:
\( 8,000 \times \frac{8}{100} = \text{Rs. } 640 \)

(ii) A sells the article to B at a profit of 20%.
The selling price to B is:
\( 8,000 + \left(8,000 \times \frac{20}{100}\right) = 8,000 + 1,600 = \text{Rs. } 9,600 \)
The tax collected by A upon selling is:
\( 9,600 \times \frac{8}{100} = \text{Rs. } 768 \)

(iii) The net VAT paid by A is:
\( 768 - 640 = \text{Rs. } 128 \)
In simple words: First, find the tax A pays when buying the item. Next, find the selling price with the 20% profit to calculate the tax collected from B. Subtract the purchase tax from the sales tax to get the VAT.

Exam Tip: Ensure that the profit percentage is added to the cost price before calculating the output tax.

Question 6. A shopkeeper purchases an article for Rs. 12,400 and sells it to a customer for Rs. 17,000. If the tax under VAT is 8%, find the VAT paid by the shopkeeper.
Answer:
The shopkeeper's purchase price is Rs. 12,400, and the customer's purchase price is Rs. 17,000. The VAT rate is 8%.
The tax paid on purchase (Input Tax) is:
\( 12,400 \times \frac{8}{100} = \text{Rs. } 992 \)
The tax collected on sale (Output Tax) is:
\( 17,000 \times \frac{8}{100} = \text{Rs. } 1,360 \)
The net VAT payable by the shopkeeper is the difference between these tax amounts:
\( 1,360 - 992 = \text{Rs. } 368 \)
In simple words: Calculate the 8% tax on both the cost price and the selling price. The VAT is the difference between these two tax values.

Exam Tip: Double-check your arithmetic subtraction when calculating the final VAT to avoid simple calculation errors.

Question 7. A purchases an article for Rs. 7,200 and sells it to B for Rs. 9,600. B, in turn, sells the article to C for Rs. 11,000. If the tax (under VAT) is 10%, find the VAT paid by A and B.
Answer:
We are given the transaction chain with a 10% tax rate:
- A buys the article for Rs. 7,200 and sells it to B for Rs. 9,600.
- B sells the article to C for Rs. 11,000.

First, let us calculate the tax collected at each stage:
- Tax charged by A = \( 7,200 \times \frac{10}{100} = \text{Rs. } 720 \)
- Tax charged by B = \( 9,600 \times \frac{10}{100} = \text{Rs. } 960 \)
- Tax charged by C = \( 11,000 \times \frac{10}{100} = \text{Rs. } 1,100 \)

The VAT payable by each trader is:
- VAT paid by A = Tax collected by A - Tax paid by A:
\( 960 - 720 = \text{Rs. } 240 \)
- VAT paid by B = Tax collected by B - Tax paid by B:
\( 1,100 - 960 = \text{Rs. } 140 \)
In simple words: To find the VAT for each person, calculate the tax on their selling price and subtract the tax they paid when buying the item.

Exam Tip: Remember that the output tax of one seller (like A) is the input tax of the next buyer (like B).

Question 8. A manufacturer buys some goods for Rs. 60,000 and pays 5% tax. He sells these goods for Rs. 80,000 and charges tax at the rate of 12%. Find the VAT paid by the manufacturer.
Answer:
The manufacturer buys raw materials for Rs. 60,000 with a tax rate of 5%.
Tax paid on purchase is:
\( 60,000 \times \frac{5}{100} = \text{Rs. } 3,000 \)
The manufacturer sells the finished goods for Rs. 80,000 with a tax rate of 12%.
Tax collected on sale is:
\( 80,000 \times \frac{12}{100} = \text{Rs. } 9,600 \)
The net VAT paid by the manufacturer is:
\( 9,600 - 3,000 = \text{Rs. } 6,600 \)
In simple words: The manufacturer pays Rs. 3,000 tax when buying materials and collects Rs. 9,600 when selling the products. The VAT is the difference between these two.

Exam Tip: Be careful as the tax rates are different at the purchase stage (5%) and the selling stage (12%). Apply each percentage to its respective value.

Question 9. The cost of an article is Rs. 6,000 to a distributor, he sells it to a trader for Rs. 7,500 and the trader sells it further to a customer for Rs. 8,000. If the rate of tax under VAT is 8%; find the VAT paid by the: (i) distributor (ii) trader
Answer:
We are given:
- Cost price to the distributor = Rs. 6,000
- Selling price of the distributor (Cost price for the trader) = Rs. 7,500
- Selling price of the trader = Rs. 8,000
- Rate of VAT = 8%

First, let's find the tax values at each point:
- Tax on distributor's cost price = \( 6,000 \times \frac{8}{100} = \text{Rs. } 480 \)
- Tax on distributor's selling price (trader's cost price) = \( 7,500 \times \frac{8}{100} = \text{Rs. } 600 \)
- Tax on trader's selling price = \( 8,000 \times \frac{8}{100} = \text{Rs. } 640 \)

(i) The VAT paid by the distributor is the difference in tax on their selling price and purchase price:
\( 600 - 480 = \text{Rs. } 120 \)

(ii) The VAT paid by the trader is:
\( 640 - 600 = \text{Rs. } 40 \)
In simple words: Calculate the 8% tax at each purchase and sale price. Subtraction gives the actual VAT paid to the government at each level of distribution.

Exam Tip: Keep track of the cost price and selling price for each specific entity in the chain to correctly identify their input and output tax.

Question 10. The marked price of an article is Rs. 10,000. A buys it at 30% discount on the marked price and sells it at 10% discount on the marked price. If the rate of tax under VAT is 5%, find the amount of VAT paid by A.
Answer:
The list price of the article is Rs. 10,000, and the tax rate is 5%.
A purchases the article at a 30% discount.
The purchase price for A is:
\( 10,000 - \left(10,000 \times \frac{30}{100}\right) = 10,000 - 3,000 = \text{Rs. } 7,000 \)
The input tax paid by A on this purchase is:
\( 7,000 \times \frac{5}{100} = \text{Rs. } 350 \)
A resells the article at a 10% discount on the original list price.
The selling price for A is:
\( 10,000 - \left(10,000 \times \frac{10}{100}\right) = 10,000 - 1,000 = \text{Rs. } 9,000 \)
The output tax collected by A on this sale is:
\( 9,000 \times \frac{5}{100} = \text{Rs. } 450 \)
The total VAT paid by A is:
\( 450 - 350 = \text{Rs. } 100 \)
In simple words: First, calculate the prices A bought and sold the item for using the respective discount rates. Then, compute the 5% tax for both prices and subtract them to find the VAT.

Exam Tip: Be sure to apply both discounts to the original list price of Rs. 10,000 as specified in the problem statement.