In this let us study several aspects of commercial Mathematics which are useful in our daily life.

**1.4.2 Percentage **

We are so familiar with the meaning of percentage. **Per cent** means **for every hundred**. Thus, 4 percent means **4 for every hundred or **^{4}/_{100 }**.** Percentage is also a fraction whose denominator is 100. the numerator of the percentage is called **rate percent**.

Thus, 12% means 12 out of hundred or ^{12}/_{100}. The concept of percentage used in business transactions, calculating interest, comparison of quantities and the like.

**Percentage is always a convenient way of comparing quantities.**

** **

**Example 1: **A man has spends 78% of his monthly income and saves Rs. 1,100. What is his monthly income?

**Solution:**

Note: Let us solve the problem by using percentage.

Let his monthly income be Rs. 100. Then his expenditure is Rs. 78. Therefore, his savings is 100 – 78 = Rs. 22

If his savings is Rs. 22, then the income is Rs. 100. If the savings is Rs. 1,100 then the income is ^{100}/_{22 }x 1000 = 5000 rupees.

His monthly income is Rs. 5000

**Verification: **785 of Rs. 5000 = ^{78}/_{100 }x 5000 = Rs. 3900_{.} Hence his savings is 5000 – 3900 = Rs. 1100.

**Example2:** An athlete won 8 events out of a number of events. If the win percentage was 40, how many events were there in total?

**Solution:**

We know, the win percentage is 40. So, 40 events are won out of 100.

Therefore 8 events are won out ^{40}/_{100 }x 8 = 20 events._{ }

Hence, There were 20 events in total.

**Example 3:** Ravi’s income is 25% more than that of Raghu. What percent is Raghu’s income less than that of Ravi?

**Solution:**

Let Raghu’s income be Rs. 100. Then, Ravi’s income is Rs. 125. Put this in the reverse way. If Ravi’s income is Rs. 125 and Raghu’s income is Rs. 100.

If Ravi’s income is Rs. 1, Raghu’s income is ^{100}/_{125}. Changing the scale to 100.

If Ravi’s income is 100 then, Raghu’s income is ^{100}/_{125 }x100 = Rs. 80

Therefore, Raghu’s income is 100 – 80 = 20% less than that of Ravi.

**Example 4:** The salary of an employee is increased by 15%. If his new salary is Rs. 12650, what is his salary before enhancement?

**Solution:**

Let the salary before enhancement be Rs. 100. Since his increment is 15%, his salary enhancement is Rs. 100 + Rs. 15 = Rs. 115.

Now, we reverse the role. If the new salary is Rs. 115, salary before enhancement is Rs 100. If the new salary is Rs. 12,650, salary before enhancement is^{100}/_{115} x 12650 = Rs 11000

Therefore, his salary before enhancement is Rs. 11000

**Exercise 1.4.2**

- In a school, 30% of students play chess, 60% play carrom and rest play other games. If the total number of students in the school is 900, find the exact number of students who play each game.

**Solution:**

Total number of students in the school= 900

30% of students play chess

Exact number of students who play chess

= ^{30}/_{100}× 900 = 270

60% of students play carrom

Exact number of students who play Carrom

=^{60}/_{100}× 900 = 540

Numbers of students who play other games = 900 – [270 + 540]

= 900 – 810

= 90

- In a school function Rs 360 remained after spending 82%of the money. How much money was there in the beginning? Verify your answer.

**Solution: **

Let the money in the beginning be =100 Rs

It is given that, 82% of the money was spent.

Therefore, exact amount spent = ^{82}/_{10o} × 100 = 82 Rs

Money remaining = Money in the beginning – Money spent

= 100 – 82

= 18 Rs.

Therefore, the amount of money remaining is 18 if the money in the beginning is 100. It is given that, the money remaining is Rs. 360 and then money in the beginning is,^{ (360×100)}/_{18} = 2000

Therefore, money in the beginning is Rs. 2000.

Verifying:

If the money in the beginning is Rs. 2000 after spending 82%, then the amount of money spent is, = 1640.

Therefore,

Remaining money = money in the beginning – amount of money spent

= 2000 – 1640

= 360.

Thus, the remaining money is Rs. 360.

- Akshay’s income is 20% less than that of Ajay. What percent is Ajay’s income more than that of Akshay?

**Solution: **

Ajay’s income be = 100Rs

Akshay’s income = 100 – 20 = 80 Rs

In reverse way, if Akshay’s income is 80Rs, Ajay’s income is 100 Rs

If Akshay’s income is Rs. 1 and Ajay’s income is 1×^{100}/_{80} =^{100}/_{80}

Changing scale to Rs. 100.

If Akshay’s income is 100 Rs, then Ajay’s income =^{100}/_{80} × 100 = 125 𝑅𝑠

∴ Ajay’s income = Rs. 125 and Akshay’s income is Rs. 100.

Therefore, Ajay’s income is 25% more than Akshay’s income.

- A daily wage employee spends 84% of his weekly earning. If he saves Rs. 384, find his weekly earning.

**Solution: **

Let the weekly earning of the employee be = 100 Rs

Amount spent = 84% of weekly earning

Exact amount spent =^{84}/_{100}× 100 = 84 𝑅𝑠

Money saved = earning – money spent

= 100 – 84

= 16

Rs. 16 is the weekly savings for Rs. 100. If weekly savings is Rs. 384, then, the his weekly earning is,^{ (384×100)}/_{16} = 2400

Therefore, his weekly earnings be Rs. 2400.

- A factory announces a bonus of 10% to its employees. If an employee gets Rs 10,780, find his actual salary.

**Solution: **

Let the salary of the employee be = 100 Rs

Bonus declared = 10%

Thus, bonus amount = ^{10}/_{100}× 100 = 𝑅𝑠 10

Therefore, total amount employee gets which includes salary and bonus = 100+10 = 110 Rs

If the total amount is 110 Rs, actual salary = 100 Rs

If the total amount is 10,780 Rs, actual salary

=^{(10780×100)}/_{110} = 9800𝑅s.

Therefore, his actual salary is 9800.

** **

**1.4.3 Profit and loss:**

The money paid to buy goods is called **cost price** and abbreviated as **C.P.** The price at which goods are sold in shops is called **selling price** and abbreviated as **S.P.**

When an article is bought, some addition expenses such as freight charges, labour charges, transportation charges, maintenance charges etc., are made before selling. These expenses are known as **overhead expenses**. These expenses have to be included in the cost price. Thus,

Real cost price = total investment

⇒ C.P. = price for + overhead

Buying goods charges

- If S.P > C.P., there is a gain or profit.
- If S.P > C.P., there is a loss.
- Profit = S.P – C.P.
- Loss = C.P – S.P.
- Gain (profit) or loss on 100 is called gain percent or loss percent.

**Profit or Loss is always calculated on C.P **

- Gain or profit % =
^{Gain}/_{P}×100 - Loss % =
^{Loss}/_{P}×100 - P.=
^{(100+Gain%)}/_{100}× C.P - P. =
^{(100-Loss%)}/_{100}x C.P - P.=
^{100}/_{(100+gain%) }× S.P - P =
^{100}/_{(100-LOSS%)}×S.P

**Example 5: **The cost price of a computer is Rs. 19500. An additional Rs. 450 was spent on installing software. If it is sold at 12% profit, find the selling price of the computer.

**Solution:**

Cost price of the computer is Rs. 19500 + Rs. 450 (overhead expenses) = Rs. 19950,

The computer is sold at a profit of 12%. Therefore,

S.P.= ^{(100+Gain%)}/_{100}× C.P

S.P. = ^{(100+12%)}/_{(100)}

= ^{112}/_{10 } × 19950

= 22344

Thus, selling price is Rs. 22344

**Example 6: ** On selling a bicycle for Rs. 4300, a dealer loses 14%. For hiw much should he sell it to gain 14%.

**Solution:**

Selling price of a bicycle is Rs. 4300 and the loss is 14%.

Therefore,

C.P =^{100}/_{(100-loss%)} x S.P

= ^{100}/_{(100-14%) } × 4300

= ^{100}/_{(84) }×S.P

= 5000

So the cost of the bicycle is 5000. Let us find out what should be the selling price to get 14% profit.

Expected gain = 14%

Therefore,

S.P.=^{ (100+gain%)}/_{(100) }× 5000

= ^{114}/_{(100) } × 5000

= 5700

Hence, the selling price of the bicycle to gain 14% profit is 5700.

**Example 7:** Two cows were sold for Rs. 12000 each, one at a gain of 20% and the other at a loss of 20%. Find the loss or gain in the entire transaction.

**Solution:**

Let us first find out the cost price of each cow and then add them to find the total amount spent. We know the total amount of money received. Comparing them we will know whether there is a loss or profit in the whole transaction.

First cow:

Selling price Rs. 12000

Gain = 20%.

Therefore, C.P.= ^{100}/_{(100+gain%) }×S.P

C.P.= ^{100}/_{(100+20%) }×12000

= ^{100}/_{(120) }×12000

= 10000

Second cow:

Selling price 12000

Loss 20%

Therefore,

C.P = ^{100}/_{(100-loss%) }× S.P

C.P = ^{100}/_{(100-20%) }× 12000

= ^{100}/_{80 }× 20000

= 15000

Total cost of the cows = 10000 + 15000 = 25000

Total selling price of both cows = 12000 + 12000 = 24000

Therefore, loss = Rs. (25000 – 24000) = Rs. 1000

Hence, loss percentage, ^{loss}/_{c.p }x 100 =^{100}/_{(250000)} x 100 = 4

Therefore, there is a loss of 4% in the whole transaction.

**Example 8: **If the cost of 21 cell phones is equal to selling price of 18 cell phones, find the profit percent.

**Solution:**

Let the c.p of each cell phones be Re. 1 . Then the c.p. of 21 cell phones is Rs. 21.

By the given data, S.P. of 18 cell phones = c.p. of 21 cell phones = Rs 21

Therefore, S.P. of 1 cell phone is ^{21}/_{(18) } . This gives,

Profit = S.P. – C.P. = ^{21}/_{18} – 1 = ^{3}/_{18} = ^{1}/_{6 }=0.166

Thus, there is a profit of Rs. ^{1}/_{6} on each phone. Now we can calculate profit percentage:

Profit percentage =^{Profit}/_{c.p }x 100 = ^{1.6}/_{1 }x100 =^{50}/_{3 } = 16 ^{2}/_{3 }

Therefore, the profit percentage is 16 ^{2}/_{3}

_{ }

**Example 9: **A dealer sells a radio at a profit of 8%. Had he sold it for Rs. 85 less, he would have lost 2%. Find the cost price of the radio.

**Solution:**

Let the cost price of the radio at a profit of the radio be x. Let us calculate the selling price with 8% profit and 2% loss, separately.

With 8% profit:

We have here,

S.P.= ^{(100+Gain%)}/_{100 } × C.P

S.P.=^{ (100+8)}/_{100 } × x

= ^{54}/_{50 } x

With 2% loss:

We obtain here,

S.P.=^{ (100-Loss%)}/_{100} C.P

S.P.= ^{(100-8)}/_{100 }x x

= ^{49}/_{50 }x

Thus, the difference in the selling price with 8% profit and 2% loss is,

^{54}/_{50 } x – ^{49}/_{50 }x = ^{5}/_{50 }x = ^{x}/_{10}

But, this difference is given to be equal to Rs. 85, so that ^{x}/_{10} = 85. This implies that, x = 850. Hence, the cost price of radio is Rs. 850.

**Exercise 1.4.3**

- Sonu bought a bicycle for Rs 3750 and spent Rs 250 on its repairs. He sold it for Rs4,400. Find his loss or profit percentage.

**Solution:**

Amount paid to buy the bicycle = Rs 3750

Amount spent on repairs = Rs 250

Total amount spent on bicycle = 3750 + 250 = 4000

Profit = S.P – C.P

= 4400 – 4000

= 400

Profit % =^{(Profitx100)}/_{c.p }= ^{(400×100)}/_{(4000) }=10%

- Shopkeeper purchases a article for Rs 3500 and pays transport charge of Rs 100. He incurred a loss of 12% in selling this. Find the selling price of the article.

**Solution: **

Amount paid to purchase the article = 3500 Rs

Amount spent on transport = 100 Rs

Total amount spent on article

= 3500 + 100

= 3600

Let the C.P be 100 Rs. It is given that, loss 12% then, the loss for Rs 100 is, Rs. 12.

{Loss = C.P – S.P = C.P – Loss}

∴ S.P = 100 – 12 = 88 Rs

If C.P is 100 Rs, S.P of article for 12% loss = 88 Rs

If C.P is 3600Rs, S.P of article for 12% loss, then,

_{=}^{ (3600×88)}/_{100 }= 3168

Therefore, the cost price is Rs. 3168.

- By selling a watch for Rs 720, Ravi loses 10%. At what price should he sell it, in order to gain 15%?

**Solution: **

It is given that,

S.P of watch = 720 Rs

Loss = 10%

Therefore,

C.P = ^{100}/_{(100-Loss%) }x S.P.

= ^{100}/_{(100-10) }x 720

= ^{100}/_{90 }x 720

= 800

C.P of watch is Rs. 800.

It is given in the problem that, Profit expected = 15%

C.P = ^{100}/_{(100+gain%) }x S.P.

= ^{100}/_{(100-15%) }x 800

= ^{100}/_{85 }x 800

=920

- Hari bought two fans for Rs. 2400 each. He sold one at a loss of 10% and the other at a profit of 15%. Find the selling price of each fan and find also the total profit or loss.

**Solution: **

Selling price of a fan Rs. 2400

Loss from 1^{st} fan = 10%

Therefore,

S.P = ^{(100-Loss%)}/_{100 }x C.P.

= ^{(100-10)}/_{100 }x 2400

= ^{90}/_{100 }x 2400

= 2160

Profit from a fan is 15%

C.P. of fan = 2400 Rs

S.P = ^{(100+Gain%)}/_{100 }x C.P.

S.P = ^{115}/_{100 } × 2400

=2760 Rs

Total cost price of two fans = 2400 + 2400 = 4800 Rs

Total selling price of two fans = 2160 + 2760 = 4920 Rs

Total profit = Total S.P.-Total C.P.

= 4920 – 4800

= 120 Rs

- A store keeper sells a book at 15% gain. Had he sold it for Rs 18 more, he would have gained 18%. Find the cost price of the book.

**Solution:**

Let the C.P be Rs. x

Profit = 15% = 15Rs.

S.P = ^{(100+Gain%)}/_{100 }x c.p.

S.P = ^{(100+15)}/_{100 }x C.P.

S.P.= ^{115}/_{100 }x X

When the profit is 18%,

S.P.= ^{(100+gain%)}/_{100 }x C.P.

= ^{118}/_{100 }X

It is given that, if he had he sold it for Rs 18 more, he would have gained 18%.

18 = ^{118}/_{100 }x – ^{115}/_{100} x

= ^{3}/_{100 }x

x = (18×100)/3 = 600

The cost price of the book is Rs. 600.

- The cost price of 12 pens is equal to selling price of 10 pens. Find the profit percentage.

**Solution:**

Let the C.P of one pen be 1 Rs

∴ C.P of 12 pens = 12 Rs

Given S.P. of pens = Cost price of 12 pens

∴ S.P of 10 pens = 12 Rs

C.P of 10 pens = 10 Rs

S.P.> C.P.

Profit = S.P. – C.P.

= 12 – 10

= 2 Rs

Profit % = ^{(profitx100)}/_{c.p }=^{(2×100)}/_{10 }= 20%

Therefore, percentage of profit is 20%.

** **

** **

**Discount:**

A reduction on the market price of articles is called discount. The following completely describe facts related to the discount:

- Discount is always given on the marked price of the article.
- Discount =Marked price – Selling price.
- Marked price is called list price.
- Discount = Rate of discount × Marked price
- Net price [S.p] = Marked price – discount.

**Example 10: **A computer marked at Rs. 18000 was sold at Rs. 15840. Find the percentage of discount.

**Solution:**

Marked price is Rs. 18000 and selling was sold at Rs. 15,480. Therefore, discount = Rs. 18000 – Rs. 15840 = Rs. 2160

Thus, for a marked price of Rs. 18000, discount is Rs. 2160.

For a marked price of Rs. 100, discount =^{2160}/_{18000 }x100 = 12%

Therefore, the percentage of discount is 12.

**Example 11: **A tape recorder is sold at Rs. 5225 after being given a discount of 5%. What is its marked price?

**Solution:**

We are given that the discount is 5%. This means that for Rs. 100, the discount is Rs. 5%.

Therefore, selling price = Rs. 100 – Rs. 5 = Rs. 95

Thus on a selling price of Rs. 95, the marked price = ^{100}/_{95}x 5225 = 5500

Therefore, marked price of the tape recorder is Rs. 5500.

**Example 13: ** A shop keeper buys an article for Rs. 500. He marks it at 20% above the cost price. If he sells it at 12% discount, find the selling price.

**Solution:**

Cost price = Rs. 500

Profit = 20% of Rs. 500 = ^{20}/_{100} x 500 = Rs. 100

Marked price = cost price + profit = 500 +100 = Rs. 600

Now the discount given is 12%.

For, Rs. 600, the discount = ^{12}/_{100 }x600 = 72 rupees

Therefore, selling price = cost price – discount

= 600 – 72

= Rs. 528

Hence, the selling price of the article is Rs. 528.

**Exercise 1.4.4**

- An article marked Rs 800 and sold for Rs. 704. Find the discount and discount percent.

**Solution:**

M.P of article = 800 Rs

S.P of article = 704 Rs

Discount = M.P. – S.P.

= 800 – 704

=96 Rs

Discount % = ^{Discount}/_{M.P }×100

Discount % =^{96}/_{800 }×100 = 12%

- A dress is sold at Rs 550 after allowing a discount of 12%. Find its marked price.

**Solution:**

Let the marked price be 100 Rs

Discount = 12% = 12 Rs

∴ S.P = M.P – Discount = 100 -12 = 88 Rs

If the S.P is 88 Rs, the M.P = 100 Rs

If the S.P is 550 Rs,

Therefore, the marked price = ^{(550×100)}/_{88} = 625_{ }

Therefore, the marked price is Rs. 625.

- A shopkeeper buys a suit piece for Rs 1400 and marks it 60% above the cost price. He allows a discount of 15% on it, find the marked price of the suit piece and also the discount given.

**Solution: **

Amount paid by the shopkeeper to buy the suit piece = 1400 Rs

Profit = 60%

then, Profit for 1400 is , = ^{Profit%}/_{100 } ×C.P

= ^{60%}/_{100 } ×1400

= x (14)

=840 Rs

15% discount on raised M.P = ^{15}/_{100 } × 2240 = 336 Rs

[ ∵ Discount = rate of discount × M.P ]

Discount = M.P – S.P

⇒ S.P = M.P. –Discount

= 2240-336

= 1904 Rs

- A dealer marks his goods 40% above the cost price and allows a discount of 10% find the profit percent.

**Solution:**

Let the C.P. be 100 Rs

The marked price = 100 + 40 = 140 Rs

[∴ Discount = 10% = 10]

If M.P. is 100 Rs, then S.P. = 90 Rs

∴ If M.P is 140 Rs, then S.P = ^{(140+90)}/_{100 }

=126

C.P = 100 Rs, S.P = 126 Rs,

S.P > C.P.

Therefore there is a profit.

Profit = S.P – C.P

= 126 – 100

= 26 Rs

Profit % = ^{(profitx100)}/_{c.p }= ^{(26×100)}/_{100 } = 26%

- A dealer is selling an article at a discount of 15% find:
- a) The selling price if the marked price is Rs 500
- b) The cost price if he makes 25% profit

**Solution:**

- a) P. = 500 Rs

Discount = 15 %

15% of 500 =^{15}/_{100 }x 500 = 75 Rs

Discount = M.P – S.P

S.P = M.P – Discount

=500-75

=425 Rs

- b) S.P = 425 Rs

Profit = 25%

C.P = ^{100}/_{(100+profit%) }× S.P

= ^{(100)}/_{(100+25%) }× 425

=^{100}/_{125 }x 425 = 340 Rs

**1.4.5 Commission**

The mediator who helps buying and selling of houses, sites, vehicles ect. Is called commission agent or broker.

The money that the broker or agent receives in the deal is called brokerage or commission.

Commission is calculated on the transaction amount in percentage.

Commission per hundred rupees is called commission rate.

Commission = Commission rate ×S.P

Selling price = ^{100}/_{commission rate } ×𝑐𝑜𝑚𝑚𝑖𝑠𝑠𝑖𝑜𝑛.

** **

**Example 14: **A real estate agent receives a commission of 1.5% in selling a land for Rs. 1,60,000. What is the commission amount?

**Solution:**

Selling price of a land is Rs. 1,60,000 and the commission rate is 1.5%

If the selling price is Rs. 100, commission = Rs. 1.5

If the selling price is Rs. 1,60,000 , commission =^{1.5}/_{100} x 160000 = 2400_{ }

Therefore the commission amount is Rs. 2400.

**Example 15: ** The price of a long book is Rs. 18. A shop keeper sells 410 note books in a month and receives Rs. 1,033.20 as commission. Find the rate of commission.

**Solution:**

Price of one long note book is Rs. 18

Price of 410 long note books = 410 x 18 = 7380 Rs.

Now the commission received for this amount is Rs. 1033.20. Hence for Rs. 100, the commission is

^{1033.20}/_{7380 } x 100 = 14

Therefore, the rate of commission is 14%.

**Example 16: ** Abdul sold his house through a broker by paying Rs. 6125 as brokerage. If the rate of brokerage is 2.5%, find the selling price of the house.

**Solution:**

Brokerage given is Rs. 6125. The brokerage rate is 2.5%

If the brokerage is Rs. 2.5, selling price would be Rs. 100.

If the commission is Rs. 6125, selling price is,

^{100}/_{2.5 } x 6125 = 245000_{ }

Thus, the selling price of the house is Rs. 2,45,000.

**Exercise 1.4.5**

- Sindhu sells her scooty for Rs 28,000 through a broker. The rate of brokerage is %. Find the commission agent gets and the net amount Sindhu gets.

**Solution: **

Commission = commission rate × S.P

= ^{2.5}/_{100} x 28000_{ }

= 700 Rs

The amount commission agent gets is Rs. 700

Net amount Sindhu gets = 28000 – 700

= 27300 Rs

[∴ Net amount = S.P – Commission]

- A share agent sells 2000 shares at Rs 45 each and gets the commission at the rate of 1.5%. Find the amount the agent gets.

**Solution:**

S.P of 1 share = 45 Rs

S.P of 2000 shares = 2000×45 = 90000 Rs

Commission rate = 1.5% =^{15%}/_{10}

Commission = Commission rate ×S.P

= ^{15}/_{(10×100) } × 90000

= 1350 Rs

- A person insures Rs 26000 through an insurance agent. If the agent gets Rs 650 as the commission, find the rate of commission.

**Solution:**

Commission for Rs 100 is called

Commission rate for Rs 26,000

Commission = 650 Rs

∴ Commission for 100 =^{(100×650)}/_{6000 }=^{5}/_{2 } = 2.5

∴ Commission rate = 2.5%

- A selling agent gets Rs 10,200 in a month. This include his monthly salary of Rs 6,000 and 6% commission for the sales find the value of goods he sold.

**Solution:**

Salary + Commission = 10,200

6,000 + Commission = 10,200

Commission = 10,200 – 6,000

Commission = 4,200 Rs

If the commission is 6 Rs, the value of goods sold = 100 Rs.

∴ If the commission is 4200 Rs, the values of goods sold = ^{(4200×100)}/_{6 }= 70,000 Rs

**1.4.6 Simple Interest**

** **

People borrow money from banks or financial institutions or money lenders for various purposes. While returning the borrowed money after a period of time, they need to pay some extra amount. This extra amount paid on the borrowed money after a period of time is called interest.

Principal : The money borrowed is called principal or sum.

Interest : The extra money paid on the principal after a period of time is called interest.

Amount : The total money paid is called amount. **Amount = Principal + Interest. **

Rate : Interest for every Rs 100 for one year is known as rate percent per annum.

Time : Time is the duration for which the borrowed money is utilised. Time is expressed in years or months or days.

Simple interest : The interest paid on the principal alone. In the world of finance (Bankers rule), time is often expressed in days also.

**Formula to find the simple interest **

Let P = principal,

R = rate of interest per annum,

T = time in years,

I = simple interest. These are related by the formula.

I = ^{(PxTxR)}/_{100 }

When the time is given in days or months, it is be expressed in years.

For calculating interest, the day on which money deposited is not counted, while the day on which money is withdrawn is counted.

**Example 17: **Calculate the interest on Rs. 800 at 6^{1}/_{2}% per annum, for 3^{1}/_{2 }years.

**Solution:**

Given: P= Rs. 800; T=3^{1}/_{2 }years , R = 6^{1}/_{2}% = ^{13}/_{2}%

We use the formula for I,

I = ^{PxTxR}/_{100} = ^{(800x 7 x13)}/_{(2x2x100)} = 2x7x13 =182

Thus, the interest is Rs. 182.

**Example: **Rs. 800 amounts to Rs. 920 in 3years at a certain rate of interest. If the rate of interest is increased by 3%, what would the amount will become?

**Solution:**

Recall, interest = Amount – Principal

Hence,

I = Rs. 920 – Rs. 800 = Rs. 120

This interest is accrued in T = 3years, for the principal Rs. 800. Using ,

I = ^{PxTxR}/_{100}, we get,

R = ^{(100xI)}/_{PxT}

R = ^{(100×120)}/_{(800×3) } = 5

Thus, the original rate of interest is 5%. After the increase of interest by 3%, the new rate of interest, which we again denote by R = 5% + 3% = 8%.

The principal P = Rs. 800 and the period T = 3 years remain the same.

Therefore,

I = ^{PxTxR}/_{100 }= ^{(800x3x8)}/_{100} = 192

Therefore, the new amount = Rs. 800 + Rs 192 = Rs. 992

**Exercise 1.4.6**

- Find the simple interest on Rs 2500 for 4 years at 6 % per annum.

**Solution:**

P = 2500 Rs

T = 4 years

R = 6^{1}/_{4} % = 6.25% = ^{25%}/_{4}

I = ^{PxTxR}/_{100 } = ^{2500x4x25}/_{100} = 625 Rs.

- Find the simple interest on Rs 3,500 at the rate of 2
^{1}/_{2}% per annum for 165 days.

**Solution:**

P = 3500 Rs

T = 165 days = 165/365 year [∴ 1 year = 365 days]

R = 2^{1}/_{2} % = 2.5%

I = ^{PxTxR}/_{100 }= ^{(3500×2.5×165)}/_{(100×365)} = ^{1443750}/_{36500 }= 39.55 Rs.

- In what period will Rs 5,200 amount to Rs 7,384 at 12% per annum simple interest?

**Solution:**

P = 5,200 Rs

A = 7,384 Rs

R = 12%

T = ?

S.I = A-P = 7,384 – 5,200 = 2,184 Rs

2184= ^{(5200xTx12)}/_{100 }

T = I = ^{(2184×100)}/_{(5200×12)}

= 3.5 years.

- Ramya borrowed a loan from a bank for buying a computer. After 4 years she paid Rs 26,640 and settled the accounts. If the rate of interest is 12% per annum, what was the sum borrowed?

**Solution:**

Let the sum borrowed be = x → P

T = 4 years 18000

A = 26,640 37 666000

R = 12% 37↓

S.I = A-P 296

S.I = 26,640-x 269

S.I = ^{PxTxR}/_{100}

(26,640 – x) = ^{(xX4X12)}/_{100}

25(26,640-x) = 12x

25 ×26,640 – 25x = 12x

6,66,000 = 12x + 25x

6,66,000 = 37x

x = ^{66600}/_{37}

x = 18,000

Therefore, the sum borrowed = 18,000 Rs

[Principal]

- A sum of money triples itself in 18 years, find the rate of interest.

**Solution:**

Let the principal amount be = x

Therefore, the amount = 3x

[∴ sum of money triples]

T = 8 years

R = ?

S.I = A – P

S.I = 3x – x

S.I = 2x

S.I = ^{PxTxR}/_{100}

2x = ^{(xx8xR)}/_{100}

2x = ^{(2xxR)}/_{25}

R = ^{(2xx25)}/_{2x} = 2x

**1.4.7 Tax**

The Government requires money for its functioning. Money required for a Government is collected from the public in the form of taxes. One such method of collecting money is sales tax.

Sales tax is the tax we pay when we buy goods/articles from a shop. Sales tax is charged by the Government on the sale of every good/article. Sales tax is called indirect tax as it is collected from the manufacturer, wholesaler and retailer (shop keeper) who in turn collects it from the customer.

Value added tax (VAT): VAT is revised version of sales tax.

Example 21: Abdul purchases a pair of clothes with a marked price Rs. 1350. If the rate of sales tax is 4%, calculate the amount to be paid by him:

**Solution:**

Marked price of the item is Rs. 1350

Sales tax is 4%

Hence the total sales tax on the item is ^{4}/_{100 }x 1350 = Rs. 54

Amount to be paid = marked price + tax

= 1350 + 54

= 1404 Rs.

Hence, the amount to be paid by him is Rs. 1404.

**Exercise 1.4.7**

- A person purchases the following items from a mall for which the sales tax is mentioned a mall for which the sales tax is mentioned against.

- a) Stationary materials for Rs 250 and sales tax of 4% there on.
- b) Electronic goods worth Rs 2,580 and sales tax of 10% there on.
- c) Groceries worth Rs 1,200 on which sales tax of 3% is levied.
- d) Medicines worth Rs 200 with sales tax of 6%.

Find the bill amount for each item.

**Solution:**

- a) M.P = 250 Rs

Sales tax = 4%

Tax amount = ^{4}/_{100} × 250 = 10 Rs

Bill amount = M.P + tax amount

= 250 + 10

= 260 Rs

- b) M.P =2,580

Sales tax = 10%

Tax amount = ^{10}/_{100} × 2580 = 258 Rs

Bill amount = M.P + tax amount

= 2,580 +258

= 2,838 Rs

- c) M.P = 1,200

Sales tax = 3%

Tax amount = ^{3}/_{100} × 1200 = 36 Rs

Bill amount = M.P + tax amount

= 1,200 +36

= 1,236 Rs

- d) M.P = 200

Sales tax = 6%

Tax amount = ^{6}/_{100} × 200 = 12 Rs

Bill amount = M.P + tax amount

= 200 +12

=212

- A person buys electronic goods worth Rs 10,000 for which the sales tax is 4% and another material worth Rs 15,000 for which the sales tax is 6%. He manufactures a gadget using all these and sells it a 15% profit. What is his selling price?

**Solution:**

M.P of electronic goods = 10,000 Rs

Sales tax = 4%

Tax amount = ^{4}/_{100} × 10000 = 400 Rs

Bill amount = 10,000 + 400 = 10,400 Rs

[ Bill amount = M.P + tax ]

M.P of other material = 15,000 Rs

Sales tax = 6%

Tax amount = ^{6}/_{100} × 15000 = 900 Rs

Bill amount = M.P + tax = 15,000 + 900

= 15,900 Rs

Total C.P of gadget = 10,400 + 15,900

Profit = 15% = 26,300 Rs

S.I. = ^{ (profit%+100)}/_{100} × C.P.

= ^{(100+15)}/_{100} × 26300 = 26,300 Rs

= ^{115}/_{100} × 26300

= 30,245 Rs

- A trader purchases 70 kg of tea at the rate of Rs 200/kg and another 30 kg at the rate of Rs 250/kg. He pays a sales tax of 4% on the transaction. He mixes both of them and sells the product at the rate of Rs 240/kg. what is the percentage gain or loss (Find approximate value)?

**Solution:**

C.P of 70 kgs of tea = 200 ×70

Tea-powder at 200/kg = 14,000 Rs

C.P of 30 kgs of tea = 250×30

Powder at 250/kgs = 7,500 Rs

C.P of 100kgs of tea powder = 14,000+7,500 = 21,500 Rs

Sales tax = 4%

Tax amount = ^{4}/_{100} ×21500 = 860 Rs

Bill amount = C.P + Tax = 21,500 + 860 – 22,360 Rs

Total C.P of 100 kgs of tea-powder = 22,360 Rs

S.P of 100 kgs of tea-powder at Rs 240/kg = 240 ×100 = 24,000 Rs

S.P > C.P

Therefore, profit is,

Profit = S.P-C.P

= 24,000-22,360

= 1,640 Rs

Profit % = ^{(profit%x100)}/_{c.p.} = I = ^{(1640×100)}/_{2360} = 7.3%

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