If person A borrows some money from another person B for a certain period, then after that specified period, the borrower has to return the money borrowed as well as some additional money. This additional money that borrower has to pay is called interest.
The actually borrowed money by A is called principal (SUM).
The interest that the borrower has to pay for every 100 rupees borrowed for every year is known as rate percent per annum. It is denoted as R% per annum =
The time for which the borrowed money has been used is called the time. It is denoted as T years.
The interest is directly proportional to the principal, the rate and time for which the borrowed sum is used. If the interest on a certain sum borrowed for a certain period is reckoned uniformly, then it is called Simple Interest and denoted as S.I.
Where P = Principal or the sum borrowed
R = Rate percent per annum
T = Number of years for which the borrowed money has been used.
The principal and the interest together is called amount.
Dr. M. I. Rajpoot wants to buy an airconditioner for his family. He went to a shop and selects an AC whose price is Rs. 12,700. The shopkeeper gave him two offers either you pay the full amount of Rs. 12,700 or pay only Rs. 4,000 and installment of Rs. 3000 per month for next 3 months just paying only Rs. 300 as interest. What is the rate of interest the shopkeeper charged Rajpoot?
Sol. In these types of problems the interest charged is always calculated on the basis of the onemonth principal, not on the amount of the loan taken. Here first of all we have to calculate the onemonth principal for every installment paid.
Principal for 1^{st} month = loan amount = Rs 8,700
Principal for 2^{nd} month = loan amount – 1^{st} installment = Rs (8,700 – 3,000) = Rs. 5,700Principal for 3^{rd} month = Rs. (5,700 – 3,000) = Rs. 2,700
Total one month principal = Rs.(8700 + 5700 + 2700) = Rs. 17,100
As discussed in the topic on ‘Simple Interest’, the principal (P) remains constant throughout the period for which the money (principal) is borrowed.
For the first year the simple interest and compound interest, both are same, but with the next following years the C.I will always be more than the S.I.
Let P is the principal invested at r% rate per annum at S.I and C.I respectively. What will be the difference between SI and CI for different years?
Let the value of each equal annual installment = Rs. A
Rate of interest = R % p.a. at CI
Number of installments per year = n
Number of years = T
∴ Total number of installments = n × T
Borrowed Amount = B
Then,
Sol. Principle becomes 3 times means, the interest is 2P.
∴ 2P = ⇒ r = 20%
Ex.2 A man took Rs. 5000 at 10% simple interest and gave it to another person at 10% compound interest, which is being compounded annually. After 3 years, how much extra money he will get?
Sol. He has to pay = 6500
He will get = 6655
So, the ans is 6655 – 6500 = Rs 155.
Ex.3 Certain money becomes double in 4 years according to simple interest. In how many years, will it become 3 times?
Sol. If P is the principle, it becomes 2P in 4 years, that means the interest earned is 2P – P = P. If it has to become 3 times means the interest has to become 2P, so, it takes double the time. i.e. 2 × 4 = 8 years.
Ex.4 Certain money doubles itself according to compound interest in 3 years. How much time does it take to become 4 times?
Sol. Since the money gets doubled in 3 years according to the Compound Interest, it again gets doubled in the next 3 years. So it gets 4 times in 6 years.
∴n = 6 years.
Ex.5 Mr. A deposited certain amount in a bank exactly 5 years back @ of 8.8% simple interest. Mr. B deposited the same amount of money exactly 2 years back, which is for compound interest, compounded annually. Now both they got same amount of money at what rate of interest, Mr. B deposited his money.
Sol. Mr. A will get P + = 1.44P
Mr. B will get
Since both got the same money.
= 1.44P
⇒ r = 20%
Ex.6 On a sum of Rs. 1000, the C.I. for 2 years is twice the S.I. for 2 years when the rate is 11%. Find the rate at which the interest is compounded annually?
Sol. S.I. at 11% for 2 years on a sum of Rs. 1000 = 220
Since C.I. is twice the S.I.
Ex.7 Two equal sums are lent at the same time at 6% and 5% simple interest respectively. The former is received 2 years earlier than the later, and the amount in each case is Rs. 2400. Find the sum?
Sol. Let the sum be Rs. x & the latest period be n years
Solving these two equations simultaneously,
n = 12, x = 1500 The periods are 10 years & 12 years, Sum is Rs. 1500.
Ex.8 Ram invests a certain amount of money and earns a Compound Interest of Rs. 420 in the second year and a C.I. of Rs. 462 in the third year. Calculate at what rate of interest did Ram invest?
Sol. C.I. on thirdyear – C.I. on secondyear = 462 – 420 = 42
Thus Rs. 42 is the interest on Rs. 420. i.e. 10% of 420Hence Rate = 10%.
Ex.9 Robin lend out Rs. 9 on the condition that the loan is payable in 10 months by 10 equal installments of Re. 1 each. Find the rate of interest per annum.
Sol. Let the rate of interest per month be r
Total amount repaid = Rs.10 interest = Re. 1
(9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1) = 1
r =
Rate of interest per annum = (100/45) 12 = 26 2/3%.
Shares & Stocks
The capital is divided into equal parts called shares.
The term stock is also applied to the various amounts of money which have been borrowed by the Government or by the corporations of cities.
To pay the interest on different sums of money borrowed from time to time, the Government sets aside certain funds.
Kinds of Stock
(1) Stock at Par: Stock is said to be at par if the market value of the stock is equal to its face value. (Market Value = Face Value)
(2) Stock at a premium (or above par): Stock is said to be at a premium if the market value of the stock is greater than its face value. (Market value = Face value + Premium)
(3) Stock at a discount (or below par): Stock is said to be at a discount if the market value of the stock is less than its face value. (Market value = Face value – discount)
For example, If Rs. 100 stock is bought for Rs. 106, the stock is at a premium of 6% or 6 above par; but if it is bought for Rs. 96, the stock is at a discount of 4% or 4 below par; and if Rs. 100 stock is bought for Rs. 100, it is said to be at par.
The total investment of a company is called stock.
Dividend
When a company makes a profit, part of that profit is divided amongst the shareholders and it is called the dividend. The dividend is always calculated on the face value of a share and is generally expressed as a percentage.
Brokerage
The stock is generally bought or sold through a broker who charges a small commission called the brokerage. A buyer has to pay the market value together with the brokerage and a seller gets market value reduced by the brokerage i.e.
Amount paid by the buyer = Market value + Brokerage.
Amount received by the seller = Market value – Brokerage.
Debentures
Ex.1. How much stock can be purchased with Rs. 52,625 at 5% above par (Brokerage ¼ %).
Sol. To purchase Rs. 100 stock we need Rs. = Rs.
If investment is Rs. ; stock purchased = Rs. 100.
If investment is Rs. 52,625; stock purchased = 100 × × 52,625 = Rs. 50,000.
Ex.2. A man buys Rs. 6000 stock at 5% discount and sells at 2% above par. Find his gain or loss. Brokerage @ %.
Sol.
Alternate Method: From above –
Income Problem
A statement such as ‘5% stock @ 95’ means.
1. Face value of stock is Rs. 100.
2. Market value of Rs. 100 stock is Rs. 95.
3. Income (Annual) from this stock is Rs. 5.
Hence we get Rs. 5 as income or dividend (Annual) by investing Rs. 95 and obtain a stock worth Rs. 100.
Note: Percentage Return on the shares purchased less than the market value will always be more than the actual return.
Ex.3. X invested an amount of Rs. 23,920 in 8% stock at 92. Find his net income if he pays 4% income tax.
Sol. Cost of Rs. 100 stock = Rs. 92
Income on Rs. 92 = Rs. 8
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