CONCEPT 1:
1. Money borrowed today is repaid with a higher amount tomorrow. This gives rise to the concept of interest.
2. Creditor is a person who lends money to someone who wishes to borrow it.
3. The person who receives the money from him is termed as debtor.
4. The amount of money which the creditor lends initially is known as the Principal (P) or Capital and the time frame for which he lends the money is known as the Time or Period (T or n).
5. The difference between the amount of money which the debtor borrows today (principal) and the amount of money which he needs to repay at the end of time period is called the Interest (I) over the principal amount. Also, the total money which he repays is termed as Amount (A).
Amount (A) = Principal (P) + Interest (I)
6. The interest is calculated based on the Rate of Interest (R), which is specified in terms of percent per annum (p.c.p.a) unless otherwise stated.
7. There are two ways in which the Interests are calculated => Simple Interest and Compound Interest.
CONCEPT 2:
SIMPLE INTEREST:
1. The interest which is calculated for the given time duration only on the original principal, is called simple interest.
2. Simple interest = (Principal * Rate * Time)/100
3. Amount = Principal + Interest = P+I = P + (P*r*T)/100
4. An example: Suppose Mukesh borrows Rs.100 from Rajesh for a period of 2 years and agrees to pay simple interest at the rate of 10% per annum. Now, the amount of interest due to Rajesh at the end of 1 year will be (100*1*10)/100 => Rs.10
So, Mukesh owes Rajesh Rs.110 at the end of the year.
5. What is the concept here? Since he has agreed to pay simple interest, he pays the interest only on the original Rs.100 for the second year too. That is, for the second year also the interest is calculated based on the original principal Rs.100 and not on Rs.110
This implies that he pays the interest of Rs.10 for the second year too. So, the cumulative interest will be Rs.20 for the time period of two years
6. This can also be calculated together for two years using the formula,
SI = (100*10*2)/100 = Rs.20 => Total amount = 100+20 = Rs.120.
7. Thus, when we calculate the simple interest, the accumulated interest is not taken into account for the purpose of calculating interest for later years. The interest remains constant every year.
Example 1:
Find the simple interest on a principal of Rs.3500 at the rate of 4% per annum for a period of 8 years.
Solution:
SI = (3500*4*8)/100 = Rs.1120
Example 2:
Calculate the simple interest on Rs. 75000 for 2.5 years at 47/3% per annum. Find the amount to be paid after 2.5 years.
Solution:
SI = (75000*2.5* (47/3))/100 = Rs. 29375.
Amount = SI + P = 29375+75000 = Rs. 104375
Example 3:
In 4 years, Rs. 6000 amounts to Rs. 8000. In what time at the same rate will Rs. 525 amount to Rs. 700?
(SNAP 2009)
Solution:
We can see that (8000/6000) = (700/525) = (4/3)
Since the ratio is same between the two, the time required for 525 to grow up to 700 will be the same as the time required for 6000 tp grow upto 8000. So, the required time period is 4 years.
CONCEPT 3:
COMPOUND INTEREST:
1. Now consider the example of Mukesh and Rajesh again. There, Mukesh owes to Rajesh Rs.110 at the end of year 1. If Mukesh pays to Rajesh an interest of 10% on Rs.110 at the end of second year, we can say that he pays compound interest. So, in this case the interest for second year is (10/100)*110 = Rs.11
So, in this case, Mukesh pays a cumulative interest of Rs.10+ Rs.11 = Rs. 21 to Rajesh at the end of 2 years.
2. Note that the interest to be paid is more when it is compounded. In the above illustration, we consider the frequency of compounding to be yearly. This frequency of compounding is specified in questions and can be quarterly, half-yearly, 2 yearly etc.
3. What is the concept of Compound Interest? When the money is lent at compound interest, at the end of a fixed period, the interest for that fixed period is added to the principal and this amount is considered to be the principal for the next year or period. This is repeated until the amount for the last period has been calculated.
Compound interest = Final amount - Original principal
Final amount = P * ( 1 + (r/100) )^n => r = rate of interest and n = number of years
In the case of Rajesh and Mukesh,
Amount = 100 * (1.1)^2 = Rs.121 => CI = 121-100 = Rs. 21
Example 1:
If Rs.12000 has been lent out at 10% rate of interest, the interest being compounded annually, then what is the interest for the third year?
Solution:
Note that the compound interest at the end of 3 years is not asked, But the interest for the third year alone is asked. This can be solved by calculating the amount which is compounded for 2 years and then finding out the interest for third year (By SI method) on the compounded amount for 2 years.
Amount at the end of second year = 12000*(1+ (10/100))^2 = Rs. 14520
The interest for the third year is calculated on Rs. 14520 (In case the interest is not compounded, then it will be calculated on Rs.1200 which is the original principal)
Therefore, Interest for the third year = 14520 * (10/100) = Rs.1452
Example 2:
If the compound interest on a certain sum for 3 years at 4% is Rs.1500, then what would be the simple interest on the same sum at the same rate and the same time period?
Solution:
Let the sum be P.
CI = P*(1+ (4/100))^3 - P = 1500
=> (0.124864)P = 1500 => P= (1500/0.124864)
SI = 0.12 * (15000/0.124864) = (1500/1.04) = Rs.1442 (app)
Example 3:
Mungeri Lal has two investment plans- A and B, to choose from. Plan A offers interest of 10% compounded annually while plan B offers simple interest at 12% per annum. Till how many years is plan B a better investment? (XAT 2009)
Solution:
Let he invested Rs.100. In plan B, SI is calculated
Plan A, Interest is compunded. For first year: (100*10)/100 = 10 => Amount =110
Second year: (110*10)/100= 11 => 121 Third year : (121*10)/100 = 12.1 => 133.1
Fourth year: (133.1*10)/100 = 13.31 => 146.41 Fifth year: (146.41*10)/100 => 14.641 => 161.051
Now, we calculate plan B for four years
(100*4*12)/100 => 48 => Amount = Rs.148 which is greater than the fourth years'value of plan A
For 5 years (100*5*12)/100 =60 => Amount = Rs.160 which is less than the fifth year's value of plan A
So, plan B is a better investment for four years.
Example 4:
Mr.Jeevan wanted to give some amount of money to his two children, so that although today they may not be using it, in the future the money would be of use to them. He divides a sum of Rs.18750 between his two sons of age 10 years and 13 years respectively in such a way that each would receive same amount at 3% p.a compound interest when he attains the age of 30 years. What would be the original share of the younger son? (IIFT 2007)
Solution:
Let P be the share of younger son and 18750-P be the share of the elder son
It is given that the amount received by the two at the end is equal.
Younger son's no. of years for maturity = 20 years and Elder son's no. of years = 17
P* (1+(3/100))^20 = (18750-P) * (1+(3/100))^17
By solving the equation, P = Rs.8959.60
CONCEPT 4:
COMPOUNDING MORE THAN ONCE A YEAR:
1. As mentioned earlier, the frequency of compounding may vary. It can be done half yearly, quarterly, monthly etc.
2. When compounding is done more than once a year, the rate of interest for the time period will be less than the effective rate of interest for the entire year.
3. For example, if the rate of interest for the whole year is 10%, then the rate of interest when the amount is compounded half-yearly will be 5% and the time period of compounding becomes doubled.
4. For half yearly rate, A = P * (1+(r/2/100))^2n
For quarterly rate , A = P * (1+ (r/4/100))^4n
For monthly rate, A = P * (1+ (r/12/100))^12n
NOTE:
The difference between the simple interest and compound interest (calculated on the same principal and with the same rate of interest) for the second year is equal to the interest calculated for one year on one year's simple interest.
In mathematical terms, CI-SI = P*(r/100) => when the P and R are same for both SI and CI
Example 1:
The compound interest on a certain sum for 2 years is Rs.360 and the simple interest on the same sum for 2 years if Rs.300. Find the principal and the rate percent.
Solution:
Since SI remains the same for all years, SI for the first year= 300/2 = Rs.150
The SI and CI remain same for the first year. So, CI for the first year = Rs.150
Now CI is more because it has an additional component of interest on the SI of the first year. So, CI for the first year = Rs.60
So, Rate of interest = (60*100)/150 = 40% => Principal =150/0.4 = Rs.375
CONCEPT 5:
POPULATION FORMULA:
1. If the original population of a town is P and the annual increase is r%, then the population is n years (P')
P' = P* (1+ (r/100))^n
For example, if the rate of growth of population of rabbits in a warren is 100% per year or if the population doubles every year, 2 rabbits will become 16 rabbits in a matter of 3 years.
2. If the annual decrease is r%, then the population in n years is given by a change of sign in the formula:
P' = P* (1- (r/100))^n
3. Depreciation of a value: The value of an asset decreases with time and this decrease is depreciation. If the rate of depreciation is r%, then the value will be P*(1-(r/100))^n
Example 1:
The population of a city is currently 30 million. The number has been increasing at a steady rate for the past 10 years. If it is observed that the rate of increase is 15% every year, then what will be the population of the city 3 years from now?
Solution:
P= 30 million r=15 and n=3 years Formula => P' = 30* (1+(15/100))^3 = 45.6 million.
CONCEPT 6:
GROWTH RATE:
1. The absolute growth rate = Final value - Initial value
2. Growth rate of a year = [(Final value - Initial value)/initial value ]*100
3. Simple annual growth rate or Average annual growth rate
=> AAGR =(Sum of the growth rates (a%) of all the years)/ Number of years
4. Compounded Annual Growth Rate (CAGR) = [(Final value/Initial value)^(1/Number of years)] -1
Example 1:
Bennett distribution company, a subsidiary of a major cosmetics manufacturer Bavlon, is forecasting the zonal sales for the next year. Zone I with current yearly sales of Rs.193.8 lakh is expected to achieve a sales growth of 7.25% Zone II with current sales of Rs.79.3 lakh is expected to grow by 8.2% and zone III with sales of 57.5 lakh is expected to increase sales of Rs. 57.5 lakh is expected to increase sales 7.15%. What is the Bennett's expected sales growth for the next year? (IIFT 2009)
Solution:
Total sales this year = 193.8+79.3+57.5 = Rs. 330.6 lakhs
Expected sales next year = (193.8*1.0725) + (79.3*1.082) + (57.5*1.0715) = Rs.355.26 lakhs.
Sales growth = ((355.26-330.6)/330.6)*100 = 7.46%
CONCEPT 7:
OFFERING LOANS ON A DISCOUNT BASIS:
1. If a bank or a company offers loan on a discount basis at a% then if the original value of the loan is Rs. 100 then the bank or the company loans out Rs. [100 - (a% of 100)] or Rs. (100-a) (In this case) and get Rs.100 back at the time of maturity.
In general, if d is the discount rate, then the effective interest rate is (d/1-d) *100
Example 1:
ICICI bank offers a 1-year loan to a company at a interest rate of 20 percent payable at maturity, while Citibank offers on a discount basis at a 19% interest rate for the same period. How much should the ICICI bank decrease/ increase the interest rate to match up the effective interest rate of Citibank? (FMS 2009)
Solution:
Assume that the ICICI bank gives a loan of Rs.100 at 20% interest=> The amount that it ll receive will be Rs.120. To calculate ICICI's effective rate such that it is competitive with Citibank's, we need to find out how much the Citibank earns if it lends out Rs.100. Citibank lends at at 19% on a discount basis.
When Citibank gives 19% interest rate on discount basis and if the original value of the loan is Rs.100, it means that the bank gets only Rs.81 right now and pays Rs.100 at the end of the year.
So, the effective rate of interest = (19/81)*100 = 23.45%
Thus for the interest rate of ICICI bank to match up the effective interest rate of Citibank. it has to be increased by (23.45-20) = 3.45%
1. Money borrowed today is repaid with a higher amount tomorrow. This gives rise to the concept of interest.
2. Creditor is a person who lends money to someone who wishes to borrow it.
3. The person who receives the money from him is termed as debtor.
4. The amount of money which the creditor lends initially is known as the Principal (P) or Capital and the time frame for which he lends the money is known as the Time or Period (T or n).
5. The difference between the amount of money which the debtor borrows today (principal) and the amount of money which he needs to repay at the end of time period is called the Interest (I) over the principal amount. Also, the total money which he repays is termed as Amount (A).
Amount (A) = Principal (P) + Interest (I)
6. The interest is calculated based on the Rate of Interest (R), which is specified in terms of percent per annum (p.c.p.a) unless otherwise stated.
7. There are two ways in which the Interests are calculated => Simple Interest and Compound Interest.
CONCEPT 2:
SIMPLE INTEREST:
1. The interest which is calculated for the given time duration only on the original principal, is called simple interest.
2. Simple interest = (Principal * Rate * Time)/100
3. Amount = Principal + Interest = P+I = P + (P*r*T)/100
4. An example: Suppose Mukesh borrows Rs.100 from Rajesh for a period of 2 years and agrees to pay simple interest at the rate of 10% per annum. Now, the amount of interest due to Rajesh at the end of 1 year will be (100*1*10)/100 => Rs.10
So, Mukesh owes Rajesh Rs.110 at the end of the year.
5. What is the concept here? Since he has agreed to pay simple interest, he pays the interest only on the original Rs.100 for the second year too. That is, for the second year also the interest is calculated based on the original principal Rs.100 and not on Rs.110
This implies that he pays the interest of Rs.10 for the second year too. So, the cumulative interest will be Rs.20 for the time period of two years
6. This can also be calculated together for two years using the formula,
SI = (100*10*2)/100 = Rs.20 => Total amount = 100+20 = Rs.120.
7. Thus, when we calculate the simple interest, the accumulated interest is not taken into account for the purpose of calculating interest for later years. The interest remains constant every year.
Example 1:
Find the simple interest on a principal of Rs.3500 at the rate of 4% per annum for a period of 8 years.
Solution:
SI = (3500*4*8)/100 = Rs.1120
Example 2:
Calculate the simple interest on Rs. 75000 for 2.5 years at 47/3% per annum. Find the amount to be paid after 2.5 years.
Solution:
SI = (75000*2.5* (47/3))/100 = Rs. 29375.
Amount = SI + P = 29375+75000 = Rs. 104375
Example 3:
In 4 years, Rs. 6000 amounts to Rs. 8000. In what time at the same rate will Rs. 525 amount to Rs. 700?
(SNAP 2009)
Solution:
We can see that (8000/6000) = (700/525) = (4/3)
Since the ratio is same between the two, the time required for 525 to grow up to 700 will be the same as the time required for 6000 tp grow upto 8000. So, the required time period is 4 years.
CONCEPT 3:
COMPOUND INTEREST:
1. Now consider the example of Mukesh and Rajesh again. There, Mukesh owes to Rajesh Rs.110 at the end of year 1. If Mukesh pays to Rajesh an interest of 10% on Rs.110 at the end of second year, we can say that he pays compound interest. So, in this case the interest for second year is (10/100)*110 = Rs.11
So, in this case, Mukesh pays a cumulative interest of Rs.10+ Rs.11 = Rs. 21 to Rajesh at the end of 2 years.
2. Note that the interest to be paid is more when it is compounded. In the above illustration, we consider the frequency of compounding to be yearly. This frequency of compounding is specified in questions and can be quarterly, half-yearly, 2 yearly etc.
3. What is the concept of Compound Interest? When the money is lent at compound interest, at the end of a fixed period, the interest for that fixed period is added to the principal and this amount is considered to be the principal for the next year or period. This is repeated until the amount for the last period has been calculated.
Compound interest = Final amount - Original principal
Final amount = P * ( 1 + (r/100) )^n => r = rate of interest and n = number of years
In the case of Rajesh and Mukesh,
Amount = 100 * (1.1)^2 = Rs.121 => CI = 121-100 = Rs. 21
Example 1:
If Rs.12000 has been lent out at 10% rate of interest, the interest being compounded annually, then what is the interest for the third year?
Solution:
Note that the compound interest at the end of 3 years is not asked, But the interest for the third year alone is asked. This can be solved by calculating the amount which is compounded for 2 years and then finding out the interest for third year (By SI method) on the compounded amount for 2 years.
Amount at the end of second year = 12000*(1+ (10/100))^2 = Rs. 14520
The interest for the third year is calculated on Rs. 14520 (In case the interest is not compounded, then it will be calculated on Rs.1200 which is the original principal)
Therefore, Interest for the third year = 14520 * (10/100) = Rs.1452
Example 2:
If the compound interest on a certain sum for 3 years at 4% is Rs.1500, then what would be the simple interest on the same sum at the same rate and the same time period?
Solution:
Let the sum be P.
CI = P*(1+ (4/100))^3 - P = 1500
=> (0.124864)P = 1500 => P= (1500/0.124864)
SI = 0.12 * (15000/0.124864) = (1500/1.04) = Rs.1442 (app)
Example 3:
Mungeri Lal has two investment plans- A and B, to choose from. Plan A offers interest of 10% compounded annually while plan B offers simple interest at 12% per annum. Till how many years is plan B a better investment? (XAT 2009)
Solution:
Let he invested Rs.100. In plan B, SI is calculated
Plan A, Interest is compunded. For first year: (100*10)/100 = 10 => Amount =110
Second year: (110*10)/100= 11 => 121 Third year : (121*10)/100 = 12.1 => 133.1
Fourth year: (133.1*10)/100 = 13.31 => 146.41 Fifth year: (146.41*10)/100 => 14.641 => 161.051
Now, we calculate plan B for four years
(100*4*12)/100 => 48 => Amount = Rs.148 which is greater than the fourth years'value of plan A
For 5 years (100*5*12)/100 =60 => Amount = Rs.160 which is less than the fifth year's value of plan A
So, plan B is a better investment for four years.
Example 4:
Mr.Jeevan wanted to give some amount of money to his two children, so that although today they may not be using it, in the future the money would be of use to them. He divides a sum of Rs.18750 between his two sons of age 10 years and 13 years respectively in such a way that each would receive same amount at 3% p.a compound interest when he attains the age of 30 years. What would be the original share of the younger son? (IIFT 2007)
Solution:
Let P be the share of younger son and 18750-P be the share of the elder son
It is given that the amount received by the two at the end is equal.
Younger son's no. of years for maturity = 20 years and Elder son's no. of years = 17
P* (1+(3/100))^20 = (18750-P) * (1+(3/100))^17
By solving the equation, P = Rs.8959.60
CONCEPT 4:
COMPOUNDING MORE THAN ONCE A YEAR:
1. As mentioned earlier, the frequency of compounding may vary. It can be done half yearly, quarterly, monthly etc.
2. When compounding is done more than once a year, the rate of interest for the time period will be less than the effective rate of interest for the entire year.
3. For example, if the rate of interest for the whole year is 10%, then the rate of interest when the amount is compounded half-yearly will be 5% and the time period of compounding becomes doubled.
4. For half yearly rate, A = P * (1+(r/2/100))^2n
For quarterly rate , A = P * (1+ (r/4/100))^4n
For monthly rate, A = P * (1+ (r/12/100))^12n
NOTE:
The difference between the simple interest and compound interest (calculated on the same principal and with the same rate of interest) for the second year is equal to the interest calculated for one year on one year's simple interest.
In mathematical terms, CI-SI = P*(r/100) => when the P and R are same for both SI and CI
Example 1:
The compound interest on a certain sum for 2 years is Rs.360 and the simple interest on the same sum for 2 years if Rs.300. Find the principal and the rate percent.
Solution:
Since SI remains the same for all years, SI for the first year= 300/2 = Rs.150
The SI and CI remain same for the first year. So, CI for the first year = Rs.150
Now CI is more because it has an additional component of interest on the SI of the first year. So, CI for the first year = Rs.60
So, Rate of interest = (60*100)/150 = 40% => Principal =150/0.4 = Rs.375
CONCEPT 5:
POPULATION FORMULA:
1. If the original population of a town is P and the annual increase is r%, then the population is n years (P')
P' = P* (1+ (r/100))^n
For example, if the rate of growth of population of rabbits in a warren is 100% per year or if the population doubles every year, 2 rabbits will become 16 rabbits in a matter of 3 years.
2. If the annual decrease is r%, then the population in n years is given by a change of sign in the formula:
P' = P* (1- (r/100))^n
3. Depreciation of a value: The value of an asset decreases with time and this decrease is depreciation. If the rate of depreciation is r%, then the value will be P*(1-(r/100))^n
Example 1:
The population of a city is currently 30 million. The number has been increasing at a steady rate for the past 10 years. If it is observed that the rate of increase is 15% every year, then what will be the population of the city 3 years from now?
Solution:
P= 30 million r=15 and n=3 years Formula => P' = 30* (1+(15/100))^3 = 45.6 million.
CONCEPT 6:
GROWTH RATE:
1. The absolute growth rate = Final value - Initial value
2. Growth rate of a year = [(Final value - Initial value)/initial value ]*100
3. Simple annual growth rate or Average annual growth rate
=> AAGR =(Sum of the growth rates (a%) of all the years)/ Number of years
4. Compounded Annual Growth Rate (CAGR) = [(Final value/Initial value)^(1/Number of years)] -1
Example 1:
Bennett distribution company, a subsidiary of a major cosmetics manufacturer Bavlon, is forecasting the zonal sales for the next year. Zone I with current yearly sales of Rs.193.8 lakh is expected to achieve a sales growth of 7.25% Zone II with current sales of Rs.79.3 lakh is expected to grow by 8.2% and zone III with sales of 57.5 lakh is expected to increase sales of Rs. 57.5 lakh is expected to increase sales 7.15%. What is the Bennett's expected sales growth for the next year? (IIFT 2009)
Solution:
Total sales this year = 193.8+79.3+57.5 = Rs. 330.6 lakhs
Expected sales next year = (193.8*1.0725) + (79.3*1.082) + (57.5*1.0715) = Rs.355.26 lakhs.
Sales growth = ((355.26-330.6)/330.6)*100 = 7.46%
CONCEPT 7:
OFFERING LOANS ON A DISCOUNT BASIS:
1. If a bank or a company offers loan on a discount basis at a% then if the original value of the loan is Rs. 100 then the bank or the company loans out Rs. [100 - (a% of 100)] or Rs. (100-a) (In this case) and get Rs.100 back at the time of maturity.
In general, if d is the discount rate, then the effective interest rate is (d/1-d) *100
Example 1:
ICICI bank offers a 1-year loan to a company at a interest rate of 20 percent payable at maturity, while Citibank offers on a discount basis at a 19% interest rate for the same period. How much should the ICICI bank decrease/ increase the interest rate to match up the effective interest rate of Citibank? (FMS 2009)
Solution:
Assume that the ICICI bank gives a loan of Rs.100 at 20% interest=> The amount that it ll receive will be Rs.120. To calculate ICICI's effective rate such that it is competitive with Citibank's, we need to find out how much the Citibank earns if it lends out Rs.100. Citibank lends at at 19% on a discount basis.
When Citibank gives 19% interest rate on discount basis and if the original value of the loan is Rs.100, it means that the bank gets only Rs.81 right now and pays Rs.100 at the end of the year.
So, the effective rate of interest = (19/81)*100 = 23.45%
Thus for the interest rate of ICICI bank to match up the effective interest rate of Citibank. it has to be increased by (23.45-20) = 3.45%
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