#### SI & CI concept

**Concept | Practice set 1 | Practice set 2**

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#### SI AND CI CONCEPT

##### SIMPLE INTERST CONCEPT

**Definition of the term Interest:** Interest is actually one of the most fundamental business terms, and without it, the financial trading of the world would come to standstill. Interest is defined as the “time value of money”.

**What exactly does this mean?**

Well, look at this way: with time, the value of money changes. Suppose you have Rs. 100 in the year 2000. Would it still be Rs. 100 or would the amount have grown? If you had deposited the money in a saving bank account, say with an annual rate of interest 4%, that money would have definitely grown by now. Can you calculate the amount you would have with you in 2013? Well, in case you can’t right now, go through these concept notes and you would know the answer.

The concept of simple and compound interest is especially applicable to the world of banking and economics. Whenever we borrow a certain sum of money (known as the principal), we pay back the original amount accompanied with a certain amount of interest on that amount. In a way, those are the charges of borrowing that sum of money. Simple interest is one method of determining the amount due at the end of loan duration. Another method of interest application is compound interest, but we study about it in next article.

**Simple Interest Tooltip1: The Definitions****Principal (P):** The original sum of money loaned/deposited. Also known as capital.**Interest (I):** The amount of money that you pay to borrow money or the amount of money that you earn on a deposit.**Time (T):** The duration for which the money is borrowed/deposited.**Rate of Interest (R):** The percent of interest that you pay for money borrowed, or earn for money deposited

**Simple Interest Tooltip 2: The Formula**

Where:

P: Principal (original amount)

R: Rate of Interest (in %)

T: Time period (yearly, half-yearly etc.)

Amount Due at the end of the time period, A = P (original amount) + SI*If you have a close look, Simple Interest is nothing else but an application of the concept of percentages.*

**Simple Interest Tooltip 3:** Basic Problems to explain the concept

* Basic Problem 1:* What is the SI on Rs. 7500/- at the rate of 12% per annum for 8 years?

Using the Basic Formula:

Simple Interest (SI) = (P x R x T)/100

P – Principal amount, T- Number of years, R – Rate of Interest

Given P = 7500, T = 8 Years, R = 12%

Simple Interest (S.I.) = (7500X12X8)/100

Simple Interest (S.I.) = 7200

* Basic Problem 2:* A man borrowed Rs 15000/- at the rate of 24% SI and to clear the debt after 6years, much he has to return:

Using the Basic Formula:

Simple Interest (SI) = (P x R x T)/100

P – Principal amount, T- Number of years, R – Rate of Interest

Given P = 15000, T = 6 Years, R = 24%

Simple Interest (S.I.) = (15000X24X6)/100= Rs 21600

Therefore, total interest = 21600

Total repayment = S.I + Principal amount = 21600 + 15000 = Rs 36600

* Basic Problem 3:* A man borrowed Rs.12000 at the rate of 10% SI, and lent the same sum toanother person at the rate of 15% what will be the gain after 5 years?

Using the Basic Formula:

Simple Interest (S.I.) = (P x R x T)/100

P – Principal amount, T- Number of years, R – Rate of Interest

The man borrowed at 10% and he lent the same sum to another person at 15%

Therefore, his gain is actually equal to the different in the interest rate (per year)

= 15 – 10 =5% for 1 year

Thus, to calculate his gain, we use this difference as the rate of interest.

Given T = 5 years and P = Rs. 12000

Amount Gained = (12000x5x5)/100 = Rs 3000

Therefore, his gain = Rs 3000/-

*Tips, Tricks & Results for Simple Interest-1*

*The purpose of this article is to provide you with some useful tips and tricks that you can use for Simple Interest questions. Various applications and formulas based on this concept are explained here.*

*Simple Interest Tips, Tricks, and Results: Tooltip 1*

** Example : **The interest on a sum of money is 1/16 of the principal, and the number of years is equal to the rate of interest. What is the rate percent?

** Example: **The rate of interest for 3 years is 4%, 5years is 6%,1 years is 5%. If a man gets interest of Rs. 4700 for 9 years, calculate the principal amount?

** Example:** A sum of money becomes 4 times in 20 years. Calculate the rate of interest.

*Tips, Tricks& Results for Simple Interest-2*

*The purpose of this article is to provide you with some useful tips and tricks that you can use for Simple Interest questions. Various applications and formulas based on this concept are explained here.*

*Simple Interest Tips, Tricks, and Results: Tooltip 1*

#### Compound interest concept

**Compound Interest Tooltip 1: The Definitions**

**Principal (P):** The original sum of money loaned/deposited. Also known as capital.**Interest (I):** The amount of money that you pay to borrow money or the amount of money that you earn on a deposit.**Time (T):** The duration for which the money is borrowed. The duration does not necessarily have to be years. The duration can be semi-annual, quarterly or any which way deemed fit.**Rate of Interest (R):** The percent of interest that you pay for money borrowed, or earn for money deposited

**Compound Interest Tooltip 2: The Basic Formula**

Amount Due at the end of the time period, A = P (1+r/100)^{t}

Where:

P: Principal (original amount)

R: Rate of Interest (in %)

T: Time period (yearly, half-yearly etc.)

Compound Interest (CI) = A- P = P (1+r/100)^{t} -P

= P {(1+r/100)^{t} -1}

**Compound Interest Tooltip 3: Basic Problems to explain the concept**

**Example 1: **Maninder took a loan of Rs. 10000 from Prashant . If the rate of interest is 5% per annum compounded annually, find the amount received by Prashant by the end of three years

**Solution**:

The following is the data given:

Principal, P= 10000

Rate = 5%

Time =3 years

Using the formula for Compound Interest:**A = P(1+R/100) ^{t}**

**So A=**10000(1+5/100)

^{3}

A = 10000(1+1/20)

^{3}

A = 10000 x 21/20 x 21/20 x21/20 =11576.25

So the total amount paid by Maninder at the end of third year is Rs.11576.25

**Example 2**: Richa gave Rs. 8100 to Bharat at a rate of 9% for 2 years compounded annually. Find the amount of money which she gained as a compound interest from Bharat at the end of second year.

**Solution**:

Principal value = 8100

Rate = 9%

Time = 2 years

So the total amount paid by Bharat

= 8100(1+9/100)^{2}

=Rs. 9623.61

The question does not probe the amount, rather, it wants to know the CI paid, that the difference between the total amount and original principal.

The Compound Interest = 9623.61 – 8100 = 1523.61

**Compound Interest Tooltip 4: Multiple Compounding in a year**

Amount Due at the end of the time period

Where:

A = future value

P = principal amount (initial investment)

r = annual nominal interest rate

n = number of times the interest is compounded per year

t = number of years money borrowed

**Amount for Half Yearly Compounding, A = P {1+(R/2)/ 100} ^{2T} **

(compound interest applied two times an year).

**Like Half Yearly Compound Interest, we can calculate the amount for Quarterly Compounding:**

**A = P {1+(R/4)/ 100} ^{4T}**

**Example 3:** Sona deposited Rs. 4000 in a bank for 2 years at 5% rate. Find the amount received at the end of year by her from the bank when compounded half yearly.

Solution:

Principal value = Rs. 4000

Rate = 5%

Time = 2 years

Since the interest is compounded half yearly so 2 years = 4 times in two years

So we have =** A = P {1+(R/2)/ 100} ^{2T}**

A= 4000{1+ (5/2)/100}

^{4}

A = 4000 x 41/40 x 41/40 x 41/40 x 41/40

A = Rs. 4415.2

So, Sona received Rs. 4415.2 from the bank after two years

**Example 4:** Manpreet lent Rs 5000 to Richa at 10% rate for 1 year. But she told her that she will take her money on compound interest. So find the amount of interest received by Manpreet when compounded quarterly?

Solution:

Principal value = Rs. 5000

Rate = 10%

Time = 1 year

Since the interest is compounded quarterly, that is 4 times in 1 year

Using the formula=** A = P {1+(R/4)/ 100} ^{4T}**

A= 5000{1+ (10/4)/100}

^{4}

A = 5000 x 41/40 x 41/40 x 41/40 x 41/40

A = Rs. 5519.064

So, Manpreet received Rs. 4415.2 from bank after two years

And the total amount of interest received by her is 5519.064 – 5000 = Rs. 519.06

**Compound Interest Tooltip 5: Difference between Simple Interest and Compound Interest**

In case the same principle P is invested in two schemes, at the same rate of interest r and for the same time period t, then in that case:

Simple Interest = (P x R x T)/100

Compound Interest = P [(1+R/100)^{T} – 1]** **So, the difference between them is

= PRT/100 – P[(1+R/100)^{T} -1]

= P [(1+r/100)^{T} -1-RT/100]

**Two shortcuts which we can use:****Difference between CI and SI when time given is 2 years = P(R/100) ^{2}**

**Difference between CI and SI when time given is 3 years = P[(R/100)**

^{3}+ 3(R/100)^{2}]**Example 5:****The difference between compound interest and simple interest is 2500 for two years at 2% rate, then find the original sum.**

**Solution: **

Given Interest is = 2500

So, Simple Interest = (P X R X T)/100

Compound Interest = P [(1+r/100)^{t} – 1]

So the difference between both of them is

= PRT/100 – P [(1+R/100)^{T} -1]

= P [(1+r/100)^{T} -1-RT/100]

So the sum is 2500 = P [{(1+2/100)^{2}-1}-4/100]

On simplification this equation the sum will be = Rs. 6250000

**We can check it by our shortcut method**

When time given is 2 years = P(R/100)^{2 }

Since we are given by the difference so

2500 = P (2/100)^{2}

=> 2500 = P (1/50)^{2}

=> 2500= P (1/2500)

=> 6250000=P

So the sum is Rs.6250000.

**Compound Interest Tooltip 6: Formula for compound interest when compounding in a year but time is in fraction**

The formula to calculate compound interest when the time given is in fractions is as follows:

A =P[(1+R/100)^{real part}{1+(Fraction part x R/100)}]

Where

A: Amount at the end of the time period

P: Principal amount

R: Rate of interest

Real Part and Fraction part: For example, the time given is two and half years, then real part would be two and half years.

**Example 1: ****Manpreet gave Rs. 1000 to Richa for 1 year 6 months. Then find the amount paid by Richa to Manpreet after this duration if rate of interest is 5% per annum compounded annually?**

**Solution :**

We have a time of 1 years and 6 months = (1 year + 1/2 years)

So to find the amount we will use

A =P[(1+R/100)^{real part}{1+(Fraction part x R/100)}]

A = 1000[(1+5/100)^{1}{1+{(6/12 x 5)/100]

A= P[(21/20) x (41/40)]

A = Rs. 1076.25

**Compound Interest Tooltip 7: Concept of Equal Installments**

How does the concept of equal installments work for Compound interests?

Well, in this case, the problem basically tells us that a certain sum of money is borrowed on compound interest for a certain period and it is returned with the help of equal installments. Lets us derive a formula for these installments.

Let us derive a formula where the amount is returned in two equal installments for a time period of two years.

Assume P to be the principal and r the rate of interest.

Step 1: P(1+r/100)=P1 (Amount for one year)

Step 2: New Principal

Now let X be the first installment. After giving the first installment, the principal value will change and the new principal will be = P1 – X

P2 = P1 –X (1+R/100)

Step 3: Amount and Interest for the second year

Now the interest charged will be charged on this amount.

Amount at the end of second year: [P(1+r/100)-X][1+r/100]

Step 4: Since the installments are equal, this new amount has to equal X.

Hence,

[P(1+r/100)-X][1+r/100]=X

On solving, we have

P [(1+R/100)^{2}-X (1+R/100)] = X

P [(1+R/100)^{2}] = X+X (1+R/100)

Divide both sides by (1+r/100)^{2}

So we left with**P= X/(1+r/100) ^{2} + X/(1+r/100)**

**Generalizing the formula for EQUAL INSTALLMENTS ****P= X/(1+r/100) ^{n} + …………………….X/(1+r/100)**

Where x is the installment and n is number of installment

**Example 2: ****Richa borrowed a sum of Rs. 4800 from Ankita as a loan . She promised Ankita that she will pay it back in two equal installments .If the rate of Interest be 5% per annum compounded annually, find the amount of each installment****Solution:**

Given that principal value = 4800

Rate =5%

Two equal installments annually = 2 years

Appling the formula = X/(1+r/100)^{n}…………………….X/(1+r/100)

So we have here two equal installments so

P= X/(1+r/100)^{2} + X/(1+r/100)

4800=X/(1+5/100)^{2} + X/(1+5/100)

On simplifying

We have x= Rs. 2581.46

So the amount of each installment is 2581.46

**Compound Interest Tooltip 8: Application of Compound Interest for concepts of population growth. ****Case 1: When population growing in a constant rate**

If the rate growth of population increased with a constant, rate then the population after T years will be = P (1+R/100)^{t}

In fact, this is nothing else but an application of the fundamentals of compound interest.

It is actually similar to finding the compound amount after time T years

Net population after T years = = P (1+R/100)^{t}

Net population increase = P [(1+R/100)^{t}– 1]

**Example 3:****The population of Chandigarh is increasing at a rate of 4% per annum. The population of Chandigarh is 450000, find the population in 3 years hence.**

Solution:

P = 450000

Rate of increasing = 4%

Time =3 years

Therefore, the total population will be

=> T = P (1+R/100)^{3}

=> T = 450000(1+4/100)^{3}

=> T = 506188

**Case 2: When the population growing with different rates and for different intervals of time **

If the rate growth of population increased with different, rate and for different intervals of time then the population after T years will be =

P (1+R1/100)^{t1 }x (1+R2/100)^{t2}…………………………….. (1+RN/100)^{tn}

Let us take an example for this concept.**Example 4:****The population of Chandigarh increased at a rate of 1% for first year, the rate for second year is 2%, and for third year, it is 3%. Then what will be the population after 3 years if present population of Chandigarh is 45000?**

**Solution:**

Since the rate growth of population increased with different rates for the three difference years, the population after T years will be =

P (1+R1/100)^{t1 }x (1+R2/100)^{t }x (1+R/100)^{tn}

45000(1+1/100)^{1} x (1+2/100)^{1} x (1+3/100)^{1}=47749.77

**Case 3: When the population is decreasing with rate R**

Population after a time period of T years=P (1-R/100)^{t}

Where T is the total population

““““““`R is rate at which the population is decreasing

**Example 5:**

The population of Chandigarh is increases at a rate of 1% for first year, it decreases at the rate of 2% for the second year and for third year it again increases at the rate of 3%. Then what will be the population after 3 years if present population of Chandigarh is 45000.

Solution:

Since the rate growth of population is increasing first and then decreasing for the second year and again it increases for third year, then the population after T years will be =

Present Population: P (1+1/100)^{t1 }x (1-2/100)^{t }x (1+3/100)^{t3}

Present Population: 45000(1+1/100)^{1} x (1-2/100)^{1} x (1+3/100)^{1}=45877.23

**Compound Interest Tooltip 8: ****Negative Compound Interest**

As we can see from the last case above, it is not necessary that there is always an increase in any quantity or amount. There can also be a reduction in the amount of something. This reduction is actually called the rate of depreciation, especially in the financial world. In this case, we do nothing else but take the interest rate to be negative. The formula for this is as follows:

Let P be the original amount.

Let P1 be the new amount at the end of t years.

P1 = P (1-R/100)^{t}

Here R is the rate of interest (negative rate).

Always remember, the rate of depreciation is nothing else but negative rate of interest.

**Example 6:****Manpreet bought a new car. The value of the car is Rs. 45000. If rate of depreciation is 10% per annum then what will be the value of the car after 2 years**

**Solution:**

P = 45000

Rate of depreciation = 10%

T = 2 years

Therefore, the value will be after 2 years

= P (1 – R/100)^{t}

= 45000(1-10/100)^{2}

= 36450