###
**In
this post, problems are solved based on Varying Rates of Interest and Difference
between Compound Interest and Simple Interest in just a couple of steps.**

In our previous post How to find Simple Interest and Compound Interest in the SBI PO, IBPS exam we solved questions using smart methods to save time. This post is divided into two parts - Module 3 and Module 4. Module 3 is based on the problems which have Varying rates of Interest and Module 4 is based on the Difference between Compound Interest and Simple Interest.

**Module 3: Varying Rates of Interest**

Do problems on Varying rates of Interest scare you?
Do you often think of skipping the problem? Stop worrying. Here are few smart methods which will help you to solve questions in less than 25 seconds.

###
**Example 1:** **Varying
Rates of Interest**

**Problem:**If 20000 is given as loan for a period of 3 years with the interest rates 5%, 7% and 9% for the 1st, 2nd and 3rd year respectively. What is the total amount that has to be paid at the end of 3 years?

###
**Regular Method**

**Solution:**

**Step 1:**

**P - 20000**

**R - 5%, 7% and 9%**

T - 1, 2 and 3
years i.e 3 years.

S - ?

A - ?

**Step 2:**

SI = PTR/100

SI = PTR/100 +
PTR/100 + PTR/100 (In case of varying rates of Interest)

SI = P(TR/100 +
TR/100 + TR/100)

SI = 20000( 1x
5/100 + 1x7/100 + 1x9/100)

SI = 2000 x
21/100

SI = 4200

**Step 3:**

A = P + SI

A =
20000 + 4200

A = 24200.

Therefore, The amount that has to be paid end
of three years is Rs.24200/-

**Smart Method**

The problem is solved using Percentage and Ratio of Equivalent
Proportionality.

As this is a case of Simple Interest we add 5% + 7% + 9% [different Rate
of Interest for 3 years] i.e. 21%.
Amount here is Principal + Rate of Interest. Principal is 100% of the
amount. So,

Amount = 100% + 21%.

Therefore Amount = 121%

Amount = 100% + 21%.

Therefore Amount = 121%

###
**Step
1:
**

As per the Ratio
of Equivalent Proportionality we cross multiply the percentage with Principal
amount and Principal Percentage with unknown amount to find the actual amount.

A = (121 x
20000)/100)

A = 24200

Therefore, the amount that has to be paid end of 3 years is Rs.24200/-

###
**Example 2:**** ****Module
3-Varying Rates of Interest**

**Problem:**An amount of Rs.10000 is taken as loan by Vivek at compound interest charging 5% for the first year,10% for the second year and 20% for the third year. What is the total interest to be paid by Vivek at the end of the third year?

###
**Regular Method**

**Solution:**

**Step 1:**

**P - 1000**

R - 5%,
10%, 20%

T - 1,
2, 3 years i.e. 3 years.

CI - ?

**Step 2:**

A = p[(1+R

_{1})^{T1 }(1+R_{2})^{T2}(1+R_{2})^{T3}
A = 10000 [(1+5)/100
x (1+10)/100 x (1+20)/100]

A = 10000 [
(105/100) x (110/100) x (120/100)]

A = 13860

**Step 3:**

CI = A - P

CI = 13860 - 10000

CI = 3860

Therefore, Compound Interest for the three years is
Rs.3860/-

###
**Smart
Method**

**Solution:**

We can solve this
problem in just one step by using Effective percentage method and the concept of Ratio and
Equivalent Proportionality.

- First, we need to find the percentage of Compound Interest for the consecutive 3 years by using effective percentage method.

Effective Percentage of first 2 years. |
(a + b + ab)/100 |
a = 5b = 10 |
5 +
10 +[(5+10)/100]= |
15.5% |

Effective Percentage of first 2 years
Interest+3rd years percentage. |
(a + b + ab)/100) |
a = 15.5b = 20 |
15.5 + 20 +[(15.5+20)/100]= |
38.6% |

Compound Interest = 38.6%

**Step 1:**

By using the concept of Ratio and
Equivalent Proportionality we cross multiply Principal Amount with Interest
Percentage and Principal Percentage with the unknown amount of Compound Interest to
find the Compound Interest paid at the end of 3 years.

Compound Interest = 3860

Therefore, Compound Interest for the 3 years is
Rs.3860/-

**Module 4: Difference between Simple Interest and Compound Interest**

Problems on Difference between Compound Interest and
Simple Interest are complex. Solving them by the regular method takes long. The
complexity arises due to not having a thorough understanding about the concept.
This module deals with the concept of Difference between Simple Interest and
Compound Interest using smart method which helps you to pick up your speed
during exam and saves time.

###
**Concept:**

Let the difference
between simple interest and compound interest for a period of T years at the R%
p.a be 'D', then the principal P is given as.

We can solve problems on this concept by simple substituting the values in the formulas.

###
**Example 1****:**** ****Module
4- Difference between Simple Interest and Compound Interest**

**Problem:**Find the difference between Simple Interest and Compound Interest at 5%p.a for two years on a principal of Rs.2000.

**Regular Method**

**Solution:**

**Step 1:**

P = 2000

R = 5%p.a

T = 2 years

D = ?

**Step 2:**

P =

**D x 100**^{2}/R^{2 }
2000 = D x 100

^{2}/5^{2}^{ }D = 5

Therefore, difference
between Compound Interest and Simple Interest is Rs.5

**Smart Method**

**Solution:**

**Step 1**

**:**

**SI = 5 + 5 ( rate of interest of two years)**

= 10%

**Step 2:**

**Compound Interest = 5 + 5 + (5 x 5)/100**

= 10.25%

**Step 3:**

Compound Interest -Simple Interest

10.25% - 0.5% = 0.25%

**Step 4:**

Cross multiplication,

We can easily avoid step
one and two by doing the calculation in our mind.

Alternatively, we know
that the actual difference between CI and SI is 5 x 5/100 so we can calculate
that by directly cross multiplying and avoid step 1 and step 2.

###
**Example 2:****
****Module 4- Difference between
Simple Interest and Compound Interest**

**Problem:**If the difference between simple interest and compound interest on some amount at 20 p.a for 3 years is Rs.48, then what must be the principal amount.

###
**Regular Method**

**Solution:**

**Step 1:**

P = ?

D= 48

R = 20

T = 3

**Step 2:**

**P = D x 100**

^{3}/R

^{2}(300+R)

P = 48 x 100

^{3}/20^{2}(300+20)**Step 3:**

**P = 300/8**

P = 375

Therefore, the Principal amount is Rs.375/-

Solving this problem about the consecutive 3 years difference between Compound Interest and Simple Interest by percentage method is a tedious job. However, the Regular method i.e Substituting values in the formulas is the simplest way to solve this problem.

Stay tuned for our next post on How to Calculate Interest Compounded Half-yearly.

Do write in the Comments section on how this blog helped you to break the complexity of the problems.

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