Friday, 21 April 2017

Interest III - Problems on Varying Rates of Interest and Difference between Compound Interest and Simple Interest.

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.

Compound Interest formula and Simple Interest

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.
Compound Interest formula 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%

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.
                                          Compound Interest formula and Simple Interest
By cross multiplication we get,
     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+R1)T1 (1+R2)T2(1+R2)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 = 5
b = 10

    5 + 10 +[(5+10)/100]=

15.5%
Effective Percentage of first 2 years Interest+3rd years percentage.



(a + b + ab)/100)


a = 15.5
b = 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 formula and Simple Interest
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.
Compound Interest formula and Simple Interest
Compound Interest and Simple Interest Formula
*Note: D is the difference between SI and CI.
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 1002/R2  
    2000 = D x 1002/52
     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,
                                      Compound Interest formula and Simple Interest
Therefore, difference between Compound Interest and Simple Interest is Rs.5/-

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 1003/R2(300+R)
   P = 48 x 1003/202(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.  

Compound Interest formula and Simple Interest




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