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CHAPTER – 11: SIMPLE AND COMPUND INTEREST

                                     
1. SIMPLE INTEREST

Simple interest is the money paid on an original amount that has been either borrowed or invested. Or interest is the money paid by borrower to lender for using his money for specified time.
TERMINOLOGIES:-

  1.     Interest = money paid by borrower to lender.

  2.     Principle = the original/actual sum borrowed.

  3.     Time = some specified time for which money is borrowed on interest 

  4.     Rate of interest (r) = rate at which principal is given and it is calculated on principal

  5.     Amount (A) = it is the sum of principal and interest 

Interest = (Principal × time ×rate of interest)/100
Interest = (p×n×r)/100       

Let’s understand above formula with example 
Q.1. Person A gave 50,000 Rs. To person B for one year at the rate of interest 10% per annum. What will be the amount received by person A at the end of the year?
Solution:    
Here, P = 50,000
n = 1 year
r = 10%
Interest = (p×n×r)/100
            = (50,000×1×10)/100
            = 5000 Rs.
Amount received = principal + interest
                         = 50000 + 5000
                         = 55000 Rs.


SOLVED PROBLEMS ON SIMPLE INTEREST

Q.1. A certain sum of money amounts to 1560 Rs. In 3 years at 2% per annum find the amount of money borrowed?
Solution:
Here, P=?
A = 1560 rs
n = 2 years
r = 2%

Here A = P + (P×n×r)/100
1560 = P (1+(2×2)/100)
1560 = P ((100+4)/100)
(1560×100)/104 = P
P = 1500 Rs.
1500 Rs. Was borrowed.

Q.2. A certain sum of money amounts to 3924 Rs. In 3 years and 4140 in 5 years find the rate of interest?
Solution:
    Here A1 = 3924     n1 = 3      r =?    And P is same
            A1 = 4140     n2 = 5
Method 1
A1 = P + (Pn_1 r)/100                                A= P + (Pn_2 r)/100
3924 = P + 3Pr/100                                   4140 = P + 5Pr/100
P = 3924 -  3Pr/100   …(1)                        P = 4140 -  5Pr/100   …(2)
From 1 and 2
3924 -  3Pr/100  = 4140 -  5Pr/100
2Pr/100 = 4140 - 3924
2Pr/100 = 216
Pr = 10800
P = 3924 - (10800×3)/100
P = 3924 – 324
P = 3600
P = 3924 -  3Pr/100
3600 = 3924 -  (3×3600×r)/100
(3×3600×r)/100 =3924 – 3600
r = (324×100)/(3600×3)
  = 3%
Above method is time taking the simpler method with less time taking is,
Method 2
A1 = P + (SI)1
A2 = P + (SI)2
A2 – A1 = (SI)2 - (SI)1
4140 – 3924 = (SI)2 - (SI)1 
216 = (SI)2 - (SI)1
Here, 216 -> SI for (n2 - n1) ‘2’ years

By Unitary Method
     2 → 216

     3 →  ?
? = interest for 3 years = (216×3)/2
   = 324

P = A – simple interest
   = 3924 – 324
   = 3600
S.I. =  Pnr/100
324 = (3600×3×r)/100
 r = 3%

Q.3. A sum of Rs. 800 amounts to Rs. 920 in 3 years at simple interest if the interest rate is increased by 3% it would amount to how much? 
Solution:
Here, P = 800
n = 3 years
A = 920
SI = 920 – 800
    = 120
(SI)1 = Pnr/100
        = (800×3×r)/100
        = 120
      r = 5%
Now, new rate = r + 3%
                      = 5% + 3%
                      = 8%
(SI)2 = (800×8×3)/100
         = 192
New amount = P + (SI)2
                   = 800 + 192
                   = 992 Rs.

Q.4. A sum of money becomes 3 times in 5 years at simple interest. In how many years, will the same sum becomes 6 times at the same rate of interest?
Solution:
Here first,
A = 3P
n = 5 years
r =?
A = P + Pnr/100
3P = P + (P×5×r)/100
3 – 1 = 5r/100
r = (100×2)/5
  = 40%
Now, secondly A = 6P, r = 40% and n =?
6P = P + (P×40×n)/100
5 = (40×n)/100
n = (100×5)/40
n = 12.5 years

Q.5. When the article is sold at 20% discount, the selling price is 24 Rs. What will be selling price when the discount is 30%?
Solution:
As we know, Discount is always given on marked price,
S.P. = (100 – D1) % of M.P.
Here, D1 = 20%
And (SP)1 = 24 Rs.
24 = 20/100× M.P.
M.P. = 30 Rs.
(SP)1 =? And    D2 = 30%
(SP)2 = (100 - D2)% M.P.
          = 70/100 × 30
(SP)= 21 Rs.

Q.6. A man invested 1000 Rs. On a simple interest at a certain rate and 1500 Rs. At 2% higher rate.  The total rate of interest in three years is 390 Rs. What is the rate of interest for 1000 Rs.?
Solution:
Here, P = 1000 Rs.
R1 = r %
R2 = (r + 2)%
By the condition given
(Pnr1)/100 + (Pnr2)/100 = 390
(100×r×3)/100 + (1500(r+2)×3)/100 = 390
30r + 45r + 90 = 390
75 × r = 300
r = 4%
Therefore, rate of interest for 1000 Rs. = 4%

2. COMPOUND INTEREST


As we studied in simple, the interest was taken out after the decided time but in case of compound interest, interest generated by principal is reinvested and interest in the next period is then counted on the sum of principal and interest earned before, 
Here is explained the difference between simple and compound interest.

              In simple interest (R = 10%)                        In compound interest (R = 10%)
                       Table – 1                                                       Table – 2 
                  Principal   Interest     Amount                    Principal   Interest     Amount
   Year1        1000         100         1100           year1       1000         100        1100
   Year2        1000         100         1200           year2       1000         100        1210    
   Year3        1000         100         1300           year3       1210         121        1330
   Total interest = 300                                     Total interest = 331 

So you can see in the above tables that interest earned by compound interest is greater than interest earned by simple interest and reason behind that is in case of simple interest the principle was same while in case of compound interest the principal amount was increasing as interest earned was being reinvested.
Terminologies used in the compound interest is same as that of simple interest only one term is added in compound interest which is,
Compounding frequency:
Compounding frequency is the number of times the interest earned is added to principal in a year.

Compounding frequency could be yearly, half yearly, quarterly etc.
Formula of the compound interest when compounded annually,
A = P ( 1+ r/100 )n
Where P = Principal
R = rate of the interest
n = time in years
When compounded half yearly
A = P (1+ r/200 )2n
When compounded quarterly,
A = P (1+ r/400 )4n
In general,
A = P ( 1+ r/(k*100))kn
Where, r = rate of interest
n = time
k = compounding frequency
P = principal

SOLVED PROBLEMS:


Q.1. Find the amount after 3 years if the sum is 2000 at rate of interest 5% annually when compounded yearly?
Solution:
Here, P = 2000       r = 5%          n = 3
When compounded yearly
A = P (1+ r/100 )n
   = 2000 (1+ 5/100)3
   = 2000 (1+ 1/20)3
   = 2000 (21/20)3
   = 2000 (1.05)3
   = 2000 × 1.157
   = 2315.25 ≈ 2315

Q.2. What will be the amount after 3 years when 5000 Rs. Is given on compound interest of 10% per annum compounded annually?  

also find,

1. Interest earned in 1st year
2. Interest earned in 2nd year
3. Interest earned in 3rd year
Solution:
Here, P = 5000
r = 10%
A =?
A = P (1+ r/100 )n
   = 5000 (1+ 10/100)3
   = 5000 (1.1)3
   = 5000 (1.331)
A = 6655 Rs.
Therefore amount after 3 years will be 6655 Rs.
let I1 be Interest earned in 1st year
Let A1 = amount after one year
A1 = P (1+ r/100 )n
     = 5000 (1+ 10/100 )1
     = 5000(1.1)
     = 5500
Interest = A1– P
             = 5500 – 5000
I1 = 500
let I2 be Interest earned in 2nd year
Let A2 = amount after 2nd year
A2 = P (1+ r/100 )n
A2 = 5000 (1+ 10/100 )2
     = 5000 (1.1)2
     = 6050
Interest earned in 2nd year = A2 – A1
                                         = 6050 – 5500
I2 = 550 Rs.
let I3 be Interest earned in 3rd year
Let A3 = amount after 3rd year
As we have already calculated
A3 = 6655 Rs.

Interest earned in 3rd year = A3 – A2
                                        = 6655 – 6050
                                        = 605 Rs.
SHORTCUT METHOD (SLAB METHOD)
      P                1st year                 2nd year              3rd year                   
   5000                  5000                   5000                    5000
                              500                     500                      500       
                                                          50                        50
                                                                                      50
                                                                                       5
Amount after 3 years = 6655 Rs.
Interest earned in 1st year = 500 Rs.
Interest earned in 2nd year = 550 Rs.
Interest earned in 3rd year = 605 Rs.

Q.3. If P = 16000, r = 10%, n = 2 years compounded annually then find the amount and compound interest?
Solution:
      A = P (1+r/100) n
         = 16000 (1+10/100) 2
         = 16000(1.1)2
      A = 19360 Rs.
And C.I. = A – P 
             = 19360 – 16000
             = 3360 Rs.   
Shortcut method 
r = 10% = 10/100 = 1/10
Now, 1/10 
     10         11
   ×10       ×11    (two times because n = 2)
100 (P)       121(A)
CI = A – P = 21
When P = 100            A = 121
P = 16000       A =?
? = 16000×121/100
   = 19360 Rs.
Same way interest can be found.

Q.4. When R = 20%, time = 2 year, and difference between compound interest and simple interest is 30 Rs. Then find the principal, SI and CI.
Solution:
With normal method it will be lengthy as well as time taking. So by doing this sum by ratio method, 
Shortcut method (Ratio method)
Here 
r = 20% = 20/100 = 1/5
Here P = 5 and n = 2 years
For two years
P = 5×5 = 25
                P         For first Year            Second Year
               25              25                                25
                                  5                                  5
                                                                      1
  CI = 5+5+1 = 11          and     SI = 5+5 = 10
                     CI – SI = 11 – 10 
                                = 1
             1 is equivalent to 30 Rs.
            25 is equivalent to X Rs.
                 X = (25×30)/1
                    = 750 Rs.
Principal = 750 Rs.

By Unitary Method,
When CI is 11,       P = 25
          CI is X         P = 750
              X = (750×11)/25     
             CI = 330 Rs.
      And SI = 330 – 30 
                 = 300 Rs.

3. BASED ON SIMPLE INTEREST AND COMPOUND INTEREST 


Q.5. The difference between compound interest and simple interest for 2 year at the rate of 10% over principal amount of Rs. X is 10 Rs. What is the value of X?
Solution:
By doing with another method then formula method,
Rate of interest over two years From CI,
                       rc = 10 + 10+(10×10)/100     …successive percentage formula
                           = 10 + 1 + 10
                       rc = 21        
                       rs = 20%
             Here, rc - rs = 1%
             Here difference = 1% of P
                                  10 = 1/100 x X
                                   X = 1000 Rs.
                                   X = P = 1000 Rs.  

Q.6. The compound interest on a sum for 2 years is 832 Rs. And simple interest on the sum at the same rate for the same period is 800 Rs. What is the rate of interest?
Solution:
Given,    CI = 832 Rs.
             SI = 800 Rs. And n = 2 years
             CI = P {(1+r/100) n – 1}                          SI = Prn/100                      
             832= P {(1+r/100) 2 – 1}                        800 = (P×r×2)/100
             832 = P [ 1+ r^2/10000 + 2r/100 – 1]         P = 40000/r ………(2)
             832 = P [r^2/10000 + 2r/100 ] ……..(1)
     From (1) and (2)
     832 = 40000/r (r^2/10000 + 2r/100)
     832 = (4r/1 + 800)
     832 – 800 = 4r
          4r = 32
            r = 8%
    Rate of interest = 8%

Q.7. How much more or less would Rs.20,000 fetch after two years, if it is put at 10% per annum compound interest payable half yearly than if it is put at 15% per annum simple interest payable yearly?

Solution:
Compound interest                                Simple interest
P = 20,000                                                 P = 20,000
r = 10%                                                       r = 15%
n = 2 years, compounded half yearly   n = 2 years
A = P (1 + r/(2×100)) 2n                                   S.I = Prn/100
   = 20000(1 + 10/200) 4                                        = (20000×2×15)/100
   = 20000(1.05) 4                                                  = 6000
   = 20000 × 1.215                                   SI – CI = 6000 – 4310
   = 24310                                                           = 1690 Rs.
CI = A – P
    = 24310 – 20000
    = 4310
∴ Money put on Compound interest would fetch 1690 Rs. Less than Money put on SI.

Posted By:

Hardik Kaneriya

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