335
Co m p o u n d I n t e r e s t
Chapter-18
Compound Interest
K KUNDAN
Introduction
You know that, if Principal = Rs P, Rate = R% per annum and Time = T (in years), then the Simple Interest (SI) in Rs is given by SI =
Clearly, compound interest at the end of certain specified period is equal to the difference between the amount at the end of the period and the original principal ie CI = Amount – Principal.
Conversion Period
P R T
100 For example, if Principal = Rs 5000 and rate of interest = 10% per annum, then
5000 10 1 = Rs 500 100
SI for 1 year = Rs
5000 10 2 = Rs 1000 100
SI for 2 years = Rs
5000 10 3 100
SI for 3 years = Rs
= Rs 1500 and so on. Clearly, in computing SI the principal remains constant throughout. But the above method of computing interest is generally not used in banks, insurance corporations, post offices and other money lending and deposit taking companies. They use a different method for computing interest. In this method, the borrower and the lender agree to fix up a certain time interval, say a year or a half year or a quarter of a year for the computation of interest and amount. At the end of first interval the interest is computed and is added to the original principal. The amount so obtained is taken as the principal for the second interval of time. The amount of this principal at the end of the second interval of time is taken as the principal for the third interval of time and so on. At the end of certain specified period, the difference between the amount and the money borrowed ie the original principal is computed and it is called the compound interest (abbreviated as CI) for that period. Thus, we may define the compound interest as follows: If the borrower and the lender agree to fix up a certain interval of time (say, a year or a half-year or a quarter of a year etc) so that the amount ( = Principal + Interest) at the end of an interval becomes the principal for the next interval, then the total interest over all the intervals calculated in this way is called the compound interest and is abbreviated as CI.
The fixed interval of time at the end of which the interest is calculated and added to the principal at the beginning of the interval is called the conversion period. In other words, the period at the end of which the interest is compounded is called the conversion period. When the interest is calculated and added to the principal every six months, the conversion period is six months. Similarly, the conversion period is three months when the interest is calculated and added quarterly. Note: If no conversion period is specified, the conversion period is taken to be one year.
Computation of Compound Interest (i) By the the method method when when the the inte interes restt is calculated and added to the principal every interval For example, the compound interest on Rs 1000 for 2 years at 4% per annum is Rs 81.60. Let us see. How? Principal for the first year = Rs 1000 Interest for the first year
K KUNDAN 1000 4 1 = Rs 40 100
= Rs
P R T Using : Interest 100
Amount at the end of first year = Rs 1000 + Rs 40 = Rs 1040 Principal for the second year = Rs 1040 Interest for the second year
1040 4 1 = Rs 41.60 100
= Rs
Amount at the end of second year = Rs 1040 + Rs 41.60 = Rs 1081.60
336
C o n c ep ep t o f A r i t h m e t i c
Compound interest = Rs (1081.60 – Rs 1000) = Rs 81.60
2
R R 1 100 100
R 100
= Rs P 1
(ii) By the Formula We have seen in the above example that it takes a lot of time to find compound interest. Hence, we explore below a formula for finding compound interest. Let principal be Rs P. Rate = R% Time = T years Interest for first year
= Rs P 1
3
2 R P 1 becomes a common factor 100
K KUNDAN P R 1 PR = Rs 100 100
= Rs
T
Thus, we get the formula f or finding the amount in case of compound interest as
PR R 1 100 100
R R 1 100 100
R = Rs P 1 100
Time
A = P 1
Amount at the end of second year or principal for third year
= Rs P 1
R – Rs P 100
R = Rs P 1 100 1
R P 1 100 R 1 = Rs 100
R PR R + Rs 1 100 100 100
T
= Rs P 1
Interest for second year
R 100
CI (Compound Interest) for T years
R PR = Rs P 1 = Rs P + Rs 100 100
= Rs P 1
T
= Rs P 1
Principal for second year
= Rs
Proceeding in the same manner, amount at the end of T years
Rate 100
Now, we solve the above example, by using the above formula
A = Rs 1000 1
4 100
2
1000 104 104 = Rs 1081.60 100 100
= Rs
CI = Rs 1081.60 – Rs 1000 = Rs 81.60
2
R P 1 100 becomes a common factor
Computation of Compound Interest when Interest is Compounded Half-Yearly or Quarterly
K KUNDAN Interest for the third year
2 R P 1 R 1 100 = Rs 100
= Rs
PR R 1 100 100
2
Amount at the end of third year
= Rs P 1
2
R PR R + Rs 1 100 100 100
2
In compound interest, the time from one specified period to the next is known as the conversion period period as stated earlier. If time is one year, there is one conversion period a year. If the time is six months, there are two conversion periods a year. If the time is three months, there are four conversion periods a year. As stated earlier the rate of interest is usually quoted as per cent per annum. Thus when the interest is calculated : ( i ) quarterly, the rate of interest per conversion 1 period is of the rate stated yearly. 4 ( i i ) half-yearly, the interest rate per conversion period is
1 2
of the rate stated yearly.
337
Co m p o u n d I n t e r e s t For example, we have to find the compound 1 interest on Rs 12000 for 1 years at 16% per 2 annum, interest being compounded (a) quarterly and (b) half-yearly. We proceed as follows: (a) Principal = Rs 12000 Rate of Interst = 16% per annum
16
=
= 4% per quarter
4
Computation of Compound Interest, when Time is not an Exact Number of Years Suppose we have to find Compound Interest (CI) for 2
1 2
years on a certain sum at a certain rate per
cent. The Compound Interest for 2
1
years will be 2 equal to the Compound Interest for 2 years at the given rate together with Compound Interest for 1 1 year at of the given rate. Thus interest for any 2 fraction of a year is the same as the interest for full year at the rate equal to the same fraction of the rate. For example, if we have to find the compound
K KUNDAN Time = 1
1 2
3 4 = 6 quarters 2
years = T
A = P 1
R 100
= Rs 12000 1
4 100
6
6
104 = Rs 15183.83 100
= Rs 12000
CI = Rs 15183.83 – Rs 12000 = Rs 3183.83 (b ) Principal = Rs 12000 Rate of Interest = 16% per annum
16 8% per half-year 2
=
Time = 1
1 2
3 2 = 3 half-years 2
years = T
A = P 1
R 100
= Rs 12000 1 = Rs
8 100
3
interest on Rs 25000 at 13% per annum for 2
1 2
year s, we procee d as foll ows: Principal = Rs 25000 Rate = 12% Time = 2
1 2
years T
A = P 1
R 100
12 2 12 = Rs 25000 1 1 2 100 100 1 [ Interest for year at 12% 2
12 6%] 2
= Interest for 1 year at
12000 108 108 108
100 100 100 = Rs 15116.54 CI = Rs 15116.54 – Rs 12000 = Rs 3116.54 Thus, if Principal = P, Time = T years and Rate = R% per annum (i) Amount (when interest compounded quarterly)
2 12 6 25000 1 1 = Rs 100 100
K KUNDAN R P 1 4 100
4T
R P 1 400
4T
(ii) Amount (when interest compounded half year ly) R P 1 2 100
2T
R P 1 200
2T
= Rs 25000
112
100
112
100
106
100
= Rs 33241.60
CI = Rs 33241.60 – Rs 25000 = Rs 8241.60 Thu s, le t P be th e pri nci pal and th e rat e of interest be R% per annum. If the interest is compounded annualy but time is the fraction of a year say 2
1 2
years, then amount A is given by
R 2 R 2 and CI = A – P A = P 1 1 100 100
338
Co n c ep t o f A r i t h m e t i c
Alternative Method In this method, we calculate Compound Interest for the exact number of years by the formula and Simple Interest for the remaining time. And this Simple Interest should be added to the Compound Interest. For example, solve the above example. Principal = Rs 25000 Rate = 12%
Here, P = Rs 4000, R1 = 5% per annum and R2 = 15% per annum. Amount after 2 years
= P 1
R R1 1 2 100 100
5 15 1 100 100
1 3 1 20 20
= Rs 4000 1
K KUNDAN Time = 2
1
years 2 Compound Interest for the first 2 years = A – P
= Rs 4000
T
R A = P 1 100
2 12 25000 1 = Rs 100
25000 112 112 = Rs 31360 100 100
= Rs
Compound Interest for the first 2 years = Rs 31360 – Rs 25000 = Rs 6360 For the next year, Rs 31360 will be Principal
Interest for next
= Rs 4000 1
1 2
year
21
20
23 20
= Rs 4830. Thus, the refrigerator will cost Rs 4830 to Ram Singh.
General Formula for Computing Amount Let P be the principal, the rate of interest be R% per annum and time be T years and the interest is compounded after each month. Interest on Rs 100 for 1 year = Rs R Interest on Rs 100 for x months
R x 12
= Rs
Interest on Rs P for x months 1 31360 12 P R T 2 = = Rs 100 100 = Rs 1881.60
Total Compound Interst = Rs 6360 + Rs 1881.60 = Rs 8241.60
Computation of Compound Interest, when Interest is Compounded Annually but Being Different for Different Years
P R x 100 12
= Rs
Amount at the end of x months
Rx = P = P 1 12 100 12 100 P R x
This amount is considered as principal for the next x months. According to the above, Amount at the end of next x months = Principal for the first x months + Interest of the next x months
K KUNDAN Let P be the principal and the rate of interest be R1% for the first year, R 2% for second year, R 3% for third year and so on and in the last Rn % for the n th year. Then the amount A and the compound interest (CI) at the end of the n years is given by
A = P 1
R R1 R 1 2 .......1 n and 100 100 100
CI = A – P For example, Ram Singh bought a refrigerator for Rs 4000 on credit. The rate of interest for the first year is 5% and of the second year is 15%. How much will it cost him if he pays the amount after two years? To solve thi s, we proceed as follow s:
Rx Rx R x = P 1 12 P 1 12 100 100 100 12 Rx Rx Rx 12 1 12 P 1 12 = P 1 100 100 100
2
339
Co m p o u n d I n t e r e s t Similarly, amount for the next x months Rx = P 1 12 100
(e) If the rate of interest is calculated annually, then x = 12
3
12 T
R 12 12 T R = P 1 Amount = P 1 12 100 100
Now, Number of x months in 1 year =
12
Thus, we can conclude that the above formula is the General Formula and it is very useful in computing Amount (Principal + Compound Interest) and Compound Interest, whether rate is calculated quarterly, four-monthly, half-yearly, nine-monthly etc. For example, to find the Compound Interest on
K KUNDAN
Number of x months in T years =
x
12 T
Rx Amount (A) for T years = P 1 12 100
x
12 T x
Rs 12000 for 1
(a) If the rate of interest is calculated quarterly, then x = 3
R 3 Amount = P 1 12 100
12 T 3
R = P 1 4 100
4 T
12 T 6
R P 1 2 = 100
2 T
(c) If the rate of interest is calculated ninemonthly, then x = 9
R 9 Amount = P 1 12 100
12 T 9
3R 4 = P 1 100
years at 12% per annum, interest 4 being compounded five-monthly, we proceed as follows: Here, P = Rs 8000, R = 12%, T = 1
1 4
=
5 4
and
x = 5
(b ) If the rate of interest is calculated half yearly, then x = 6
R 6 Amount = P 1 12 100
1
4 T 3
Rx P 1 12 Amount (A) = 100
12 T x
125 12 5 45 12 = Rs 8000 1 100
= Rs 8000 1
3 5 100
K KUNDAN
(d ) If the rate of interest is calculated fourmonthly, then x = 4 12 T R 4 4
Amount = P 1 12 100
R 3 = P 1 100
3 T
21 21 21 = Rs 8000 20 20 20 = Rs 9261
Compound Interest = Rs 9261– Rs 8000 = Rs 1261
340
Co n c ep t o f A r i t h m e t i c
Solved Examples Ex. 1:
Soln:
F i n d t h e c om p o u n d i n t er e s t o n R s 1 2 0 0 0 f o r 3 y e a r s a t 1 0 % p er a n n u m c om p o u n d e d a n n u a l l y .
25 1000
1 40
= Rs 64000 × 1
We know that the amount A at the end of T years at the rate of R% per annum when the interest is compounded annually is given by
= Rs 64000 × 1
3
3
K KUNDAN
A = P 1
41 40
T
R 100
Here, P = Rs 12000, R = 10% per annum and T = 3 years R Amount A after 3 years = P 1 100
10 100
1
= Rs 12000 × 1
= Rs 12000 × 1
3
3
Soln:
3
= Rs 12000 ×
11
11
10 10 10 = Rs (12 × 11 × 11 × 11) = Rs 15972 Now, Compound Interest = A – P Compound Interest = Rs 15972 – Rs 12000 = Rs 3972. Ex. 2:
41
40
41
40
F i n d t h e c om p o u n d i n t e r e st a t t h e r a t e o f 1 0 % p er a n n u m f o r f o u r y e a r s o n t h e p r i n c i p a l w h i c h i n f o u r y ea r s a t t h e r a t e o f 4 % p er a n n u m g i v e s Rs 1 6 0 0 a s si m p l e i n t e r e s t .
Let Rs P be the principal. This principal giv es Rs 1600 as SI in four years at the rate of 4% per annum.
P=
41
40
= Rs 68921 Hence, compound interest payable after 3 years = Rs 68921 - Rs 64000 = Rs 4921. Ex. 3:
3
10
11 10
= Rs 64000 ×
= Rs 12000 ×
11
3
= Rs 64000 ×
SI 100 R T
or, P = Rs
1600 100
= Rs 10000 4 4 Now, we have P = Rs 100000 R = 10% and T = 4. T
Amount after 4 years = P 1
V i j a y o bt a i n s a l o a n o f R s 6 4 0 0 0 a g a i n s t h i s f i x e d d ep o s i t s . I f t h e r a t e o f i n t e r e s t b e 2 . 5 p a i s e p er r u p e e p e r a n n u m , c a l c u l a t e t h e c om p o u n d i n t er e st p a y a b l e a f t e r 3 y ea r s .
10 = Rs 10000 × 1 100
R 100
4
K KUNDAN Soln:
1 = Rs 10000 × 1 10
Here, P = Rs 64000, T = 3 years, and R = 2.5 paise per rupee per annum = (2.5 × 100) paise per hundred rupees per annum = =
11 10
2.5 100 per hundred rupees 100
= Rs 10000 ×
per annum 2. 5% per a nnum.
2.5
100
11 10
11 10
11 10
11 10
= Rs 14641.
Compound interest
R Amount A after 3 years = P 1 100
4
= Rs 10000 ×
Rs
= Rs 64000 × 1
4
= Rs 14641 – Rs 10000 = Rs 4641.
3
Ex. 4:
C om p u t e t h e c om p o u n d i n t e r e st o n R s 12000 f or 2 years at 20% per annum w h en c o m p o u n d e d h a l f -y e a r l y .
Soln:
Here, Principal P = Rs 12000,
3
341
Co m p o u n d I n t e r e s t Rate = R = 20% per annum =
Soln:
20
= 10% per half-year 2 T = 2 years = 2 × 2 = 4 half-years
=
20
= 5% per quarters 4 T = 1 year = 1 × 4 = 4 quarters
T
A = P 1
Here, Principal = P = Rs 320000 Rate = R = 20% per annum
R 100
Amount (A) after 1 year = P 1
4 10 = Rs 12000 1 100
T
R 100
K KUNDAN
4 5 = Rs 320000 1 100
4 1 = Rs 12000 1 10
= Rs 12000
4
21 = Rs 320000 20 4
11 11 11 11 10 10 10 10
320000 21 21 21 21 20 20 20 20
= Rs
14641
= Rs 388962
10000 = Rs 17569.20 Compound interest = Rs 17569.20 – Rs 12000 = Rs 5569.20 Alternative Method: We can solve this question by General Formula also. Rx Amount (A) for T years = P 1 12 100
1 = Rs 320000 1 20
4 11 12000 = Rs 10
= Rs 12000
Compound interest = Rs 388962 – Rs 320000 = Rs 68962. Alternative Method:
Rx Amount (A) for T years = P 1 12 100
12 T x
12 T x
Here, P = Rs 320000 R = 20% per annum T = 1 year and x = 3
Here, P = Rs 12000 R = 20% per annum T = 2 years and x = 6
121 20 3 3 = Rs 320000 1 12 100
K KUNDAN 122 20 6 6 = Rs 12000 1 12 100 4 1 = Rs 12000 1 10
F i n d t h e c om p o u n d i n t er e s t o n R s 3 2 0 0 0 0 f o r o n e y ea r a t t h e r a t e o f 2 0 % p er a nnum . If t he i n t er e s t is c o m p o u n d e d q u a r t er l y .
4 5 100
Now, do as the above. Ex. 6:
Now, do as the above. Ex. 5:
= Rs 320000 1
Soln:
W h a t s u m o f m o n ey a t c o m p o u n d i n t er e s t w i l l a m o u n t t o Rs 2 2 4 9 . 5 2 i n 3 y e a r s , i f t h e r a t e of i n t e r e s t i s 3 % f o r t h e f i r s t y e a r , 4 % f o r t h e sec on d y ea r a n d 5 % f or t h e t h i r d y e a r ?
The general formula for such question is:
A= P 1
R1 R R R 1 2 1 3 ... 1 n 100 100 100 100
342
Co n c ep t o f A r i t h m e t i c Compound interest
Where A = Amount, P = Principal and R 1, R2, R3, ...., R n are the rates of interest for different years. In the above case,
2249.52 = P 1
= Rs (133.10 - 100) = Rs 33.10 Now, If compound interest is Rs 33.10, principal = Rs 100 If compound interest is Re 1, principal =
3 4 5 1 1 100 100 100
2249.52 = P (1.03) (1.04) (1.05)
100
Rs
2249 .52
33.10
If compound interest is Rs 331, principal
P = (1.03) (1.04) (1.05) = Rs 2000
K KUNDAN Ex. 7:
Find
t he
24000
at
c om p o u n d 1 5 % p er
i n t er e s t
annum
on
for
100 331 = Rs 1000 33.10
= Rs
Rs
1 2 3
Hence, principal = Rs 1000. Alternative Method: Rate = 10% per annum Time = 3 years Amount = Rs 331
years.
Soln:
Here, P = Rs 24000 R = 15% per annum and Time = 2
1 3
years.
1
Amount after 2 3 years
P =
1 R 2 R 3 = P 1 1 100 100
1 15 2 15 3 = Rs 24000 1 1 100 100
3 1 1 20 20 2
= Rs 24000 1
R 100
A
331
T
R 1 100
10 1 100
3
331 10 10 10 = Rs 1000 11 11 11
= Ex. 9:
W h a t s u m w i l l b e co m e Rs 9 8 2 6 i n 1 8 m o n t h s i f t h e r a t e o f i n t e r es t i s 5 % p er annu m and t he i n t er e s t is c o m p o u n d e d h a l f -y ea r l y ?
Soln:
Let the required sum ie the principal, be Rs P. We have, Principal = Rs P, Amount = Rs 9826, R = 5% per annum
2 23 21 20 20
= Rs 24000
T
A = P 1
=
5 2
% per half-year and
K KUNDAN = Rs 33327 Compound interest = Rs (33327 – 24000) = Rs 9327.
Ex. 8:
Soln:
F i n d t h e p r i n c i p a l , i f t h e c om p o u n d i n t e r e s t c om p o u n d ed a n n u a l l y a t t h e r a t e o f 1 0 % p er a n n u m f o r t h r e e y e a r s is Rs 331.
Let the principal be Rs 100. Then, Amount after three years 3 10 = Rs 100 1 100 3 11 = Rs 100 100
= Rs 133.10.
18
T = 18 months =
12
years =
3 2 3 half-years 2
=
A = P 1
T
R 100
5 or, 9826 = P 1 2 200
or, 9826 = P 1
1 80
3
3
3
2
years.
343
Co m p o u n d I n t e r e s t
81 80
3
T
or,
882 21 800 20
or, P = 9826 ×
or,
441 21 400 20
Hence, required sum = Rs 9466.54. Alternative Method: Let the required sum be Rs 100. Then,
or,
or, 9826 = P
3
80 = 9466.54 81
2
T
21 21 20 20
2
K KUNDAN the amount after 18 months ie
3
T = 2.
years at
2 the rate of 5% compounded half-yearly, is given by 3 2 5 2 Amount = Rs 100 1 2 200
81 80
= Rs 100
= Rs
Hence, required time is 2 years. Ex. 11: I n w h a t t i m e w i l l R s 6 4 0 0 0 a m o u n t t o R s 6 8 9 2 1 a t 5 % p er a n n u m , i n t e r e s t b ei n g c o m p o u n d e d h a l f -y ea r l y ?
Soln:
=
5
% per half-year 2 Let the time be T years = 2T half-years The re fo re ,
3
T
A = P 1
R 100
531441 5120
5 68921 64000 1 2 100
Now, If amount is Rs
Here, Principal (P) = Rs 64000 Amount (A) = Rs 68921 Rate (R) = 5% per annum.
531441
, 5120 then principal = Rs 100 If the amount is Re 1, then principal
or,
100 5120 531441
= Rs
68921 1 1 64000 40 3
41 41 40 40
2T
2T
2T
or,
If the amount is Rs 9826, then principal
100 5120 9826 = Rs 9466.54 531441
or, 2T = 3
= Rs
3
1
K KUNDAN T =
Hence, required sum = Rs 9466.54.
Ex. 10: I n w h a t t i m e Rs 8 0 0 a m o u n t t o Rs 8 8 2 at 5% per annually?
Soln:
annum
c o m p o u n d ed
Here, Amount (A) = Rs 882, Principal (P) = Rs 800 and Rate (R) = 5% per annum.
A = P 1
T
R 100
T
or,
years.
2
c o m p o u n d i n t e r e st w i l l Rs 1 0 0 0 0 a m o u n t t o R s 1 3 3 1 0 i n t h r e e y ea r s ?
Soln:
Let the rate be R% per annum. We have, P = Principal = Rs 10000, A = Amount = Rs 13310 and T = 3 years.
A = P 1
T
R 100
or, 13310 10000 1
T
1 1 800 20 882
5 100
years = 1
Ex. 12: A t w h a t r a t e p e r c e n t p e r a n n u m
or, 882 = 800 1
2
R 1 or, 10000 100 13310
R 100
3
3
344
Co n c ep t o f A r i t h m e t i c
or,
1331 R 1 1000 100 113
R 1 or, 3 100 10 or, 1
R 100
3
or,
3
or,
11 10 R
200
10
R 200
11 10
1
1 10
200 = 20% 10
R =
11
1
Hence, the rate of interest = 20% per annum.
K KUNDAN or,
or,
R
100 R
100
or, R =
11
10
1
Ex. 14: D et e r m i n e t h e r a t e o f i n t er e s t f o r a s u m
1
t h a t b e c om e s
10
100 10
= 10
Ex. 13: Ni k h i l i n v e st e d Rs 6 0 0 0 i n a c om p a n y a t c o m p o u n d i n t e r e st c o m p o u n d e d s em i -a n n u a l l y . H e r e cei v e s Rs 7 9 8 6 a f t e r 1 8 m o n t h s f r o m t h e c om p a n y . F i n d t h e r a t e o f i n t e r e st p e r a n n u m .
T = 18 months =
3 2
Let the principal be Rs P and the rate of interest be R% per annum compounded annually. It is given that the amount at the end of 3 years must become
216 R P P 1 125 100
We have, P = Principal = Rs 6000 A = Amount = Rs 7986 and
3 2 3 half-years 2
or,
Let the rate of interest be R% per annum
R
P. Therefore,
3
216 R 1 125 100 3
3
R 6 1 5 100
% per half-year.
3
or,
2 (Since the interest is compounded semiannually)
or,
T
216 125
T R Using A P 1 100
years.
=
=
t i m e s of i t s el f i n 3
years, compounded annuall y.
Soln:
Hence, rate = 10% per annum.
Soln:
2 1 6
1 2 5
R A = P 1 100
or,
6 5
1
R 100
R 100 6 5
1
K KUNDAN R or, 7986 6000 1 2 100
or, 7986 6000 1
or,
or,
7986 6000
R 200
R 1 200
1331 R 1 1000 200 3
or,
3
3
R
1
100 5 or, R = 20 Hence, the rate of interest is 20% per annum.
3
Ex. 15: T h e d i f f e r e n c e b et w ee n t h e c o m p o u n d i n t er e s t a n d s i m p l e i n t e r e st o n a c e r t a i n s u m o f m o n ey a t 1 0 % p er a n n u m f o r 2 y e a r s i s R s 5 0 0 . F i n d t h e su m w h e n t h e i n t e r e st i s co m p o u n d e d a n n u a l l y .
3
R 11 1 10 200
or,
3
Soln:
Let the sum be Rs 100. Computation of compound interest: We have, Principal = Rs 100, R = 10% per annum, and T = 2.
345
Co m p o u n d I n t e r e s t 2 10 100 1 Amount = Rs 100 2 11 100 = Rs 10
15
or, 2 1
R 100
....(i)
Putting A 2P and T 15 in T R A P 1 100
= Rs 121.
CI = Rs 121 – Rs 100 = Rs 21.
Suppose the money becomes 8 times ie 8P in T years. Then,
Computation of simple interest : We have, Principal = Rs 100, R = 10% and Time = 2 years.
K KUNDAN or, 8 1
3
[Using (i)]
T
R R 1 100 100 or, T = 45 Hence, the money will become 8 times in 45 years. Ex. 17: A f a r m er w a n t s t o d i v i d e R s 3 9 0 3 0 0 b et w ee n h i s t w o d a u g h t e r s w h o a r e 1 6 y e a r s a n d 1 8 y e a r s ol d r es p ec t i v el y i n s u c h a w a y t h a t t h e s u m i n v e st e d a t t h e r a t e o f 4 % p e r a n n u m , co m p o u n d e d annually wi ll give the same amount to ea c h , w h e n t h e y a t t a i n t h e a g e o f 2 1 y ea r s . H o w s h o u l d h e d i v i d e t h e s u m ?
2 P 1 10 P = Rs 100
11 2 = Rs P 10 P
21P 100
45
or, 1
P 10 2 P = Rs 100 5
= Rs
R 100
15 T R R 1 1 or, 100 100
=
P 11 11 P 10 10
T
100
= Rs
R 100
3 or, 2 1
P R T
Compound Interest = Amount – Principal
R 100 T
Thus, Differ ence in CI and SI = Rs (21 – 20) = Re 1 Now, If difference between CI and SI is Re 1, Sum = Rs 100 If difference between CI and SI is Rs 500, Sum = Rs (100 × 500) = Rs 50000. Alternative Method: Let the sum be Rs P. Simple Interest =
T
8P P 1
100 10 2 = Rs 20. SI = Rs 100
Soln:
Suppose the farmer gives Rs P to 16 years old daughter and the remaining Rs (390300 – P) to 18 years old daughter. At the age of 21 years, each daughter gets the same amount. This means that the amount of Rs P invested for 5 years is same as the amount of Rs (390300 – P) invested for 3 years ie
K KUNDAN Now, according to the question,
21P
100
or,
P 5
= Rs 500
P 1
21P - 20P
= Rs 500 100 P = Rs (500 × 100) = Rs 50000
5
4 4 390300 P 1 100 100 2
4 390300 P 100
1 390300 P 25
or, P 1
Ex. 16: A s u m o f m o n e y d o u b l e s i t s e l f a t c om p o u n d i n t er e s t i n 1 5 y e a r s . I n h o w m a n y y e a r s w i l l i t b ec om e ei g h t t i m es ?
Soln:
Let the sum of money be Rs P invested at the rate of R% per annum. It is given that the money doubles itself in 15 years. The re fo re ,
2P P 1
15
R 100
or, P 1
2
2
26 390300 P 25 26 2 or, P 25 1 390300 or, P
3
346
Co n c ep t o f A r i t h m e t i c 676 625 390300 625
R 100
or, P
9680 P 1
1301 or, P 390300 625
10648 P 1
or, P =
...(i) and 3
390300 625
2
R 100
...(ii)
Now, dividing equation (ii) by equation (i), we have
= 187500 1301 Therefore, the daughter aged 16 years gets Rs 187500 and the daughter aged 18 years gets Rs (390300 – 187500) = Rs 202800. Alternative Method: Let the equal amount in each case be Rs 100 and P 1, P2 be the principals for the two daughters. In case of the first daughter, A = Rs 100, T = 5 years, R = 4%
R 100
3
P 1
R 100
2
K KUNDAN 5
4 26 P1 100 P1 1 100 25 or, P1
100
26 25
5
10648 9680
or,
5
or,
100 (25)5
or,
(26)5
100 P2 1 100
26 25
9680 R
100 R 100
or, R =
In case of the second daughter, A = Rs 100, T = 3 years, R = 5%
or, P2
10648
3
5 100
R
1
100
10648
968
9680
9680
100 = 10
Putting R = 10 in 9680 P 1
3
10 100
1
9680 P 1
3
or, 9680 P 1
(26)5
2
R , we 100
get
25 100 26
100 (25)5
1
9680
968
Ratio between their parts is P1 : P2 =
P 1
25 : 100 26
2
10
11 10
3
2
2
or, 9680 P
10 11
2
or, P = 9680
2
25 :1 26
=
or, P = 9680
100
K KUNDAN = 252 : 262 = 625 : 676 We shall divide Rs 390300 in the ratio of 625 : 676 Daughter aged 16 years old gets 390300 625 Rs 625 676 = Rs
Ex. 19: A s u m o f m o n ey i s p u t a t c o m p o u n d i n t er e s t f o r 2 y e a r s a t 2 0 % p er a n n u m . I t w o u l d f et c h R s 4 8 2 m o r e , i f t h e i n t er e s t w e r e p a y a b l e h a l f -y ea r l y t h a n i f i t w er e p a y a b l e y ea r l y . Fi n d t h e su m .
390300 625
= Rs 187500 1301 and the daughter aged 18 years old gets = Rs (390300 – 187500) = Rs 202800. Ex. 18: A su m a m o u n t s t o R s 9 6 8 0 i n 2 y e a r s a n d t o R s 1 0 6 4 8 i n 3 y ea r s c o m p o u n d e d a n n u a l l y . F i n d t h e su m (p r i n c i p a l ) a n d t h e r a t e of i n t e r es t p e r a n n u m .
Soln:
= 8000 121 Hence, principal = Rs 8000 and rate of interest = 10% per annum.
Let the sum (principal) be Rs P and the rate of interest be R% per annum. Then,
Soln:
Let the required sum of money be Rs P. Case I: When interest is payable yearly. In this case, let the amount be A 1. Then,
A1 P 1
2
2
20 36P 6 P 100 25P 5
Case II: When interest is payable half yea rly .
347
Co m p o u n d I n t e r e s t In this case, Principal = P, R = 20% per annum =
R or, 110 P 1 100 1 and PR = 5000 2
20
= 10% per half-year
2
T = 2 years = 2 × 2 = 4 half-years Let A2 be the amount at the end of 2 years. Th en ,
2R R2 110 P 1 1 and or, 100 10000 PR = 5000
T
R 100
10 100
2PR PR 2 and PR = 5000 100 10000
K KUNDAN A 2 P 1
or, A 2 P 1
11 or, A 2 P 10
or, 110
4
4
It is given that A2 – A1= 482 4
or, 110
5000
11 4 12 2 P 482 or, 10 10
11 11 12 12 482 P or, 10 10 10 10 2
241 1 482 100 100
or, P
50
50
PR
10000
110 = 100 +
or, 10 =
11 4 6 2 P 482 or, 5 10
P =
PR
or,
2
11 6 P 482 or, P 10 5
2
or, 110
R and PR = 5000
5000
R 10000 [Putting PR = 5000] R 2
R
2 or, R = 20. Putting R = 20 in PR = 5000, we get 20P = 5000 P = 250. Hence, principal = Rs 250 and rate = 20% per annum. Alternative Method: Difference between Compound Interest and Simple Interest = Rs 110 – Rs 100 = Rs 10 Simple Interest for 2 years = Rs 100
Simple Interest for 1 year = Rs
482 100 100
= 20000 241 Hence, the sum of money was Rs 20000. Ex. 20: R ee n a b o r r o w e d f r o m K a m a l c e r t a i n
100
2 = Rs 50 Because interest is reckoned yearly Compound Interest and the Simple Interest for the first year will be the same. Rs 10 is the interest on Rs 50 for 1 year . Principal = Rs 50, T = 1 year and SI = Rs 10
K KUNDAN s u m f o r t w o y ea r s a t s i m p l e i n t e r es t . Reen a l e n t t h i s s u m t o H a m i d a t t h e s a m e r a t e f o r t w o y ea r s a t c o m p o u n d i n t e r es t . A t t h e en d o f t w o y e a r s s h e r ec ei v e d R s 1 1 0 a s co m p o u n d i n t e r e st b u t p a i d R s 1 0 0 a s s i m p l e i n t er e s t . Fi n d t h e su m a n d r a t e o f i n t e r e st .
Soln:
Let the principal be Rs P and the rate of interest be R% per annum. We have, CI = Rs 110, SI = Rs 100 and Time = 2 years. 2
R P and 110 P 1 100
100
or, 110 P 1
PR2 100
PR R 1 and 100 50 100 2
Rate =
SI 100 P T
=
10 100 50 1
= 20% per annum Now, SI = Rs 100, R = 20% and T = 2 years
P=
SI 100 R T
100 100 = Rs 250 20 2
=
Hence, principal = Rs 250 and rate = 20% per annum. Ex. 21 : The simple interest on a sum at 4% per annum for 2 years is Rs 80. Find the compound interest on the same sum for the same period.
348 Soln:
Co n c ep t o f A r i t h m e t i c Let the sum be Rs P. P R T SI = 100 SI 100
P =
80 100
=
R T
Now, 1
2 4
R 100
T
CI = P 1
R P 100
43
12
or, 1
= Rs 1000
R 100
or, 4800 1
3
125 5 64 4
125 75 64 75
9375 4800
12
R 100
9375
K KUNDAN 2 4 1000 1 1000 = Rs 100
The above equat ion shows tha t Rs 4800 becomes Rs 9375 after 12 years.
Ex. 24: F i n d t h e p r e sen t v a l u e o f R s 4 0 9 6 0 d u e
1000 26 26 1000 25 25
3 y e a r s h e n ce a t
= Rs
676 1 = Rs 1000 625 = Rs
1000 51
Soln:
Present value 1
Present value = Rs
o f m o n e y f o r 2 y e a r s a t 1 0 % p er a n n u m i s R s 4 2 0 . F i n d t h e s i m p l e i n t er e s t a t t h e sa m e r a t e a n d f o r t h e s a m e t i m e.
= Rs 40960
Let the sum be Rs P.
2
10 P 100
11 11 P 10 10
P =
21 100
P
420 100 21
20 1 3 100 15 16
15 16
3
15
16
d e bt o f R s 8 1 1 6 d u e i n 3 y e a r s a t 8 % p e r a n n u m c o m p o u n d i n t e r e st ?
or, 420 P or, 420
40960
= Rs 33750 Ex. 25: W h a t a n n u a l p a y m e n t w i l l d i s ch a r g e a
T
R P CI = P 1 100 or, 420 P 1
3
20 = Rs 40960 3 100
Ex. 22: T h e co m p o u n d i n t er e s t o n a c er t a i n s u m
Soln:
annum
c o m p o u n d i n t er e s t .
= Rs 81.6
625
2 6 % p e r 3
= Rs 2000
Soln:
Let Rs x be the amount of each instalment. Then the instalment s of Rs x are paid at the end of 1 year, 2 years and 3 years respectively. Present values of these instalments are
x x x 2 3 8 , and 8 1 8 1 1 100 100 100 Total present value of these instalments
K KUNDAN Simple Interest =
P R T 100
2000 10 2 = Rs 100
25 625 15625 50725x = Rs 19683 27 729 19683
= Rs x
Also, present value of Rs 8116 due 3 years hence
8116 3 = Rs 8 1 100
= Rs 400 Ex. 23: R s 4 8 0 0 b ec o m e s Rs 6 0 0 0 i n 4 y e a r s a t a c er t a i n r a t e of c o m p o u n d i n t er e s t . W h a t w i l l b e t h e su m a f t e r 1 2 y e a r s ?
Soln:
Let the rate of interest be R%. Now, we have
8116 15625 19683
= Rs
4
R 4800 1 6000 100
8116 15625 = 19683 19683 x = Rs 2500
or, 1
4
R 6000 5 100 4800 4
50725x
349
Co m p o u n d I n t e r e s t
Practice Exercise 1.
2.
Abhay lent Rs 8000 to his friend for 3 years at the rate of 5% per annum compound interest. What amount does Abhay get after 3 years? Find the compound interest on Rs 1000 at the rate of 10% per annum for 18 months when interest is compounded half-yearly. What will be compound interest of Rs 24000 1 for 2 years at the rate of 15% per annum? 3 Ramesh deposited Rs 7500 in a bank which pays him 12% interest per annum compounded quarterly. What is the amount which he receives after 9 months? At what rate per cent per annum of compound interest will Rs 1600 amount to Rs 1852.20 in 3 years? Find the sum of money which will amount to Rs 26010 in 6 months at the rate of 8% per annum when the interest is compounded quarterly. Govardhan deposited Rs 7500 in a bank for 6 months at the rate of 8% per annum, interest compounded quarterly. Find the amount he received after 6 months. A certain sum invested at 4% per annum compounded semi-annually amounts to Rs 7803 at the end of one-year. Find the sum. Rs 16000 invested at 10% per annum compounded semi-annually amounts to Rs 18522. Find the time period of investment. Th e di ff er en ce in si mp le an d co mp ou nd interest on a certain sum for 2 years at 5% per annum compounded annually is Rs 75. Find the sum. The sim ple int ere st on a sum of mon ey at some rate for 3 years is Rs 225 and the compound interest on the same sum of money and at the same rate for 2 years is Rs 153. Find the sum and the rate per cent per annum. A principal sum of money is lent out at compound interest compounded annually at the rate of 20% per annum for 2 years. It would give Rs 2410 more if the interest is compounded half-yearly. Find the principal sum. A money-lender borrows a certain sum of money at 3% per annum simple interest and lends it at 6% per annum compound interest compounded half-yearly. If he gains Rs 618 in a year, find the sum of money borrowed by him. The compound int erest on a certai n sum of money for 2 years at 5% per annum is Rs 102.50. What will be the compound interest on the same sum of money for the same period at 4% per annum.
15. Two partners A and B together lend Rs 2523 at 5% compound interest compounded annually. The amount A gets at the end of 3 years is the same as B gets at the end of 5 years. Determ ine the share of each. 16. Two partners A and B t ogether lends Rs 84100 at 5% compound interest compounded annually. The amount which A gets at the end of 3 years is the same as what B gets at the end of 5 years. Determine the ratio of the shares of A and B. 17. A sum of money put at compound interest amounts to Rs 8820 in two years and to Rs 9261 in three years. Find the sum and the annual rate of interest. 18. Find the present value of Rs 4913 due 3 years 1 hence at 6 % per annum compound interest. 4 19. A person borrowed Rs 4000 at 5% per annum compound interest compounded annually. After 2 years, he repaid Rs 2210 and then after 2 more years, he repaid the balance with interest. Find the total interest paid by him. 2 0 . A sum of money lent out at compound interest increases in value by 50% in 5 years. A person wants to lend three different sums of money X, Y and Z for 10, 15 and 20 years respectively at the above rate in such a way that he gets back equal sums of money at the end of their respective periods. Find the value of X : Y : Z. 21 . A person closes his account in a bank by withdrawing Rs 110000. One year earlier, he had withdrawn Rs 65000. Two years earlier, he had withdrawn Rs 125000. How much money had he deposited in the bank at the time of opening the account three years ago if the annual interest rate was 10% compounded annually? 2 2 . Compound interest and simple interest on a certain sum for 2 years are Rs 104 and Rs 100 respectively. Find the rate per cent and the principal. 23 . The difference between compound and simple interests on a certain sum of money at the 1 interest rate of 10% per annum for 1 years 2 is Rs 183, when the interest is compounded semi-annually. Find the sum of money. 24 . In how many years will a sum of Rs 800 at 10% per annum compound interest, compounded semi-annually, becomes Rs 926.10. 25 . If the difference between compound interest, compounded half-yearly and simple interest on a sum of money at 8% per annum for 1 year is Rs 30. Find the sum.
K KUNDAN 3.
4.
5.
6.
7.
8.
9.
10 .
11.
K KUNDAN 12 .
13.
14.
350
Co n c ep t o f A r i t h m e t i c
Answers and explanations 1.
Here, P = Rs 8000, R = 5% per annum and R = 3 years. R Amount after 3 years = P 1 100
3.
Here, Principal (P) = Rs 24000 Rate (R) = 15%
3
Time (T) = 2
1 3
years
R 2 R 1 3 A = P 1 100 100
K KUNDAN = Rs 8000 1
5 100
= Rs 8000 1
1 20
21 20
3
3
15 2 15 1 3 = 24000 1 100 100
3
= Rs 8000
21 21 21 = Rs 9261. 20 20 20 Thus , Abh ay get s Rs 9261 at th e end of 3 year s. Here, P = Rs 1000, R = 10% per annum = Rs 8000
2.
=
10
% = 5% per half-year
2
18 3 years 12 2
T = 18 months =
3 = 2 3 half-years 2 Now, from the formula
2
23 21 20 20
= 24000
23 21 = Rs 33327 20 20 20 Compound Interest = Rs 33327 – Rs 24000 = Rs 9327 Alternative Method: Principal (P) = Rs 24000 Rate (R) = 15% = 24000
Time (T) = 2
3 5 = Rs 1000 1 100
1
years 3 Compound Interest for the first 2 years = Amount – Principal
T
R A = P 1 100
23
A = P 1
T
R 100
= Rs 24000 1
2 15 100
K KUNDAN
= Rs 1000 1
3
3
21 20
= Rs 1000
1 20
1000 21 21 21 20 20 20
24000 23 23 = Rs 31740 20 20
= Rs
Compound Interest for the first 2 years
= Rs 31740 – Rs 24000 = Rs 7740 For the next year Rs 31740 will be principal
= Rs
= Rs 1157.625 Hence, Compound Interest = Amount – Principal = Rs 1157.625 – Rs 1000 = Rs 157.625
1
Interest for next
=
P R T 100
3
years
31740 15 1 = Rs 1587 3 100
= Rs
Total Compound Interest = Rs 7740 + Rs 1587 = Rs 9327
351
Co m p o u n d I n t e r e s t 4.
Here, Principal = Rs 7500 Rate = 12% per annum
A 26010 P = = Rs T 2 R 2 1 1 100 100
12 3% per quarter and 4 9 3 year Time = 9 months = 12 4 =
26010 50 50 = Rs 25000 51 51
3 4 3 quarters 4
= Rs
=
K KUNDAN 7.
T
A = P 1
R 100
Principal (P) = Rs 7500
6 1 year 12 2
Time (T) = 6 months =
Amount after 9 months
3 3 7500 1 = Rs 100
Rate (R) = 8% per annum
7500 103 103 103
= Rs
5.
1 = 4 2 quarters 2
8 2% per quarter 4
=
100 100 100 = Rs 8195.45 Let the rate be R% per annum. We have Principal (P) = Rs 1600 Amount (A) = Rs 1852.20 and Time (T) = 3 years
2 51 7500 = Rs 50
3
3
or, 1
or, 1
R
7500 51 51 = Rs 7803 50 50
R 1852.20 = = 1.157625 = (1.05) 3 100 1600
R 100
= Rs 8.
= 1.05
R 100
2 2 7500 1 = Rs 100
R 100
R or, 1852.20 = 1600 1 100
or,
Amount = P 1
T
A = P 1
T
A = Rs 7803 R = 4% per annum
4 2% per half-year 2
=
= (1.05 – 1) = 0.05
K KUNDAN 6.
100 R = 0.05 × 100 = 5% Here, Amount (A) = Rs 26010 Rate (R) = 8% per annum
8 = 2% per quarter 4
6 1 year 12 2
Time (T) = 6 months =
1 = 4 2 quarters 2 Now,
A = P 1
T
R ; 100
where P = Principal or required sum
T = 1 year = (1 × 2 =) 2 half-years (Since the interest is compounded semiannually) T
A = P 1
R 100
A 7803 = Rs P = T 2 R 2 1 1 100 100
7803 50 50 = Rs 7500 51 51
= Rs
Thus, the sum inves ted is Rs 7500.
352 9.
Co n c ep t o f A r i t h m e t i c We have, Principal (P) = Rs 16000 Amount (A) = Rs 18522 Rate (R) = 10% per annum
Difference in CI and SI =
10 = 5% per half-year 2
=
41x 400
x 10
41x 40x
x
400 400 But it is given that the difference of CI and SI is Rs 75.
Let the time be T years = (2 × T) half-years Now, T
x
75 400 or, x = 75 × 400 = Rs 30000. The sum is Rs 30000. Simple interest for 3 years is Rs 225
R A = P 1 100
K KUNDAN
or, 18522 16000 1
or,
or,
21 16000 20 18522
9261 21 8000 20 3
5 100
2T
11.
Simple interest for 1 year is
2T
Simple interest for 2 years is
2T
21 21 or, 20 20
2T
1 = 1 years 2 2 Theref ore, time period of invest ment is oneand-a-half year. 10 . Let the sum be Rs x Simple interest on Rs x for 2 years at 5% =
3
x 2 5 100
= Rs
x
Then,
and
10
Compound Interest = P 1
Time
Rate 100
1
Compound interest on Rs x for 2 years at
75 R 100
PR 100
or, P =
3
= Rs 75
225 3
2
= Rs 150 and, compound interest for 2 years is Rs 153. Difference between compound interest and simple interest for 2 years = 153 – 150 = Rs 3. As we know that, difference between compound interest and simple interest for 2 years = Interest on simple interest for 1 year. Let the rate of interest be R%.
or, 2T = 3
T=
225
3 or R = 4%
75 or
P4 100
75
75 100
= Rs 1875 4 Sum = Rs 1875 and rate per cent = 4% per annum. 12 . Let the sum be Rs P. Compound interest when it is compounded annually,
K KUNDAN 5%
2 5 x 1 1 = 100
21 = x 20 1 2
441
= x
400
=
=
1
x 441 400 400 41x 400
= P 1
20
2
P 100
Compound interest when it is compounded half-yearly, 20 2 = P 1 100
= P 1
22
P 4
10 P 100
Now, according to the question, we have 4 2 10 20 P 1 P P 1 P = 2410 100 100
353
Co m p o u n d I n t e r e s t
or, P 1
or, P 1
4
2
10609 1 10000
10 20 P 1 = 2410 100 100 4
10 20 1 100 100
= Rs P
2
= Rs
= 2410
609P
10000 Now, according to the question,
11 4 12 2 = 2410 10 10
or, P
3P 609P = Rs 618 10000 100
K KUNDAN 2 2 2 11 12 = 2410 or, P 10 10
11 12 12 11 = 2410 or, P 10 10 10 10 2
2
121 12 121 12 or, P = 2410 100 10 100 10
609P - 300P = Rs 618 10000
or,
14.
or, 309P = Rs (618 × 10000) P = Rs 20000 Let the principal be Rs P. Then, Compound Interest
= Rs P 1
2
21 P 20
= Rs P
1 241 = 2410 100 100
or, P
or, P =
13.
2
441 P 400
= Rs P
2410 100 100
= 100000 241 Hence, the required sum = Rs 100000 Let the required sum of money be Rs P. Case I: Simple Interest =
2
5 P 100
P R T 100
P 3 1 3P = Rs = Rs 100 100 Case II: Compound Interest = Amount – Principal
= Rs
41P
400 Now, according to the question, 41P 400
= Rs 102.50
102.50 400 = Rs 1000 41
or, P = Rs Now,
K KUNDAN
= P 1
T
R P 100
Compound Interest = Rs 1000 1
4 1 100 2
Here, R = 6% per annum
6 3% per half-year 2
=
T = 1 year = 2 half-years (Since the interest is compounded half-yearly) Compound Interest 2 3 P 1 P = Rs 100
103 2 P = Rs P 100
26 2 1 25
= Rs 1000
676 1 = Rs 1000 625 676 - 625 625
= Rs 1000
= Rs 1000 = Rs 81.60
51 625
354 15.
Co n c ep t o f A r i t h m e t i c Let the share of A be Rs x and share of B be Rs (2523 – x ). According to the question, Amount received by A after 3 years at 5% interest 3
5 21 x 1 x 100 20
5
21 x 20 or, y 21 3 20
3
2
x 21 441 or, y 20 400
Similarly, amount received by B after 5 years
K KUNDAN
= 2523 x 1
2523 x
21
5 100
The required ratio = 441 : 400.
5
17.
5
R 8820 P1 100
20
=
Let the Principal be Rs P and the rate be R% per annum.
Again, according to the question, 3
21 21 2523 x 20 20
and 9261 P 1
5
x
5
21 x 20 or, 2523 x 21 3 20
or,
441 21 or, 2523 x 20 400
......... (ii)
equation
(i) gives
R 100
441 8820
441 100
= 5
8820
Using equation (i)
2
8820 P 1
or, 400x 441x 2523 441
or, 841x 2523 441 or, x =
100
3
R = 5%
or, 400x 2523 441 441x
2523 441
R
or, R =
2
x
1
8820
......... (i)
R 100
Now, equation (ii)
9261
2
or, 8820
1323
841 The share of A = Rs 1323 The share of B = Rs 2523 – 1323 = Rs 1200 16. Solve as the Q.No. 15. Try yourself. Alternative Method: Let the amount lent by A be Rs x and amount lent by B be Rs y . Now, according to the question, Amount received by A after 3 years at 5% interest
18.
2
5 21 P 100 20
441
P 400 or, P = Rs 8000 Principal = Rs 8000 and Rate = 5% We know that
T
A P 1
R 100
K KUNDAN
= x 1
5 100
3
21 20
3
5
5 21 = y = y 1 100 20
[where P is the present worth]
21 21 y 20 20
5
or, 4913 P 1
25 400
3
5
Again, according to the question, 3
3
25 or, 4913 P 1 4 ; 100
= x
Similarly, amount received by B after 5 years at 5% interest
x
4913 4913 or, P = 3 3 1 17 1 16 16
355
Co m p o u n d I n t e r e s t
4913 16 16 16
or, P = 19.
= 4096
17 17 17 Hence the present worth is Rs 4096. After 2 years amount at CI
2
= 4000 1
5 1 4000 1 100 20
Suppose X
3Y 2 2
X = k , Y =
3
2
4
k 4
k and Z =
X : Y : Z = k :
21 21 = Rs 4410 20 20 After 2 years the person repaid Rs 2210, hence the amount borrowed = Rs (4410 – 2210) = Rs 2200 Now, compound interest on Rs 2200 for 2 years at 5% per annum is
9Z
2 3
k :
4 9
9
k
k
K KUNDAN = 4000
21 20
X
3X
2 2 This becomes the principal for next years and after 10 years the X will be 2
3 3X
3 X 2 2 2
Similarly, after 15 years Y will be 3
3 3 3
10 100000 11
(100000 65000)
= Rs (2425.50 – 2200) = Rs 225.50 Total inter est paid by him = Rs (225.50 + 410) = Rs 635.50 2 0 . Rate of interest = 50% in 5 years Now, according to the question, X is lent for 10 years, Y for 15 years and Z for 20 years.
4
Two years earli er the person had
2200 20
X
:
(multiplying each term by 9) 21 . The pers on wit hdr aws Rs 110000. One year earlier the person had
21
After 5 years X becomes =
2
3 9 X : Y : Z = 9 : 6 : 4
Rs 110000
2
5 2200 Rs 2200 1 100 = Rs 2200
or, X : Y : Z = 1 :
3 Y 2 2 2 2
10
= Rs 150000 11 At the time of opening the account, the person ha d
(150000 125000)
10
= Rs 250000 11 2 2 . Difference between CI and SI = Rs 104 – Rs 100 = Rs 4 SI for 2 years = Rs 100
SI for 1 year = Rs
100
= Rs 50
2
Because interest is reckoned yearly, CI and SI for the first year will be the same. Rs 4 is the interest on Rs 50 for 1 year. Principal = Rs 50, T = 1 year and SI = Rs 4
Rate =
4 100
% = 8% 50 1 Now, SI = Rs 100 R = 8% and T = 2 years
K KUNDAN and after 20 years Z will be
3 3 3 3
4
3 Z 2
Z
2 2 2 2
Now, according to the question, 2
3
4
3 3 3 X Y Z 2 2 2 or,
9X 4
27Y 8
100 100 = Rs 625 P = Rs 8 2
23 . Let the principal be Rs P. Rate = R = 10% per annum =
16
Time = T = 1
9 4
) =
or, X
2
9Z 4
2
81Z
(dividing each term by
3Y
10
1 2
3 2
= 5% per half-year
years =
3 2
years
2 = 3 half-years
356
Co n c ep t o f A r i t h m e t i c
T
Compound Interest (CI) = A = P 1
or, CI = P 1
R P 100
R 1 100
or, 926.10 = 800 1
5 100
T
or,
3 5 1 = P 1 100
926.10 800
1 = 1 20
21 or, = 8000 10
2T
3
2T
9261
2T
2T
K KUNDAN 21 10
9261 - 1 = P 8000
or,
21 10
=
or, 2T = 3
1261 and = Rs P 8000
SI =
P R T 100
P 10 =
3 2
100
T =
= Rs
3P 20
Now, according to the question,
1261P 8000 or,
3P 20
= Rs 183
1261P 1200P 8000
3 2
1
2
Hence the required time = 1
1
years. 2 25 . Let the sum of money be Rs P and rate of interest = 8% per annum. Rate of interest compounded half-yearly = 4% Difference between CI and SI 2 P R 2 R = P 1 100 1 100
= Rs 183
183 8000 = Rs 24000 61
2R R 2R = P 1 100 100 100 1 2
or, P = Rs
required sum = Rs 24000 24 . Here, principal (P) = Rs 800 Rate (R) = 10% per annum =
1
R 100
2
= P
10
= 5% per half-year 2 Amount (A) = Rs 926.10 Let the time be T years. Time = 2T half-years Time
Rate Amount = Principal 1 100
4 100
2
or, Rs 30 = P
or, Rs 30 =
P 1
625 P = Rs (30 × 625) = Rs 18750
K KUNDAN