Concept: Simple & Compound Interest

Example:A sum of Rs 4,500 is lent for 6 years at 8% p.a. The S.I. and amount respectively are
(a) Rs. 2160, Rs. 6660
(b) Rs. 1775, Rs. 6275
(c) Rs. 1600, Rs. 6100
l(d) Rs. 1120, Rs. 5620

Solution:
Interest = $\frac{\mathrm{4500\; \times \; 8\; \times \; 6}}{100}$ = 2160

Amount = P + I = 4500 + 2160 = 6660

Example:In what time, a sum of money will triple itself at the rate of 10% p.a., interest calculated as S.I.
(a) 12 years
(b) 17 years
(c) 16 years
(d) 20 years
(e) None of these

Solution:
Let principal = P. Then, amount = 3P and interest = 2P.

⇒ Amount due = P$\left(1+\frac{\mathrm{r}\times \mathrm{T}}{100}\right)$

⇒ 3P = P$\left(1+\frac{10\times \mathrm{T}}{100}\right)$

⇒ 3 = 1 + $\frac{\mathrm{T}}{10}$

⇒ T = 20 years

Alternately,
1 year’s interest = 10% of P = 0.1P.

It would take for interest to become 2P = 2P/0.1P = 20 years.

Example:At what rate per cent, the simple interest on Rs. 2375 will be Rs. 475 in 5 years?
(a) 8%
(b) 6.5%
(c) 7%
(d) 3.5%
(e) 4%

Solution:
Simple Interest = $\frac{\mathrm{P\; \times \; r\; \times \; T}}{100}$

475 = $\frac{\mathrm{2375\; \times \; R\; \times \; 5}}{100}$, gives R = 4%.

Example:A sum of Rs 3600 is lent out in two parts in such a way that the interest on one part at 7 % for 4 years is equal to that on another part at 2% for 10 yrs.  Find the sum lent out at 7%.
(a) Rs 1500
(b) Rs 2100
(c) Rs 2000
(d) None of these

Solution:
Let P be the sum lent out at 7% and (3600 - P) be the sum lent out at 2%.

So, P × 7 × 4 = (3600 - P) × 2 × 10.

⇒ 7P = 18000 - 5P

⇒ P = Rs. 1500

Example:The simple interest on a sum of money is 1/4th of the principal and the number of years is equal to the rate % p.a. Then the rate % p.a. is
(a) 7.5
(b) 6
(c) 5
(d) None of these

Solution:
$\frac{\mathrm{P}}{4}$ = $\frac{\mathrm{P\; \times \; R\; \times \; R}}{100}$

⇒ R² = 25

⇒ R = 5%

Example:A sum becomes 6 times in 6 years at S.I. The same sum will become 26 times in how many years?
(a) 26 years
(b) 48 years
(c) 30 years
(d) 57 years
(e) None of these

Solution:
Let the principal be Rs. P. Now, amount due is 6P in 6 years

⇒ 6P = P$\left(1+\frac{\mathrm{r}\times 6}{100}\right)$

∴ r = 250/3%

Now, amount due will be 26P in let’s say t years

⇒ 26P = P$\left(1+\frac{\mathrm{r}\times \mathrm{t}}{100}\right)$

Substituting r = 250/3%

⇒ 26P = P$\left(1+\frac{250\times \mathrm{T}}{\mathrm{3\times 100}}\right)$

∴ t = 30 years.

Example:A sum becomes 6 times in 6 years at S.I. The same sum will become 26 times in how many years?
(a) 26 years
(b) 48 years
(c) 30 years
(d) 57 years
(e) None of these

Solution:
Let the principal be Rs. P. 

Interest accumulate in 6 years = 5P

∴ 5P interest in accumulated in 6 years

Hence, 25P interest will be accumulated in 6 × 5 = 30 years.

Hence, P will become 26P in 30 years.

This method can be applied here because in case of SI interest every year is same.

Example:A sum triples itself in nine years at SI. In how many years will the sum become nine times of itself at the same rate of interest?

Solution:
Let the principal be Rs. P. Interest accumulate in 9 years = 2P

∴ 2P interest in accumulated in 9 years

8P interest will be accumulated in 9 × 4 = 36 years.

Hence, P will become 9P in 36 years.

Example:What sum of money will yield Rs. 60 as simple interest at 6% p.a. in 5 years?

Solution: 
Total Interest = PRT/100

⇒ 60 = $\frac{\mathrm{P\; \times \; 6\; \times \; 5}}{100}$

⇒ Required sum = Rs.$\frac{100\times 60}{6\times 5}$ = Rs. 200

Example:At what rate of interest p.a. will a sum triple itself in 10 years?

Solution: 
AT = P$\left(1+\frac{\mathrm{r}\times \mathrm{T}}{100}\right)$

⇒ 3P = P$\left(1+\frac{\mathrm{r}\times 10}{100}\right)$

⇒ 3 = 1 + $\frac{\mathrm{r}}{10}$

∴ r = 20%

Example:An amount of money lent against simple interest at 4$\frac{1}{2}$% p.a. amounts to Rs. 381 after 6 years. Find the sum

Solution: 
381 = P$\left(1+\frac{4.5\times 6}{100}\right)$ = $\left(1+\frac{27}{100}\right)$ = $\frac{127}{100}$

∴ Required sum = Rs. 300

Example:A person had Rs. 2000 a part of which he lent at 5% p.a. and the rest at 4% p.a. The total interest he received annually was Rs. 96. How much did he lend at 4%?

Solution: 
Let the part lent at 4% be Rs. x

∴ the part lent at 5% is Rs. (2000 – x)

Now, 96 = $\frac{\mathrm{x\; \times \; 4\; \times \; 1}}{100}$ + $\frac{\mathrm{(2000\; -\; x)\; \times \; 5\; \times \; 1}}{100}$

⇒ 96 = $\frac{\mathrm{4x\; +\; 10000\; -\; 5x}}{100}$

⇒ 9600 = 10,000 - x

⇒ x = Rs. 400

Example:A certain sum of money at simple interest amounts to Rs. 379.50 in 3 years and to Rs. 453.75 in 7$\frac{1}{2}$ years, the rate being the same in the two cases. Find the rate of interest p.a.

Solution: 
Amount due after 3 years = Rs. 379.5 = P + $\frac{\mathrm{P\; \times \; r\; \times \; 3}}{100}$ ...(1)
and after 4 years = Rs. 453.75

⇒ Interest for 4.5 years = 435.75 - 379.50 = 74.25

∴ Interest for 3 year = 74.25 × $\frac{3}{4.5}$ = 49.5 ...(2)

Solving (1) and (2) we get P = Rs. 330

Using (2) we get 49.5 = $\frac{\mathrm{330\; \times \; r\; \times \; 3}}{100}$

⇒ r = 5%

Example:A man invests an amount of Rs. 15,860 in the names of his three sons A, B and C in such a way that they get the same amount after 2, 3 and 4 years respectively. If the rate of simple interest is 5% then find the ratio in which the amount was invested for A, B and C?

Solution: 
Let the three amounts be x, y and z respectively. 

According to the question:

x$\left(1+\frac{5\times 2}{100}\right)$ = y$\left(1+\frac{5\times 3}{100}\right)$ = z$\left(1+\frac{5\times 4}{100}\right)$

⇒ 110x = 115y = 120z

⇒ 22x = 23y = 24z

Dividing all the terms by LCM of 22, 23 and 24, we get

⇒ $\frac{\mathrm{x}}{552}$ = $\frac{\mathrm{y}}{528}$ = $\frac{\mathrm{z}}{506}$

∴ x : y : z = 552 : 528 : 506 = 276 : 264 : 253

Example:The simple interest on a certain sum of money at 4% p.a. for 4 years is Rs. 80 more than the interest on the sum for 3 years at 5% p.a. Find the sum.

Solution: 
Let the sum be Rs. x, then at 4% rate for 4 years the simple interest = $\frac{\mathrm{x\; \times \; 4\; \times \; 4}}{100}$ = Rs. $\frac{\mathrm{4x}}{25}$

At 5% rate for 3 years the simple interest = $\frac{\mathrm{x\; \times \; 5\; \times \; 3}}{100}$ = Rs. $\frac{\mathrm{3x}}{20}$

Now, We have, $\frac{\mathrm{4x}}{25}$ - $\frac{\mathrm{3x}}{20}$ = 80

⇒ 16x - 15x = 8000

∴ x = Rs. 8,000

Example:The difference between the interest received from two different banks on Rs. 500 for 2 years is Rs. 2.5. Find the difference between their rates.

Solution: 
I₁ = 500 × 2% × r₁ = 10r₁

I₂ = 500 × 2% × r₂ = 10r₂

I₁ - I₂ = 10r₁ - 10r₂ = 2.5

⇒ r₁ - r₂ = 0.25%

Example:What is the amount of Rs. 4,000 at 4% simple interest from Jan 10, 1984 to April 11, 1985?

Solution: 
Let us calculate the number of days from Jan 10, 1984 to April 11, 1985

Number of days from January 10, 1984 to December 31, 1984 = 366 – 10 = 356 days

(Since 1984 is a leap year)

For 1 Jan, 1985 till 11 April, 1985 = 31 (Jan) + 28 (Feb) + 31 (March) + 11 (April) = 101 days

∴ Total no. of days = 356 + 101 = 457

Simple interest = Principal × Rate% × Time

⇒ Simple Interest = $\frac{4000\times 4\times {\displaystyle \frac{457}{365}}}{100}$(Assuming 365 days in a year)

∴ S.I. = 14,624/73

∴ Amount = Principal + Simple Interest = 4000 + 200.33 = Rs. 4,200.33

Example:Three equal principals amount to Rs. 3,720 after 3, 4 and 5 years at simple interest 6% p.a., Find the principal.

Solution: 
Let each principal be Rs. P

Formula : Amount Due = $\left(1+\frac{\mathrm{r}\times \mathrm{T}}{100}\right)$

Amount after 3 years = P$\left(1+\frac{6\times 3}{100}\right)$ = P × $\frac{118}{100}$

Amount after 4 years = P$\left(1+\frac{6\times 4}{100}\right)$ = P × $\frac{124}{100}$

Amount after 5 years = P$\left(1+\frac{6\times 5}{100}\right)$ = P × $\frac{130}{100}$

∴ Total amount due = P × $\left(\frac{118}{100}+\frac{124}{100}+\frac{130}{100}\right)$ = P × $\frac{372}{100}$

⇒ 3720 = P × $\frac{372}{100}$

⇒ P = Rs. 1,000

∴ Total Principal = Rs. 3000

Example:What will be the C.I. on Rs. 2,000 for 3 yrs at 20% p.a.?
(a) Rs. 1331
(b) Rs. 1456
(c) Rs. 1900
(d) Rs. 1855
(e) None of these

Solution:
Amount due after 3 years = 2000${\left(1+\frac{20}{100}\right)}^3$ = Rs. 3456

CI = Amount due - Principal = 3456 - 2000 = Rs. 1456

Example:If C.I. for a certain sum for 2 years at 5% p.a. be Rs. 2050, what is the principal?
(a) Rs. 20,000
(b) Rs. 32,000
(c) Rs. 22,000
(d) Rs. 26,400
(e) None of these

Solution:
CI = 2050 = P${\left(1+\frac{5}{100}\right)}^2$ - P

⇒ 2050 = P[(1.05)² - 1]

⇒ 2050 = 1.1025P - P = 0.1025P

⇒ P = Rs. 20,000

Example:At what rate percent compounded yearly will Rs. 80,000 amount to Rs. 88,200 in 2 years?

Solution:
We have 80,000${\left(1+\frac{\mathrm{r}}{100}\right)}^2$ = 88,200

⇒ ${\left(1+\frac{\mathrm{r}}{100}\right)}^2$ = $\frac{882}{800}$ = $\frac{441}{400}$

⇒ 1 + $\frac{\mathrm{r}}{100}$ = $\frac{21}{20}$

∴ r = 5%

Example:What sum will become Rs. 6,690 after three years and Rs. 10,035 after six years on compound interest?

Solution: 
We know amount increases by r% every year.

6690${\left(1+\frac{\mathrm{r}}{100}\right)}^3$ = 10,035

⇒ ${\left(1+\frac{\mathrm{r}}{100}\right)}^3$ = $\frac{10035}{6690}$ = $\frac{3}{2}$ ...(1)

Now dividing the second equation by the first equation, we get :

Now, 6690 is the amount due after three years,
∴ 6690 = P${\left(1+\frac{\mathrm{r}}{100}\right)}^3$

⇒ 6690 = P × $\frac{3}{2}$

∴ P = Rs. 4,460

Example:If a sum placed at compound interest doubles itself in 4 years. In how many years will it be eight times?
(a) 10
    (b) 13
(c) 12
    (d) 11

Solution:
Amount due after T years = P${\left(1+\frac{\mathrm{r}}{100}\right)}^{\mathrm{T}}$

Amount doubles in 4 years, 

⇒ 2P = P${\left(1+\frac{\mathrm{r}}{100}\right)}^4$

∴ 2 = ${\left(1+\frac{\mathrm{r}}{100}\right)}^4$    …(1)

Now, amount has to be 8 times in let’s say t years.

⇒ 8P = P${\left(1+\frac{\mathrm{r}}{100}\right)}^{\mathrm{t}}$

⇒ 8 = ${\left(1+\frac{\mathrm{r}}{100}\right)}^{\mathrm{t}}$

⇒ 2³ = ${\left(1+\frac{\mathrm{r}}{100}\right)}^{\mathrm{t}}$

⇒ ${\left[{\left(1+\frac{\mathrm{r}}{100}\right)}^4\right]}^3$ = ${\left(1+\frac{\mathrm{r}}{100}\right)}^{\mathrm{t}}$

⇒ ${\left(1+\frac{\mathrm{r}}{100}\right)}^{12}$ = ${\left(1+\frac{\mathrm{r}}{100}\right)}^{\mathrm{t}}$

∴ t = 12 years

Note: There is not need to find the value of r separately in this question.

Example:If a sum placed at compound interest doubles itself in 4 years. In how many years will it be eight times?
(a) 10
    (b) 13
(c) 12
    (d) 11

Solution:
Let the sum be Rs 100

Here, 2 = ${\left(1+\frac{\mathrm{r}}{100}\right)}^4$ means the amount doubles every 4 years.

∴ in 4 yrs Rs 100 becomes Rs 200, and in next four years Rs 200 becomes Rs 400, and again in next four years Rs 400 becomes Rs 800.

So clearly, in 12 years, the sum becomes 8 times.

Example:Rs 4800 becomes Rs. 6000 in 4 years at a certain rate of compound interest. What will be the sum after 12 years?

Solution: 
We have 4800${\left(1+\frac{\mathrm{r}}{100}\right)}^4$ = 6000

⇒ ${\left(1+\frac{\mathrm{r}}{100}\right)}^4$ = 6000/4800 = 5/4

Hence, the amount becomes 5/4 times in 4 years.

In 12 (= 4 + 4 + 4) years it will become 5/4 × 5/4 × 5/4 = 125/64 times

∴ Amount due = 125/64 × 4800 = Rs. 9375

Example:If the rate of interest is 20% per annum compounded annually in what time approximately money doubles itself?
(a) 3 years
(b) 4 years
(c) 5 years
(d) 6 years

Solution:
Let us assume that we invest Re 1

So, after 1 year, it will be Rs 1.2

After two years, it will be Rs 1.44 = (1.2)²

After three years, it will be Rs 1.728 = (1.2)³

After four years, it will be Rs 2.0736 = (1.2)⁴

So clearly, after 4 years, the money almost doubles itself

Example:If CI for 5th year is Rs. 10,000 on a certain sum at the rate of 10% p.a. compounded annually, then what will be the compound interest for 9th year?

Solution:
We know in case of CI, interest for any particular year increases by r% every year. Since there are 4 compounding periods between 9th and 5th year.

⇒ CI for 9th year = CI for 5th year × ${\left(1+\frac{10}{100}\right)}^4$ = 10,000${\left(1+\frac{10}{100}\right)}^4$ = 14,641

Example:Divide Rs 21866 into two parts such that the amount of one in 3 years is same as the amount of second in 5 years. The rate of compound interest being 5% per annum.
(a) 11466, 10400
(b) 14466, 7400
(c) 14166, 7700
(d) None of these

Solution:
Let the amount lent for 5 years be x, hence amount lent for 3 years is (21866 - x)

∴ (21866 - x)${\left(1+\frac{5}{100}\right)}^3$ = x${\left(1+\frac{5}{100}\right)}^5$

⇒ 21866 - x = x${\left(1+\frac{5}{100}\right)}^2$

⇒ 21866 - x = 1.1025x

⇒ x = Rs. 10,400

∴ Amount lent for 5 years is Rs. 10,400 and amount lent for 3 years = Rs. 11,466

Example:Find compound interest on Rs. 100000 at 20% p.a. for 2 years 3 months compounded annually.

Solution: 
Amount due after 2 year = 100000${\left(1+\frac{20}{100}\right)}^2$ = 144000

Amount due after another 3 months = 144000 × $\left(1+\frac{20\times {\displaystyle \frac{3}{12}}}{100}\right)$ = 144000 × $\frac{21}{20}$ = 151200

∴ Compound Interest = 1,51,200 – 1,00,000 = Rs. 51,200.

Example:Find the C.I. on Rs. 1200 for 1$\frac{1}{2}$ years @ 12% p.a.

Solution: 
A = 1200$\left(1+\frac{12}{100}\right)$$\left(1+\frac{6}{100}\right)$ = 1424.64

∴ C.I. = 1424.64 – 1200 = Rs. 224.04

Example:What is the compound interest on Rs. 2000 for two years at 5% interest p.a. for the first year and at 10% interest p.a. for the 2nd year?

Solution:
Compound Interest = 2000$\left(1+\frac{5}{100}\right)$$\left(1+\frac{10}{100}\right)$ - 2000

= 2000 × $\frac{21}{20}$ × $\frac{11}{10}$ - 2000 = Rs. 310

Example:What sum of money at compound interest will amount to Rs. 2249.52 in 3 years, if the rate of interest is 3% for the first year, 4% for the second year, and 5% for the third year?

Solution: 
The general formula for such question is :A = P$\left(1+\frac{{\mathrm{r}}_1}{100}\right)$$\left(1+\frac{{\mathrm{r}}_2}{100}\right)$$\left(1+\frac{{\mathrm{r}}_3}{100}\right)$……

Where A = Amount, P = Principal and r₁, r₂, r₃ are the rate of interest for different years.

In the above case: 2249.52 = P$\left(1+\frac{3}{100}\right)$$\left(1+\frac{4}{100}\right)$$\left(1+\frac{5}{100}\right)$

⇒ 2249.52 = P × 1.03 × 1.04 × 1.05

⇒ P = $\frac{2249.52}{\mathrm{1.03\; \times \; 1.04\; \times \; 1.05}}$ = Rs. 2000

Example:The difference between the simple interest and the compound interest is Rs 8 for 2 years at the rate of 20% per annum. What is the sum?
(a) Rs. 350
(b) Rs. 470
(c) Rs. 200
(d) 190
(e) None of these

Solution:
Difference in total interest for 2 years in case of CI and SI = $\frac{\mathrm{P}{\mathrm{r}}^2}{{100}^2}$

⇒ 8 = $\frac{\mathrm{P}{20}^2}{{100}^2}$

∴ P = Rs. 200

Example:If the difference between SI for 2 years and CI for 2 years on the same sum and at the same rate of interest is 120. The difference between SI for 3 years and CI for 3 years on the same sum and at the same rate of interest is Rs366. Find the rate of interest?
(a) 10 %
(b) 5 %
(c) 20 %
(d) 12 %

Solution:
We know interest in case of SI is same for every year, while in case of CI increases by r% every year.

If x is the interest for first year, then we can make the following table for interest for each of the three years.

​​​​​​​

Different between CI and SI for 1st year for same sum at same rate is zero.

Difference between CI and SI for 2nd year = (x + r% of x) - x = 120

⇒ r% of x = 120 ...(1)

Difference between CI and SI for 3rd year = (x + 120 + r% of (x + 120) - x

Also, as per given in the question, difference in CI and SI for 3rd year = 366 – 120 = 246.

⇒ x + 120 + r% of (x + 120) - x = 246

⇒ x + 120 + 120 + r% of 120 - x = 246 [From (1): r% of x = 120]

⇒ r% of 120 = 6

⇒ r = 5%

Example:Find the difference between S.I. and C.I. on Rs. 1200 at the rate of 10% for 3 yrs.
(a) Rs. 18.75
(b) Rs. 37.20
(c) Rs. 23.80
(d) Rs. 28.50
(e) None of these

Solution:
CI for 3 years = 1200${\left(1+\frac{10}{100}\right)}^3$ - 1200 = 397.2

SI for 3 years = (1200 × 10 × 3)/100 = 360

∴ CI - SI = 397.2 - 360 = Rs. 37.2

Example: If the CI on a certain sum for 2 years at 3% be Rs. 101.50. what would be the S.I.?

Solution: 
Let the prinicipale be P

CI for 2 years = P${\left(1+\frac{3}{100}\right)}^2$ - P = 101.50

⇒ 101.50 = 1.0609P - P = 0.0609P

⇒ P = Rs. 5000/3

Simple interest = $\frac{\mathrm{5000/3\; \times \; 3\; \times \; 2}}{100}$ = Rs. 100