Compound Interest Questions - 1

What is the interest payable on a sum of Rs 2000, compounded annually at 18% p.a for 2 years?
  • Rs.27848
  • Rs.26848
  • Rs.6848
  • Rs.7848
  • Rs.8432
Explanation   

Principal = Rs.20000, Rate of interest = 18%, Time = 2 years.

When Interest Compounded Annually:

Amount = P $\displaystyle \left( 1 + \frac{\text{R}}{100}\right)^n $

= 22000 $\displaystyle \left( 1 + \frac{18}{100}\right)^2 = \left(\frac{118}{100}\right)^2 $

= 22000 $\displaystyle \left(\frac{59}{50}\right)^2 = 22000 \frac{59}{50} \times \frac{59}{50}$ = 27848.

Amount = Rs.27848

Interest payable = Amount – Principal = Rs.27848 – Rs.20000 = Rs.7848.

[Or] Direct method:


Principal
 
=
 
20000
 
 
1st year@ 18%
 
=
 
3600
 
 
23600
 
 
2nd year@18%
 
 
4248
 
 
Amount
 
=
 
27848

=> Interest payable = Rs.27848 – Rs.20000 = Rs.7848

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What is the compounded annual interest on a sum of Rs 18000 at 10% p.a. compounded half yearly for two year [Approx] ?
  • Rs.21879
  • Rs.20780
  • Rs.22344
  • Rs.22800
  • Rs.21800
Explanation   

P = Rs.18000; R = 10% half yearly; T = 2 years.

When Interest Compounded Half Yearly:

Amount = P $\displaystyle \left( 1 + \frac{\frac{\text{R}}{2}}{100}\right )^{2n} $

=> 18000 $\displaystyle \left( 1 + \frac{\frac{10}{2}}{100}\right )^{2 \times 2} $

=> 18000 $\displaystyle \left( 1 + \frac{5}{100}\right )^4 => 18000 \left(\frac{105}{100}\right )^{4} $

18000 $\displaystyle \times \frac{21}{20} \times \frac{21}{20} \times \frac{21}{20} \times \frac{21}{20} = \frac{1750329}{80}$ = 21879.125 or Rs.21879.

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What is the difference between the amount payable at simple interest and the amount payable in compound interest on a sum of Rs 25000 at 15% for two years ?
  • Rs.400
  • Rs.265
  • Rs.562.5
  • Rs.150
  • No difference
Explanation   

Simple Interest = $\displaystyle \frac{25000 \times 15 \times 2}{100}$ = 7500.

Total amount = 25000 + 7500 = Rs.32500


Compound interest
 
=
 
25000
 
 
1st year @15%
 
=
 
3750
 
 
28750
 
 
2nd year@15%
 
=
 
4312.5
 
 
Amount
 
=
 
33062.5

Difference => C.I – S.I = Rs.33062.5 – Rs.32500 = Rs.562.5

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At what rate of compound interest will a certain amount of money be 1.69 times in two years ?
  • 23%
  • 25%
  • 20%
  • 13%
  • 30%
Explanation   

R = $\displaystyle \left((1.69)^{\frac{1}{2}} - 1 \right) \times 100$ = [1.3 – 1] x 100 = 30 or 30%

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A certain sum of money at compound interest becomes Rs.22400 in 1 year and Rs.25088 in two years. What is the above mentioned sum ?
  • Rs.21000
  • Rs.20000
  • Rs.18000
  • Rs.19800
  • Rs.20500
Explanation   

Difference = Rs.25088 – Rs.22400 = Rs.2688.

Rate = $\displaystyle \frac{2688}{22400} \times$ 100 = 12% => $\displaystyle \frac{22400}{100 + 12} \times 100$

Amount = $\displaystyle \frac{22400}{112} \times$ 100 = 20000.

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At what rate of interest will a sum of money become $\displaystyle \frac{125}{64}$ in 3 years compounded annually ?
  • 15%
  • 35%
  • 20%
  • 5%
  • 25%
Explanation   

Let Principal be 100.

Amount = P $\displaystyle \left(1 + \frac{\text{R}^3}{100}\right) = \frac{125}{64} = \left(\frac{5}{4}\right)^3 $

I. When Interest Compounded Annually:

1 + $\displaystyle \frac{\text{R}^3}{100} = \left(1 + \frac{\text{R}}{100}\right) = \frac{5}{4}$

$\displaystyle \frac{100\text{R}}{100} = \frac{5}{4}$ = 400R = 500 = R = $\displaystyle \frac{500}{400} \times$ 100 = 125.

As amount = Principal + interest

Rate = Amount – Principal = 125 – 100 = 25 or 25%

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A sum of money lent at compound interest for two years at 20% p.a. fetches Rs.723 more if the interest is payable half yearly rather than annually. Find the sum of money ?
  • Rs.25000
  • Rs.30000
  • Rs.70000
  • Rs.35000
  • Rs.45000
Explanation   

Let Principal be "P".

P $\displaystyle \left(1 + \frac{20}{100}\right)^2 - \left(1 + \frac{\frac{20}{2}}{100}\right)^4$ = Rs.723.

P $\displaystyle \left(\frac{12}{10}\right)^2$ - P $\displaystyle \left(\frac{11}{10}\right)^4$ = Rs.723.

$\displaystyle \frac{144\text{P}}{100} - \frac{121\text{P}}{100} \times \frac{121\text{P}}{100}$ = Rs.723.

Difference = 146.41 – 144 = 2.41 = Rs.723.

Sum = $\displaystyle \frac{723}{2.41} \times$ 100 = Rs.30000.

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The difference between simple interest and compound interest at 15% p.a for two years on a sum of money is Rs.900. Find the amount ?
  • Rs.44000
  • Rs.30000
  • Rs.45000
  • Rs.33000
  • Rs.40000
Explanation   

Principal = $\displaystyle \frac{\text{Difference}}{\text{R}^2} \times 100^2$

Principal or Sum = $\displaystyle \frac{900}{15 \times 15} \times 100 \times 100$ = Rs.40000.

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A sum of money amounts to Rs.43200 in two years and Rs.51840 in 3 years at compound interest. Find the sum ?
  • Rs.24000
  • Rs.28000
  • Rs.32000
  • Rs.30000
  • Rs.34000
Explanation   

Let principal be "P'.

I. When Interest Compounded Annually: (for two years)

Amount = P $\displaystyle \left(1 + \frac{\text{R}}{100}\right)^2$ = Rs.43200.

When Interest Compounded Annually: (for three years)

Amount = P $\displaystyle \left(1 + \frac{\text{R}}{100}\right)^3$ = Rs.51840.

$\displaystyle \frac{51840}{43200} = \frac{P \left(1 + \frac{\text{R}}{100}\right)^3}{P \left(1 + \frac{\text{R}}{100}\right)^2} = \frac{6}{5} = 1 + \frac{\text{R}}{100}$

$\displaystyle \frac{\text{R}}{100} = \frac{6}{5} - 1 = \frac{6 – 5}{5} = \frac{1}{5}$ or 20%.

Sum = P $\displaystyle \left(1 + \frac{20}{100}\right)^2$ = 43200.

= P $\displaystyle \left(\frac{120}{100}\right)^2$ = 43200.

= P $\displaystyle \left(\frac{6}{5}\right)^2$ = 43200 = P = $\displaystyle \frac{5}{6} \times \frac{5}{6} \times$ 43200 = Rs.30000.

[or] Direct:

51840 – 43200 = 8640

$\displaystyle \frac{8640}{43200} \times$ 100 = 20%, Compounded amount in third year is Rs.43200, then compounded amount in 2nd year =

$\displaystyle \frac{43200}{120} \times$ 100 = 36000.

Principal or Sum = $\displaystyle \frac{36000}{120} \times$ 100 = Rs.30000.

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A sum of money doubles itself in 6 years at certain rate of compound interest. In how many years will the sum become 16 times its original value ?
  • 16 years
  • 30 years
  • 24 years
  • 42 years
  • 36 years
Explanation   

Let the amount be ‘P’, as it gets doubled for every 6 years, then

2 times in 6 years => 2P => in 12 years => 4P => 18 years => 8P

24 years => 16P or 16 times.

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Compound Interest Formulas


P = Principal => Is the amount Lent or Borrowed.

R = Rate of Interest => The extra amount payable on the Principal, from time to time or at a certain intervals.

n = Time => Is the period for which the amount is Lent or Borrowed. The time may be yearly, half yearly or quarterly.

Compound Interest: Interest accrued on Principal is again added to the principal and Rate of interest is calculated on accumulated or compounded amount, i.e this compounded amount again becomes principal.

A => is the total amount at the end of ‘n’ years.

Basic and Important Formulas:

I. When Interest Compounded Annually:

Amount = P $\displaystyle \left( 1 + \frac{\text{R}}{100}\right)^n $

II. When Interest Compounded Half Yearly:

Amount = P $\displaystyle \left( 1 + \frac{\frac{\text{R}}{2}}{100}\right)^{2n} $

III. When Interest Compounded Quarterly:

Amount = P $\displaystyle \left( 1 + \frac{\frac{\text{R}}{4}}{100}\right)^{4n} $

IV. When Interest Compounded Annually but time is in fraction, say 2 $\displaystyle \frac{1}{2}$ years.

Amount = P $\displaystyle \left( 1 + \frac{\text{R}}{100}\right)^{2} \times \left( 1 + \frac{\frac{1}{2} R}{100}\right) $

V. When Rates are different for different years. Say R1%, R2%, R3% for 1st, 2nd and 3rd year respectively.

Amount = P $\displaystyle \left( 1 + \frac{\text{R}_1}{100}\right) \ \left( 1 + \frac{\text{R}_2}{100}\right) \ \left( 1 + \frac{\text{R}_3}{100}\right) $

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