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Important Formulas(Part 1) - Compound Interest

Introduction

In simple interest, interest is calculated on the initial principal and interest remains same each year. In compound interest, interest for each period is added to the principal before interest is calculated for the next period. These periods can be of any time duration. In other words, the principal grows as the interest gets added to it. Following examples make these concepts clear.

Note: Whenever the term compound interest is used without specifying the period in which the interest is compounded, it is assumed that interest is compounded annually.

Examples

  • Find the amount and simple interest on $₹100$ at $20\%$ per annum for $2$ years.

    Simple interest for first year $=20\%\text{ of }100=20$
    Simple interest for second year $=20\%\text{ of }100=20$

    Simple interest for $2$ years $=20+20=40$
    Amount after $2$ years $=100+40=140$

  • Find the amount and compound interest on $₹100$ at $20\%$ per annum for $2$ years, when the interest is compounded annually.

    Initial amount $=100$

    Simple interest for first year $=\dfrac{100×20×1}{100}=20$
    Amount after first year $=100+20=120$

    Simple interest for second year $=\dfrac{120×20×1}{100}=24$
    Amount after second year $=120+24=144$

    Compound interest for $2$ years $=144-100=44$
    (or, $20+24=44$)

  • Find the amount and compound interest on $₹100$ at $20\%$ per annum for $2$ years, when the interest is compounded half-yearly.

    Initial amount $=100$

    Simple interest for $6$ months $=\dfrac{100×20×\dfrac{1}{2}}{100}=10$
    Amount after $6$ months $=100+10=110$

    Simple interest for next $6$ months $=\dfrac{110×20×\dfrac{1}{2}}{100}=11$
    Amount after $1$ year $=110+11=121$

    Simple interest for next $6$ months $=\dfrac{121×20×\dfrac{1}{2}}{100}=12.1$
    Amount after $1$ year $6$ months $=121+12.1=133.1$

    Simple interest for next $6$ months $=\dfrac{133.1×20×\dfrac{1}{2}}{100}=13.31$
    Amount after $2$ years $=133.1+13.31=146.41$

    Compound interest for $2$ years $=146.41-100=46.41$
    (or, $10+11+12.1+13.31=46.41$)

Formulas

Let principal = P, rate = R% per annum, time = T years, amount due after T years = A

  • If interest is compounded annually,
    $\text{A}=\text{P}\left(1+\dfrac{\text{R}}{100}\right)^\text{T}$
    Note: $\text{T}$ must be a whole number
  • If interest is compounded half-yearly,
    $\text{A}=\text{P}\left(1 + \dfrac{\text{R/2}}{100}\right)^\text{2T}$
    Note: $2\text{T}$ must be a whole number
  • If interest is compounded quarterly,
    $\text{A}=\text{P}\left(1 + \dfrac{\text{R/4}}{100}\right)^\text{4T}$
    Note: $4\text{T}$ must be a whole number
  • If interest is compounded monthly,
    $\text{A}=\text{P}\left(1 + \dfrac{\text{R/12}}{100}\right)^\text{12T}$
    Note: $12\text{T}$ must be a whole number
  • If rates are different for different years, say $\text{R}_1\%,\text{R}_2\%$ and $\text{R}_3\%$ for $1\text{st},2\text{nd}$ and $3\text{rd}$ year respectively,
    $\text{A}=\text{P}\left(1+\dfrac{\text{R}_1}{100}\right)\left(1+\dfrac{\text{R}_2}{100}\right)\left(1+\dfrac{\text{R}_3}{100}\right)$
Compound Interest(CI) $=\text{A}-\text{P}$

Examples(Using Formulas)

  • Find the amount and simple interest on $₹100$ at $20\%$ per annum for $2$ years.

    Simple interest for $2$ years $=\dfrac{100×20×2}{100}=40$
    Amount after $2$ years $=100+40=140$

  • Find the amount and the compound interest on $₹100$ at $20\%$ per annum for $2$ years, if the interest is compounded annually.

    Amount after $2$ years $=100\left(1+\dfrac{20}{100}\right)^2=144$
    Compound interest $=144-100=44$

  • Find the amount and the compound interest on $₹100$ at $20\%$ per annum for $2\dfrac{1}{2}$ years, if the interest is compounded annually.

    Interest is compounded annually and we need to find amount after $2\dfrac{1}{2}$ years. Therefore, we will first find amount after $2$ years using the formula and then add it with the simple interest for remaining $\dfrac{1}{2}$ year.

    Amount after $2$ years $=100\left(1+\dfrac{20}{100}\right)^2=144$

    Simple interest for $\dfrac{1}{2}$ year $=\dfrac{144×20×\dfrac{1}{2}}{100}=14.4$

    Amount after $2\dfrac{1}{2}$ years $=144+14.4=158.4$
    Compound interest $=158.4-100=58.4$

  • Find the amount and the compound interest on $₹100$ at $20\%$ per annum for $2$ years, if the interest is compounded half-yearly.

    Amount after $2$ years $=100\left(1+\dfrac{20/2}{100}\right)^{2×2}=146.41$
    Compound interest $=146.41-100=46.41$

  • Find the amount and the compound interest on $₹100$ at $20\%$ per annum for $2\dfrac{1}{2}$ years, if the interest is compounded half-yearly.

    Amount after $2\dfrac{1}{2}$ years $=100\left(1+\dfrac{20/2}{100}\right)^{2×\frac{5}{2}}=161.05$
    Compound interest $=161.05-100=61.05$

  • Find the amount and the compound interest on $₹100$ at $20\%$ per annum for $2\dfrac{3}{4}$ years, if the interest is compounded half-yearly.

    Interest is compounded half-yearly and we need to find amount after $2\dfrac{3}{4}$ years (that is, $2$ year $9$ months). Therefore, we will first find amount after $2\dfrac{1}{2}$ years using the formula and then add it with the simple interest for remaining $3$ months.

    Amount after $2\dfrac{1}{2}$ years $=100\left(1+\dfrac{20/2}{100}\right)^{2×\frac{5}{2}}=161.05$

    Simple interest for $3$ months $=\dfrac{161.05×20×\dfrac{3}{12}}{100}=8.05$

    Amount after $2\dfrac{3}{4}$ years $=161.05+8.05=169.1$
    Compound interest $=169.1-100=69.1$

Present Worth

The formulas explained above can also be used to find present worth of an amount.

Let present worth of $x$ due $\text{T}$ years hence $=\text{PW},$ rate $=\text{R}\%$ per annum

  • If interest is compounded annually,
    $\text{PW}=\dfrac{x}{\left(1+\dfrac{\text{R}}{100}\right)^\text{T}}$
    Note: $\text{T}$ must be a whole number
  • If interest is compounded half-yearly,
    $\text{PW}=\dfrac{x}{\left(1+\dfrac{\text{R}/2}{100}\right)^\text{2T}}$
    Note: $2\text{T}$ must be a whole number
  • If interest is compounded quarterly,
    $\text{PW}=\dfrac{x}{\left(1+\dfrac{\text{R}/4}{100}\right)^\text{4T}}$
    Note: $4\text{T}$ must be a whole number
  • If interest is compounded monthly,
    $\text{PW}=\dfrac{x}{\left(1+\dfrac{\text{R}/12}{100}\right)^\text{12T}}$
    Note: $12\text{T}$ must be a whole number

Comments(3)

profileshiv dhiman
2015-04-03 05:29:00 
An electronic type writer worth Rs 12000 deprecates @ 10% P.A.ultimately it was sold for Rs 200.Estimate its effective life during which it was in use ?
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profileManish
2016-10-29 20:23:48 
Value of type writer becomes 200 after 38.9 years. 
200=12000 * (90/100 )^n
1/60=(9/10)^n
Apply log both sides, we get
log(1/60) = n * log(9/10)
-1.7781 = n * -0.0457
38.9 = n
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profileDev
2015-04-08 18:44:03 
Question is not clear. Straight line depreciation or Diminishing value depreciation ?
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