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joseftan
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Tyro
Asked: 2021-04-07T11:50:29+00:00 In: ,

Compound Interest

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If Lovely received a total amount of ₱67,500 five years and three months after depositing ₱42,950 in an account, at what rate of interest converted monthly did her money earn?

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  2. acedstud
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    This answer was edited.

    The value (Fv) of an investment (P) made at an annual interest rate (r) compounded t times annually for a period of n years is given by Fv=P\left(1+\dfrac{r}{t}\right)^{nt}.

    Here, Fv = ₱67,500, P = ₱42,950, n = 5 years and 3 months = 5 + 3/12 years = 5.25 years, t = 12 because it is compounded monthly, that is 12 times a year. We want to find the rate, r.

    Thus, 67,500=42,950\left(1+\dfrac{r}{12}\right)^{5.25\times12}=42,950\left(1+\dfrac{r}{12}\right)^{63}.

    First, we divide both sides by 42950 to get \dfrac{67500}{42950}=\left(1+\dfrac{r}{12}\right)^{63}\Rightarrow\left(1+\dfrac{r}{12}\right)^{63}\approx1.571594878

    Taking the log of both sides, we have \ln\left(1+\dfrac{r}{12}\right)^{63}\approx\ln1.571594878\Rightarrow63\ln\left(1+\dfrac{r}{12}\right)\approx0.452090949.

    Next, we divide both sides by 63 to get \ln\left(1+\dfrac{r}{12}\right)\approx\dfrac{0.452090949}{63}\approx0.007176047.

    Next, we remove the log by taking the exponential of both sides to get e^{\ln\left(1+\dfrac{r}{12}\right)}\approx e^{0.007176047}\Rightarrow1+\dfrac{r}{12}\approx1.007201856.

    Next, we subtract 1 from both sides to get \dfrac{r}{12}\approx0.007201856.

    Finally, we multiply both sides by 12 to get r\approx12\times0.007201856\approx0.0864=8.64\%.

    Therefore, the rate of interest is 8.64%.

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