Question

In what time the compound interest on 10,000 becomes 10404 at 6% per annum, when the
compounded quarterly?​

Answer 1

10,000 becomes 10,404 in 0.665 years at 6% compound interest per annum when compounded quarterly

Given

Principal = 10,000

Compound interest per annum = 6%

Period of compounding = Quarterly

Final Accrued amount = 10,404

To Find

By what time the principal amount becomes the accrued amount

Solution

The formula for calculating the final accrued amount using compound interest is $A=P(1+\frac{r}{n})^{nt}$ where

$A$ = Final accrued amount = 10,404

$P$ = Principal = 10,000

$r$ = Annual rate of interest = 6% = $\frac{6}{100}$

$n$ = Period of compounding = 4 (Since it is quarterly)

$t$ = Number of years

Here $t$ is the unknown and hence we are solving for $t$

Inputting the values, the equation becomes

$10404=10000(1+\frac{6}{400})^{4t}$

$1.0404=(1+\frac{6}{400} )^{4t}$

$1.0404=\frac{406}{400}^{4t}$

$1.0404=\frac{203}{200}^{4t}$

Now, above is an exponential equation. To solve this, we need to apply logarithm on both sides of the equation using the base as $\frac{203}{200}$. That is,

$log\frac{203}{200} (1.0404) = 4t$

Converting log of the base $\frac{203}{200}$ into the log of base 10,

$\frac{log 1.0404}{log\frac{203}{200}} = 4t$

$\frac{0.0172}{0.00646} = 4t$

$t=0.665$

Hence the amount 10,000 becomes 10,404 in 0.665 years at 6% compound interest per annum when compounded quarterly