Question

How long will it take money to double if it is invested at 8​% compounded semiannually question mark 7.6 % compounded​ continuously?

Answer 1

Answer with explanation:

Let A be the Initial Amount.

Initial rate of Interest =8 %

Let , t = Time Period.

Since rate of Interest is compounded semiannually.

So, New rate of Interest = 4 %

New time period =2 t

Formula for Amount

 $A=P\times [1 +\frac{R}{100}]^n$

Present value of Money at 8​% compounded semiannually

  $A=P \times [ 1+\frac{4}{100}]^{2 t}$

                         -------------------------------------------------(1)

Let after ,Time period =h, the value of money doubles at 7.6 % compounded​ continuously.

      $A=2P \times [ 1+\frac{7.6}{100}]^{h}$

                         -------------------------------------------------(2)

Equating (1) and (2)

$\Rightarrow A \times [ 1+\frac{4}{100}]^{2 t}=2A \times [ 1+\frac{7.6}{100}]^{h}\\\\\Rightarrow (1.04)^{2 t}=2 \times (1.076)^h$

where,t and h are integers.

Taking log on both sides

$\rightarrow 2 t \log (1.04)=\log 2 +h \log (1.076)\\\\ t=\frac{\log 2 +h \log (1.076)}{2 \log (1.04)}$

Replacing , t by y ,and h by x we get

$y=\frac{\log 2 +x \log (1.076)}{2 \log (1.04)}$

Substituting the values of log 2, log (1.076), and log (1.04) in the above equation

$y=\frac{0.6932 +x \times 0.07325}{2 \times 0.0393}\\\\ 0.786 y=0.6932+0.07325 x\\\\786 y=693.2 +73.25 x$

So, for distinct values of x we get distinct values of y.