# Question bfa82

Jan 30, 2018

Please see the step process blow;

#### Explanation:

Given;

"Rate" rArr r = 4.5%

$\text{Principal} \Rightarrow p = p$

$\text{Compund} \Rightarrow n = \frac{1}{4}$ (quarterly $= \frac{3}{12}$)

$\text{Compund Interest} \Rightarrow A = 2 p$ (if the amount of d money doubles)

"No of years" rArr t = ?yrs#

Hence we solve;

$A = p {\left(1 + \frac{r}{n}\right)}^{n t}$

Inputing the values above;

$2 p = p {\left(1 + 4.5 \times 4\right)}^{\frac{1}{4} \times t}$

$2 p = p {\left(1 + 18\right)}^{\frac{t}{4}}$

$2 p = p {\left(19\right)}^{\frac{t}{4}}$

Divide both sides by $p$

$\frac{2 p}{p} = \frac{p {\left(19\right)}^{\frac{t}{4}}}{p}$

$\frac{2 \cancel{p}}{\cancel{p}} = \frac{\cancel{p} {\left(19\right)}^{\frac{t}{4}}}{\cancel{p}}$

$2 = {19}^{\frac{t}{4}}$

Multiply both sides by the power of $\frac{4}{t}$..

${2}^{\frac{4}{t}} = {19}^{\frac{t}{4} \times \frac{4}{t}}$

${2}^{\frac{4}{t}} = 19$

Take $\log$ of both sides..

$\log {2}^{\frac{4}{t}} = \log 19$

$\frac{4}{t} \log 2 = \log 19$

Divide both sides by $\log 2$

$\frac{\frac{4}{t} \log 2}{\log} 2 = \log \frac{19}{\log} 2$

$\frac{\frac{4}{t} \cancel{\log} 2}{\cancel{\log}} 2 = \log \frac{19}{\log} 2$

$\frac{4}{t} = \log \frac{19}{\log} 2$

$\frac{4}{t} = 4.248$

Cross multiplying..

$\frac{4}{t} = \frac{4.248}{1}$

$4 \times 1 = 4.248 \times t$

$4 = 4.248 t$

Divide both sides by $4.248$

$\frac{4}{4.248} = \frac{4.248 t}{4.248}$

$\frac{4}{4.248} = \frac{\cancel{4.248} t}{\cancel{4.248}}$

$\frac{4}{4.248} = t$

$t = 0.94 y r s \to \text{to the nearest tenth as required}$

Hope this helps!