Question

Emmanuel invests $3600 and Kelsey invests $2400. Both investments eam 3.8%annual interest. How much longer will it take Kelsey's investment to reach $10,000than Emmanuel's investment?

Answer 1

EXPLANATION

We can build the following equations in order to represent the Emmanuel earnings:

$y_1=3600(1+0.038)^x$$y_2=10000$

Matching both expressions:

$10000=3,600(1+0.038)^x$$\mathrm{Switch\: sides}$$3600\mleft(1+0.038\mright)^x=10000$$\mathrm{Divide\: both\: sides\: by\: }3600$$\frac{3600\left(1+0.038\right)^x}{3600}=\frac{10000}{3600}$$\mathrm{Simplify}$$\mleft(1+0.038\mright)^x=\frac{25}{9}$$\mathrm{If\: }f\mleft(x\mright)=g\mleft(x\mright)\mathrm{,\: then\: }\ln \mleft(f\mleft(x\mright)\mright)=\ln \mleft(g\mleft(x\mright)\mright)$$\ln \mleft(\mleft(1+0.038\mright)^x\mright)=\ln \mleft(\frac{25}{9}\mright)$$\ln \mleft(\mleft(1+0.038\mright)^x\mright)=x\ln \mleft(1+0.038\mright)$$x\ln \mleft(1+0.038\mright)=\ln \mleft(\frac{25}{9}\mright)$$\mathrm{Divide\: both\: sides\: by\: }\ln \mleft(1.038\mright)$$\frac{x\ln\left(1+0.038\right)}{\ln\left(1.038\right)}=\frac{\ln\left(\frac{25}{9}\right)}{\ln\left(1.038\right)}$

Simplify:

$x=\frac{\ln\left(\frac{25}{9}\right)}{\ln\left(1.038\right)}$

In decimal form, this is equivalent to 27.39 years.

Now, applying the same reasoning to the Kelsey investment:

$y_1=2400(1+0.038)^x$$y_2=10000$

Matching both expressions:

$10000=2400(1+0.038)^x$$Switch\: sides$$2400\mleft(1+0.038\mright)^x=10000$$\mathrm{Divide\: both\: sides\: by\: }2400$$\frac{2400\left(1+0.038\right)^x}{2400}=\frac{10000}{2400}$$\mathrm{Simplify}$$\mleft(1+0.038\mright)^x=\frac{25}{6}$

Applying the exponent rule:

$\mathrm{If\: }f\mleft(x\mright)=g\mleft(x\mright)\mathrm{,\: then\: }\ln \mleft(f\mleft(x\mright)\mright)=\ln \mleft(g\mleft(x\mright)\mright)$$\ln \mleft(\mleft(1+0.038\mright)^x\mright)=\ln \mleft(\frac{25}{6}\mright)$

Apply log rule:

$\ln \mleft(\mleft(1+0.038\mright)^x\mright)=x\ln \mleft(1+0.038\mright)$$x\ln \mleft(1+0.038\mright)=\ln \mleft(\frac{25}{6}\mright)$$\mathrm{Divide\: both\: sides\: by\: }\ln \mleft(1.038\mright)$$\frac{x\ln\left(1+0.038\right)}{\ln\left(1.038\right)}=\frac{\ln\left(\frac{25}{6}\right)}{\ln\left(1.038\right)}$

Simplify:

$x=\frac{\ln\left(\frac{25}{6}\right)}{\ln\left(1.038\right)}$

Expressing in decimal form:

$x=38.26$

This is equivalent to 38.26 years to reach 10000 to the Kesley investment.

Comparing both investments:

Kesley Investment - Emmanuel Investment =

= 38.26 - 27.39 = 10.87

In conclusion, It will take 10.87 more years for Kelsey to reach $10,000