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You deposit $100 at the end of each quarter in a sinking fund earning 4% compounded quarterly. How many quarterly deposits must you make in order to reach your goal of saving $10,000? Round your answer off to the nearest whole number.

Guest Sep 16, 2014

Best Answer 

 #1
avatar+92805 
+5

This is the furure value of an ordinary annuity problem

$$\\S=R\left[\frac{(1+i)^n-1}{i}\right]\\\\
R=100\;\;i=0.04/4=0.01\;\;S=10000,\;\;n=? \;quarters\\\\\\
10000=100\left[\frac{(1.01)^n-1}{0.01}\right]\\\\
1=(1.01)^n-1\\\\
2=(1.01)^n\\\\
log2=log(1.01)^n\\\\
log2=nlog(1.01)\\\\
n=\frac{log2}{log1.01}\\\\$$

 

$${\frac{{log}_{10}\left({\mathtt{2}}\right)}{{log}_{10}\left({\mathtt{1.01}}\right)}} = {\mathtt{69.660\: \!716\: \!893\: \!574\: \!830\: \!3}}$$

 

It will take 70

 

70/4=17.5 years

Melody  Sep 16, 2014
 #1
avatar+92805 
+5
Best Answer

This is the furure value of an ordinary annuity problem

$$\\S=R\left[\frac{(1+i)^n-1}{i}\right]\\\\
R=100\;\;i=0.04/4=0.01\;\;S=10000,\;\;n=? \;quarters\\\\\\
10000=100\left[\frac{(1.01)^n-1}{0.01}\right]\\\\
1=(1.01)^n-1\\\\
2=(1.01)^n\\\\
log2=log(1.01)^n\\\\
log2=nlog(1.01)\\\\
n=\frac{log2}{log1.01}\\\\$$

 

$${\frac{{log}_{10}\left({\mathtt{2}}\right)}{{log}_{10}\left({\mathtt{1.01}}\right)}} = {\mathtt{69.660\: \!716\: \!893\: \!574\: \!830\: \!3}}$$

 

It will take 70

 

70/4=17.5 years

Melody  Sep 16, 2014

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