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[tex]K=\$10,000\ \ \ and\ \ \ Sum=K\cdot(1+ \frac{p}{100})^n\\----------------------- \\1)\ \ \ 7\%\ quarterly\\\\ \ \Rightarrow\ \ \frac{1}{4} \cdot7\% =1.75\%\ \ annually\ \ \Rightarrow\ \ p=1.75\\\\ quarterly\ \Rightarrow\ \ 4\ times\ annually\ \Rightarrow\ \ 16\ times\ in\ 4\ years\ \Rightarrow\ \ n=16\\\\Sum(7\%)=\$10,000\cdot(1+ \frac{1.75}{100})^{16}=\\ \\.\ \ \ \ \ \ \ \ \ \ \ \ =\$10,000\cdot(1+0,0175)^{16}\approx\$13199.29\\------------------------\\[/tex]
[tex]2)\ \ \ 6.94\%\ daily\\\\ \ \Rightarrow\ \ \frac{1}{365} \cdot6.94\% \approx0.019\%\ \ annually\ \ \Rightarrow\ \ p=0.019\\\\ daily\ \Rightarrow\ 365\ times\ annually\ \Rightarrow\ 1420\ times\ in\ 4\ years\ \Rightarrow\ n=1420\\\\Sum(6.94\%)=\$10,000\cdot(1+ \frac{0.019}{100})^{1420}=\\ \\.\ \ \ \ \ \ \ \ \ \ \ \ =\$10,000\cdot(1+0,00019)^{1420}\approx\$13096.69\\---------------------------\\\$13,199.29 > \$13,096.69\\\\Ans.\ the\ larger\ amount\ gives\ the\ compounded\ quarterly.[/tex]
[tex]2)\ \ \ 6.94\%\ daily\\\\ \ \Rightarrow\ \ \frac{1}{365} \cdot6.94\% \approx0.019\%\ \ annually\ \ \Rightarrow\ \ p=0.019\\\\ daily\ \Rightarrow\ 365\ times\ annually\ \Rightarrow\ 1420\ times\ in\ 4\ years\ \Rightarrow\ n=1420\\\\Sum(6.94\%)=\$10,000\cdot(1+ \frac{0.019}{100})^{1420}=\\ \\.\ \ \ \ \ \ \ \ \ \ \ \ =\$10,000\cdot(1+0,00019)^{1420}\approx\$13096.69\\---------------------------\\\$13,199.29 > \$13,096.69\\\\Ans.\ the\ larger\ amount\ gives\ the\ compounded\ quarterly.[/tex]
7% compounded quarterly > 6.94% compounded continuously.
What is compound interest?
Interest earned on the principal amount and the interest itself is known as compound interest. These increases exponentially.
How to solve?
1) 7% compounded quarterly
[tex]\frac{7}{4}% = 1.75%[/tex] %= 1.75% annually
quarterly for 4 years => 4*4 = 16 times
Accumulated value = present value * [tex](1+\frac{r}{100})}^n[/tex]
where present value = $10,000 , r = 1.75, n = 16 times
substituting values:
AV = [tex]10000*{(1+\frac{1.75}{100})}^{16} = $13199.295[/tex]
Thus, the value of $10,000 after 4 years at 7% compounded quarterly is $13199.295
2) 6.94% compounded continuously
[tex]\frac{6.94}{365}[/tex]% = 0.01904% per annum
365 days for 4 years => 365*4 = 1460 times
Accumulated value = present value * [tex](1+\frac{r}{100})}^n[/tex]
Where present value = $10,000, r = 0.019014 , n = 1460 times
Substituting values:
[tex]10000*{(1+\frac{0.019014}{100})}^{1460} = $ 13199.29085[/tex]
Thus, the value of $10,000 after 4 years at % compounded quarterly is $13199.29085
since $13199.295 > $13199.29085
Both values are approximately the same but the value of 7% compounded quarterly is comparatively more than 6.94% compounded continuously.
Formula used:
Accumulated value = present value * [tex](1+\frac{r}{100})}^n[/tex]
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