Answer:
2.96% (2 d.p.) difference
Step-by-step explanation:
Compound Interest Formula
[tex]\large \text{$ \sf I=P\left(1+\frac{r}{n}\right)^{nt} -P$}[/tex]
where:
- I = total interest
- P = principal amount
- r = interest rate (in decimal form)
- n = number of times interest applied per time period
- t = number of time periods elapsed
Interest earned in Year 1 (Compounded semi-annually)
Given:
- P = 80000
- r = 12% = 0.12
- n = 2 (semi-annually)
- t = 1 year
Substitute the values into the formula:
[tex]\implies \sf I=80000\left(1+\frac{0.12}{2}\right)^{2(1)} -80000[/tex]
[tex]\implies \sf I=80000\left(1.06\right)^{2} -80000[/tex]
[tex]\implies \sf I=80000\left(1.1236\right) -80000[/tex]
[tex]\implies \sf I= 89888-80000[/tex]
[tex]\implies \sf I= 9888[/tex]
Interest earned in Year 2 (Compounded semi-annually)
Given:
- P = 80000 + 9888
- r = 12% = 0.12
- n = 2 (semi-annually)
- t = 1 year
Substitute the values into the formula:
[tex]\implies \sf I=89888\left(1+\frac{0.12}{2}\right)^{2(1)} -89888[/tex]
[tex]\implies \sf I=89888\left(1.06\right)^{2} -89888[/tex]
[tex]\implies \sf I=89888\left(1.2636\right) -89888[/tex]
[tex]\implies \sf I= 100998.156...-89888[/tex]
[tex]\implies \sf I = 11110.16[/tex]
Interest earned in Year 2 (compounded annually)
Given:
- P = 80000 + 9888 = 89888
- r = 12% = 0.12
- n = 1 (annually)
- t = 1 year
Substitute the values into the formula:
[tex]\implies \sf I=89888\left(1+\frac{0.12}{1}\right)^{1(1)} -89888[/tex]
[tex]\implies \sf I=89888\left(1.12\right) -89888[/tex]
[tex]\implies \sf I=100674.56 -89888[/tex]
[tex]\implies \sf I=10786.56[/tex]
Difference between second year interests
Interest earned in Year 2 (Compounded semi-annually) = 11110.16
Interest earned in Year 2 (compounded annually) = 10786.56
Percentage difference:
[tex]\sf p=\dfrac{|a-b|}{(a+b) \div 2} \times 100[/tex]
where:
[tex]\sf \implies p=\dfrac{|11110.16-10786.56|}{(11110.16+10786.56) \div 2} \times 100[/tex]
[tex]\sf \implies p=\dfrac{323.6}{10948.36} \times 100[/tex]
[tex]\sf \implies p=2.95569...[/tex]
[tex]\implies \sf p=2.96\% \:\: (2 \:d.p.)[/tex]
Learn more about compound interest here:
https://brainly.com/question/27747709
https://brainly.com/question/27806277
