Answer:
40%.
Step-by-step explanation:
We have been given that an amount of $100 compounded annually is increased to $196 in 2 years. We are asked to find the interest rate.
We will use compound interest formula to solve our given problem.
, where,
A = Final amount,
P = Principal amount,
r = Annual interest rate in decimal form,
n = Number of times interest is compounded per year,
t = Time in years.
Upon substituting our given values in above formula, we will get:
![196=100(1+\frac{r}{1})^{1*2}](https://tex.z-dn.net/?f=196%3D100%281%2B%5Cfrac%7Br%7D%7B1%7D%29%5E%7B1%2A2%7D)
![196=100(1+r)^{2}](https://tex.z-dn.net/?f=196%3D100%281%2Br%29%5E%7B2%7D)
![\frac{196}{100}=\frac{100(1+r)^{2}}{100}\\\\1.96=(1+r)^2](https://tex.z-dn.net/?f=%5Cfrac%7B196%7D%7B100%7D%3D%5Cfrac%7B100%281%2Br%29%5E%7B2%7D%7D%7B100%7D%5C%5C%5C%5C1.96%3D%281%2Br%29%5E2)
![(1+r)^2=1.96](https://tex.z-dn.net/?f=%281%2Br%29%5E2%3D1.96)
Take positive square root of both sides:
![\sqrt{(1+r)^2}=\sqrt{1.96}](https://tex.z-dn.net/?f=%5Csqrt%7B%281%2Br%29%5E2%7D%3D%5Csqrt%7B1.96%7D)
![1+r=1.4\\\\1-1+r=1.4-1\\\\r=0.4](https://tex.z-dn.net/?f=1%2Br%3D1.4%5C%5C%5C%5C1-1%2Br%3D1.4-1%5C%5C%5C%5Cr%3D0.4)
Since interest rate is in decimal, form, so we will convert it into percentage as:
![0.4\times 100\%=40\%](https://tex.z-dn.net/?f=0.4%5Ctimes%20100%5C%25%3D40%5C%25)
Therefore, the interest rate was 40%.