Question

Bank A offers an investment account where interest is earned continuously. The bank advertises that money invested in the account will double in \[8\] years.
(a)A certain amount of money is invested in the account at the start of the year \[2015\].For what year is the amount in the account three times the initial investment?
(b)A different bank B offers an investment account where interest is compounded (earned) every \[3\] months. The bank advertises that money in their investment account will grow at the same rate as the investment account at bank A. What is the annual rate (\[3\] significant figures) for the investment account at bank B.

Answer 1

Hint: In this question, we have to find out the required value from the given particulars.
We need to first use the simple interest formula to find the rate of interest for the bank A account so that we can find out the annual rate (\[3\] significant figures) for the investment account at bank B.

Formula used: i). \[S.I. = \dfrac{{P \times r \times t}}{{100}}\]
Where,
P =principle amount
r= rate of interest
t=time
ii). Effective annual interest formula
\[{i_a} = {\left( {1 + \dfrac{{\dfrac{r}{{100}}}}{M}} \right)^M} - 1\]
Where,
r=nominal interest rate
\[{i_a}\]=effective annual interest rate
M = number of interest period per year

Complete step-by-step answer:
(a) It is given that bank A offers an investment account where interest is earned continuously. The bank advertises that money invested in the account will double in \[8\] years.
We need to find out the year when the amount in the account will be three times the initial investment, invested in \[2015\].
Since the money invested in the account will double in \[8\] years, the interest amount after \[8\] years is \[2P - P = P\], Where, P =principle amount.
Thus it will take another \[8\] years to get the amount three times the initial investment.
Therefore the total years needed to get the amount three times the initial investment is \[8 + 8 = 16years\]
Hence in the year \[2015 + 16 = 2031\] the amount will be three times the initial investment.
(b) It is given that a different bank B offers an investment account where interest is compounded (earned) every \[3\] months. The bank advertises that money in their investment account will grow at the same rate as the investment account at bank A.
Now, we have the interest amount in bank A, after \[8\] years is \[2P - P = P\],
Where, P =principle amount.
Here, S.I = P after \[8\] years.
Therefore using the simple interest formula we get,
\[P = \dfrac{{P \times r \times 8}}{{100}}\]
\[r = \dfrac{{100}}{8} = 12.5\]
Thus for bank B the rate of interest is \[12.5\] compounded after every \[3\] months. .
Therefore the effective annual interest is
\[ \Rightarrow {\left( {1 + \dfrac{{\dfrac{{12.5}}{{100}}}}{{\dfrac{{12}}{3}}}} \right)^{\dfrac{{12}}{3}}} - 1\]
Simplifying we get,
\[ \Rightarrow {\left( {1 + \dfrac{{0.125}}{4}} \right)^4} - 1\]
Dividing the terms,
\[ \Rightarrow {\left( {1 + 0.03125} \right)^4} - 1\]
Adding the terms,
\[ \Rightarrow {\left( {1.03125} \right)^4} - 1\]
Taking the power value for the term,
\[ \Rightarrow 1.1309824 - 1\]
Subtracting the terms
\[ \Rightarrow 0.1309824\]
Hence, the annual rate (\[3\] significant figures) for the investment account at bank B is \[13.098\].

Note: The effective interest rate, effective annual interest rate, annual equivalent rate or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate and expressed as the equivalent interest rate if compound interest was payable annually in arrears.
Simple interest is calculated on the principal amount. Compound interest is calculated on the principal amount as well as the interest.