Question

Elizabeth deposits a total of $750 into her savings account each year. The account earns 2% interest compounded annually. The table below shows details for the first three years the account was open. How much interest was earned the 4th year?​

Answer 1

Answer:

Elizabeth earns $ 154.86 due to interests from the beginning to the fourth year and $ 62.75 due to interest only in the fourth year.

Step-by-step explanation:

Elizabeth deposits each year the same amount, which becomes greater due to composite interest. From statement, we notice the following formulas of recurrence:

Initial amount ($t = 0$)

$C = C_{o}$ (1)

First year ($t = 1$)

$C_{1} = C_{o}\cdot (1+r)+C_{o}$ (2)

Second year ($t = 2$)

$C_{2} = C_{1}\cdot (1+r)+C_{o}$ (3)

Third year ($t = 3$)

$C_{3} = C_{2}\cdot (1+r)+C_{o}$ (4)

Fourth year ($t = 4$)

$C_{4} = C_{2}\cdot (1+r)+C_{o}$ (5)

Where:

$C_{o}$ - Initial amount of Elizabeth's account, measured in US dollars.

$r$ - Interest rate ratio, dimensionless.

If we know that $r = 0.02$ and $C_{o} = \ \,750$, then the interest earned by Elizabeth in the 4th year is:

First year

$C_{r,1} = r\cdot C_{o}$ (6)

$C_{r,1} = (0.02)\cdot (\ \,750)$

$C_{r,1} = \ \,15$

Second year

$C_{1} = C_{o}\cdot (1+r)+C_{o}$

$C_{r,2} = r\cdot C_{1}$ (7)

$C_{1} = (\ \,750)\cdot (1+0.02)+\ \,750$

$C_{1} = \ 1515$

$C_{r,2} = (0.02)\cdot (\ \,1515)$

$C_{r,2} = \ \,30.3$

Third year

$C_{2} = C_{1}\cdot (1+r)+C_{o}$

$C_{r,3} = r\cdot C_{2}$ (8)

$C_{2} = (\ \,1515)\cdot (1+0.02)+\ \,750$

$C_{2} = \ \,2340.75$

$C_{r,3} = (0.02)\cdot (\ \,2340.75)$

$C_{r,3} = \ \,46.81$

Fourth year

$C_{3} = C_{2}\cdot (1+r) +C_{o}$

$C_{r,4} = r\cdot C_{3}$ (9)

$C_{3} = (\ \,2340.75)\cdot (1+0.02)+\ \,750$

$C_{3} = \ \,3137.56$

$C_{r,4} = (0.02)\cdot (\ \,3137.56)$

$C_{r,4} = \ \,62.75$

The money gained due to interest is determined by the following sum, that is:

$C_{r} = C_{r,1}+C_{r,2}+C_{r,3}+C_{r,4}$

$C_{r} = \ \,15+\ \,30.30+\ \,46.81+\ \,62.75$

$C_{r} = \ \,154.86$

Elizabeth earns $ 154.86 due to interests from the beginning to the fourth year and $ 62.75 due to interest only in the fourth year.