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slega [8]
2 years ago
9

Elizabeth deposits a total of $750 into her savings account each year. The account earns 2% interest compounded annually. The ta

ble below shows details for the first three years the account was open. How much interest was earned the 4th year?​
Mathematics
1 answer:
just olya [345]2 years ago
7 0

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.

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