Lisa Richter deposited $5,000 at 4% compounded semiannually for three years. At the beginning of the fourth year, Lisa deposited $2500 What would her balance earning 4% compounded semiannually?

a) $8,78726

b) $8,801.05

c) $8,800

d) $8,424

Answer is option (d) $8,424

Explanation:

The total amount Lisa Richter will receive after the end of the first 3 years can be expressed using the formula for calculating the total amount after 3 years with interest compounded semiannually is as follows;

A = P (1 + r/n) ^ (nt)

where;

A = the future value of the initial investment

P = initial investment amount/principal amount

r = the annual interest rate

n = the number of times that interest is compounded per unit t

t = the time the money is invested for

In our case;

P=$5,000

r=4/100=0.04

n=interest is compounded semiannually which is twice a year=2

t=3 years

Replacing values in the formula;

A=5,000 (1+0.04/2) ^ (2*3)

A=5,000 (1+0.02) ^6

A=5,000 (1.02) ^6

A=$5,630.81

Total amount after the end of the first 3 years=$5,630.81

To calculate the total principal amount for the fourth year;

Total principal amount=Total amount after 3 years+Amount deposited at the beginning of the fourth year

where;

Total amount after 3 years=$5,630.81

Amount deposited at the beginning of the fourth year=$2500

replacing;

Total principal amount = (5,630.81+2500) = 8,130.81

Using the formula;

A = P (1 + r/n) ^ (nt)

P=$8,130.81

r=4/100=0.04

n=compounded semiannually=2

t=1 year

replacing;

A=8,130.81 (1+0.04/2) ^ (2*1)

A=8,130.81 (1.02) ^2

A=8,459.29

Option (d) $8,424 is closer to 8,459.29, take answer as (d) $8,424