Final answer:
The amount in the savings account on July 1 is $2,060. By January 1 of the next year, the amount is $4,121.80. The compound interest earned over the time period is $121.80.
Step-by-step explanation:
The subject is dealing with the concept of compound interest, which is a common financial concept covered in secondary mathematics.
For part a, the amount in the savings account on July 1, after the initial $2,000 has been compounded for six months, can be calculated using the formula for compound interest: A = P(1 + r/n)^(nt), where:
- P is the principal amount ($2,000),
- r is the annual interest rate (0.06 for 6%),
- n is the number of times the interest is compounded per year (2 for semiannual),
- t is the time the money is invested for, in years (0.5 years for 6 months).
The amount on July 1 will be: $2,000(1 + 0.06/2)^(2*0.5) or $2,000(1.03), which equals $2,060.
For part b, the amount on January 1 a year later would need to incorporate the second deposit of $2,000 made on July 1. The first $2,000 will have been compounded for one full year, and the second $2,000 will have compounded for six months. The total amount will be the sum of these two amounts: $2,000(1 + 0.06/2)^(2*1) for the first deposit and $2,000(1 + 0.06/2)^(2*0.5) for the second deposit. The calculation will be $2,000(1.03)^2 + $2,000(1.03), which equals $2,000*1.0609 + $2,000*1.03 or $4,121.80.
For part c, the compound interest can be found by subtracting the total principal amount ($4,000) from the total amount in the account on January 1 a year later ($4,121.80): $4,121.80 - $4,000 = $121.80.