Question

A sum of $1000 was invested for 4 years, and the interest was compounded semiannually. If this sum amounted to $1459.54 in the given time, what was the interest rate? (Round your answer to two decimal places.)

Answer 1

The interest rate is 7.25%.

Step-by-step explanation:

To solve this problem, we can use the compound interest formula:

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

Where:

• A = the final amount
• P = the initial investment
• r = the interest rate
• n = the number of times interest is compounded per year
• t = the number of years

In this case, we have:

• P = $1000
• A = $1459.54
• n = 2 (since interest is compounded semiannually)
• t = 4 years

Substituting these values into the formula:

Taking the 8th root of both sides:

1 + r/2 = 1.03627

Subtracting 1 from both sides:

r/2 = 0.03627

Finally, solving for r:

r = 0.07254

Multiplying by 100 to get the percentage:

r = 7.25%

Answer 2

`A=P(1+ (r)/(n))^(nt)`
A=furuter aount
P=present amount
r=rate in decimal
n=number of times per year it is compounded
t=time in years


A=1459.54
P=1000
r=r
n=2 (semianually is 2 times per year)
t=4


`1459.54=1000(1+ (r)/(2))^((2)(4))`

`1459.54=1000(1+ (r)/(2))^(8)`
divide both sides by 1000

`1.45954=(1+ (r)/(2))^(8)`

`1.45954=((2+r)/(2))^(8)`

`1.45954=((2+r)^8)/(2^8)`

`1.45954=((2+r)^8)/(256)`
times both sides by 256
373.64224=
`(2+r)^8`
take the 8th root of both sides

`\sqrt[8]{373.64224} =2+r`
minus 2 from both sides

`\sqrt[8]{373.64224} -2=r`
aprox
0.0968=r
9.67%