We deposited 7,500 dollars into a savings account which pays 3.6 percent annual interest, compound quarterly.How long after the money is deposited will the account balance be 9,750 dollars?What initial deposit will generate a balance of 12000 dollars in exactly 4 years and 6 months?
To answer the first question, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (balance)
P = the initial principal (deposit)
r = annual interest rate (in decimal)
n = number of times the interest is compounded per year
t = time in years
We know the following information:
P = $7,500
r = 0.036 (3.6% expressed as a decimal)
n = 4 (compounded quarterly)
A = $9,750
We need to find t, so we can rearrange the formula:
t = (log(A/P))/(n * log(1 + r/n))
Plugging in the values, we get:
t = (log(9,750/7,500))/(4 * log(1 + 0.036/4))
Using a calculator, we can find the value of t to be approximately 3.7 years.
Therefore, the account balance will be $9,750 around 3.7 years after the money is deposited.
Now, let's move on to the second question.
To calculate the initial deposit required to reach a balance of $12,000 in exactly 4 years and 6 months, we can rearrange the formula for compound interest:
P = A / (1 + r/n)^(nt)
Given the following information:
A = $12,000
r = (unknown)
n = (unknown)
t = 4.5 years
We are looking to find P, so we can rearrange and solve the formula:
P = 12,000 / (1 + r/n)^(4.5n)
Since we have two unknowns (r and n), we'll need more information to determine the exact values.
If you provide additional information, such as the compounding frequency (n) or an interest rate (r), we can calculate the initial deposit required.
To find out how long it will take for the account balance to reach $9,750, we can use the formula for compound interest:
A = P (1 + r/n)^(nt)
Where:
- A represents the account balance
- P represents the initial deposit
- r represents the annual interest rate (as a decimal)
- n represents the number of times interest is compounded per year
- t represents the number of years
In this case, we have:
A = $9,750
P = $7,500
r = 3.6% = 0.036 (as a decimal)
n = 4 (since it compounds quarterly)
t = ?
Substituting the given values into the formula, we have:
$9,750 = $7,500 (1 + 0.036/4)^(4t)
Now we can solve for t:
(1 + 0.009)^4t = $9,750/$7,500
1.009^(4t) = 1.3
Taking the natural logarithm (ln) of both sides:
ln(1.009^(4t)) = ln(1.3)
4t * ln(1.009) = ln(1.3)
4t = ln(1.3) / ln(1.009)
t = (ln(1.3) / ln(1.009)) / 4
Using a calculator, we find that t is approximately 10.89 years.
Therefore, it will take approximately 10.89 years for the account balance to reach $9,750.
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To find out what initial deposit will generate a balance of $12,000 in exactly 4 years and 6 months, we can use the same formula for compound interest:
A = P (1 + r/n)^(nt)
Where:
- A represents the account balance
- P represents the initial deposit
- r represents the annual interest rate (as a decimal)
- n represents the number of times interest is compounded per year
- t represents the number of years
In this case, we have:
A = $12,000
P = ?
r = same as previously mentioned (3.6% = 0.036)
n = same as previously mentioned (compounds quarterly)
t = 4 years and 6 months
To convert 4 years and 6 months to years, we divide by 12 months/year:
t = 4 + 6/12 = 4.5 years
Now we can substitute the given values into the formula:
$12,000 = P (1 + 0.036/4)^(4 * 4.5)
Simplifying further:
$12,000 = P (1.009)^18
P = $12,000 / 1.009^18
Using a calculator, we find that P is approximately $10,080.41.
Therefore, an initial deposit of approximately $10,080.41 will generate a balance of $12,000 in exactly 4 years and 6 months.