It takes 10 years for a $100 monthly bank deposit to amount to $15,528.23 when invested at 5% interest compounded monthly. Round to the nearest place value, then express the amount $15,528.23 in the form of a x 10^n where a is single digit
To solve this problem, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (monthly deposit)
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years
In this case, the principal amount is $100, the annual interest rate is 5% (or 0.05 as a decimal), interest is compounded monthly (so n = 12), and the investment period is 10 years (t = 10).
Substituting these values into the formula, we can solve for A:
A = 100(1 + 0.05/12)^(12*10)
A = 100(1 + 0.00416)^(120)
A = 100(1.00416)^(120)
A ≈ 15528.23
Therefore, the final amount after 10 years would be approximately $15,528.23.
Expressing this amount in the form of a x 10^n, we can see that the value of a is 1, and n is 4 (since we move the decimal point 4 places to the right):
$15,528.23 ≈ 1.552823 x 10^4
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future amount (15,528.23 in this case)
P = the principal or initial deposit ($100 in this case)
r = the interest rate (0.05 or 5% in this case)
n = the number of times interest is compounded per year (12 for monthly compounding in this case)
t = the number of years (10 in this case)
Plugging in the given values into the formula, we have:
15,528.23 = 100(1 + 0.05/12)^(12*10)
Simplifying the equation further:
155.2823 = (1 + 0.004167)^(120)
Taking the natural logarithm (ln) of both sides to isolate the exponent:
ln(155.2823) = ln[(1 + 0.004167)^(120)]
Using a calculator to compute the natural logarithm on the left side:
ln(155.2823) ≈ 5.043558
Now, divide both sides by 120:
5.043558/120 ≈ 0.042030
This is the value of (1 + 0.004167), so subtract 1 to find the value of r/n:
0.042030 - 1 ≈ -0.957970
Now, divide both sides by 0.004167:
-0.957970/0.004167 ≈ -230.374707
Multiplying both sides by -1, we have:
230.374707 ≈ 1/[(1 + 0.004167)^(12*10)]
Take the reciprocal on the right side:
1/230.374707 ≈ (1 + 0.004167)^(12*10)
This shows that (1 + 0.004167)^(12*10) is approximately equal to 0.004341260.
Subtracting 1, we find that the value of r/n is approximately 0.003341260.
Now, multiply r/n by 12 to find the value of r:
0.003341260 * 12 ≈ 0.040095
Thus, the interest rate r is approximately 0.040095 or 4.01%.
Therefore, 15,528.23 can be expressed as $1.552823 x 10^4, where a = 1.
To find the future value of a monthly bank deposit, we can use the formula for compound interest:
FV = P * (1 + r/n)^(nt)
Where:
FV = Future Value (the amount we are trying to find)
P = Principal (the monthly deposit)
r = Annual interest rate (expressed as a decimal)
n = Number of times interest is compounded per year
t = Time in years
In this case, we have:
P = $100
r = 5% = 0.05 (decimal)
n = 12 (compounded monthly)
t = 10 years
Substituting the values into the formula:
FV = 100 * (1 + 0.05/12)^(12*10)
Now we can calculate the future value:
FV = 100 * (1 + 0.00416667)^(120)
To round to the nearest place value, we will round the final answer to two decimal places.
FV ≈ 100 * (1.00416667)^(120) ≈ 15528.23
So, the future value of the monthly bank deposit after 10 years is approximately $15,528.23.
To express this amount in the form of a × 10^n:
We move the decimal point to the left until we have a single digit in front of it, while multiplying the number by 10 each time we move the decimal point. In this case, we need to move the decimal point four places to the left.
15,528.23 = 1.552823 × 10^4
Therefore, the amount $15,528.23 can be expressed as 1.552823 × 10^4, where 'a' is the single digit 1.