it takes 10 years for a $100 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 a single digit
To find the interest rate, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the maturity value (amount) after time t
P = the principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, we know:
P = $100
A = $15,528.23
n = 12 (compounded monthly)
t = 10 years
So the formula becomes:
15,528.23 = 100(1 + r/12)^(12*10)
Now we need to solve for r.
Dividing both sides of the equation by 100 and raising both sides of the equation to the power of (1/(12*10)), we can isolate r:
(1 + r/12) = (15,528.23/100)^(1/(12*10))
(1 + r/12) = 1.055
Subtracting 1 from both sides of the equation:
r/12 = 1.055 - 1
r/12 = 0.055
Multiplying both sides of the equation by 12:
r = 0.055 * 12
r = 0.66
Therefore, the annual interest rate is 0.66, or 66% when expressed as a percentage.
To express 15,528.23 in the form of a x 10^n, we need to count the number of decimal places in the number.
15,528.23 has 2 decimal places.
So the number can be expressed as:
1.552823 x 10^4, where a = 1.
To find the amount 10 years from now, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future amount
P = the principal (initial deposit)
r = annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the principal (P) is $100, the interest rate (r) is 5% (or 0.05 as a decimal), the interest is compounded monthly (n = 12), and the number of years (t) is 10.
Let's plug in the values into the formula:
A = 100(1 + 0.05/12)^(12*10)
Calculating this expression, we find:
A ≈ 100(1.00416666667)^(120)
A ≈ 100(1.6470097359)
Rounding this to the nearest place value, we get:
A ≈ 100 * 1.65
A ≈ 165
Therefore, the amount after 10 years of compound interest is approximately $165.
Now, let's express $15,528.23 in the form of a x 10^n, where a is a single digit:
$15,528.23 can be written as 1.552823 × 10^4.
Note: The exponent is 4 because when converting a number to scientific notation, we move the decimal point to the right until there is one non-zero digit to the left of the decimal point. In this case, there are four places we had to move the decimal point.