it takes 10 years for $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 times 10^n where a is single digit
To find the value of "a" in the form of a times 10^n, we first need to calculate the value of "a" by rounding 15,528.23 to the nearest place value.
Rounding 15,528.23 to the nearest place value gives us 15,528.
Now, we need to find "n". We know that the formula for calculating the future value of an investment with monthly compounding is:
FV = P * (1 + r)^n,
where FV is the future value, P is the monthly deposit, r is the interest rate per period (in this case, per month), and n is the number of periods.
Substituting the given values, we have:
15,528 = 100 * (1 + 0.05)^120.
Dividing both sides of the equation by 100, we get:
155.28 = (1.05)^120.
Taking the logarithm of both sides, we have:
log(155.28) = log((1.05)^120).
Using logarithm properties, we can bring down the exponent:
log(155.28) = 120 * log(1.05).
Dividing both sides by log(1.05), we get:
120 = log(155.28) / log(1.05).
Calculating this on a calculator, we find that:
n ≈ 91.24.
Since "n" represents the number of months, it should be a whole number. In this case, we can consider "n" as 91 months (ignoring decimal places).
Therefore, the value of "a" is 15,528 and the value of "n" is 91. So, the amount 15,528.23 can be expressed in the form of a times 10^n as:
15,528.23 ≈ 1.553 x 10^4.
To find the value of a monthly bank deposit after a certain number of years, we can use the formula for compound interest:
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 interest is compounded per year
t = number of years
In this case, the monthly deposit is $100, the interest rate is 5% (or 0.05 as a decimal), the interest is compounded monthly (n = 12), and the number of years is 10.
Plugging in these values, we have:
A = 100(1 + 0.05/12)^(12*10)
A = 100(1 + 0.0041667)^(120)
A = 100(1.0041667)^(120)
A ≈ 100(1.64701076767)
A ≈ 164.701076767
Rounding this amount to the nearest place value gives us $164.70.
Now, to express the amount $15,528.23 in the form of a times 10^n, we need to determine the values of a and n.
To do this, we can write 15,528.23 as:
15,528.23 = a * 10^n
To find a, we divide 15,528.23 by 10^n and round to the nearest single-digit value:
a ≈ 1.55
Now, we need to determine the value of n. We can do this by counting the number of places the decimal point must be moved to the left to get a value between 1 and 10.
In this case, we need to move the decimal point 4 places to the left:
15,528.23 → 1.552823
Therefore, we have:
15,528.23 ≈ 1.55 * 10^4
So, the amount 15,528.23 can be expressed as 1.55 * 10^4.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the future value of the investment
P is the principal amount (initial deposit)
r is the annual interest rate (expressed as a decimal)
n is the number of times that interest is compounded per year
t is the number of years
In this case, the principal amount is $100, the annual interest rate is 5% (or 0.05 as a decimal), the interest is compounded monthly (so n = 12), and the number of years is 10.
Plugging in these values into the formula, we can solve for A:
A = 100(1 + 0.05/12)^(12*10)
A = 100(1 + 0.0041667)^(120)
A = 100(1.0041667)^(120)
A = 100(1.647009)
A = 164.7009
Rounding to the nearest place value, the future value of the investment is $164.70.
To express this amount (15,528.23) in the form of a times 10^n, we need to determine the exponent to which we need to raise 10 to get a number close to 15,528.23.
We can do this by taking the logarithm of the number and counting the number of digits before the decimal point.
Using a scientific calculator or software, we can find that the logarithm of 15,528.23 is approximately 4.19. This means that 15,528.23 is equivalent to 4.19 times 10^n for some value of n.
To find the single-digit coefficient, a, we can divide 15,528.23 by 10 raised to the power of 4:
a = 15,528.23 / 10^4
a = 1.552823
Since we are looking for a single-digit coefficient, we can round this to 1.6 (to one decimal place).
Therefore, the amount 15,528.23 can be expressed as 1.6 times 10^4.