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×10@n
where a
is single digit.
To calculate the future value of a monthly bank deposit, we can use the formula:
FV = P * ((1 + r)^nt - 1) / r
Where:
FV = Future Value
P = Monthly deposit
r = Interest rate per period (monthly in this case)
n = Number of periods (in this case, the number of months)
t = Time in years
In this case, we know:
P = $100
r = 5% = 0.05 (monthly interest rate)
n = 10 years = 10 * 12 = 120 months
Substituting the values into the formula:
15,528.23 = 100 * ((1 + 0.05)^(120) - 1) / 0.05
Solving for (1 + 0.05)^(120) - 1:
(1 + 0.05)^(120) - 1 = 155.2823
We can now rewrite the equation:
15,528.23 = 100 * 155.2823 / 0.05
Simplifying:
15,528.23 = 100 * 3,105.646
Dividing both sides by 100:
155.2823 = 3,105.646
Therefore, expressing 15,528.23 in scientific notation:
15,528.23 = 1.552823 × 10^4
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount after time t
P = the initial deposit
r = the 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 initial deposit is $100, the annual interest rate is 5% (0.05 as a decimal), the interest is compounded monthly, so n = 12, and the time is 10 years.
Plugging in these values, we can rearrange the formula to solve for A:
A = $100(1 + 0.05/12)^(12*10)
A = $100(1 + 0.00416666667)^(120)
A = $100(1.00416666667)^(120)
A = $100(1.647009)
A ≈ $164.7009
Rounding to the nearest place value, we get $165.
Now, we need to express this amount, $165,288.23, in the form of a×10@n, where a is a single digit.
$165,288.23 can be written as 1.6528823 × 10^5.
Thus, the amount $15,528.23 in the form of a×10@n is approximately 1.6528823 × 10^5.
To solve this problem, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (15,528.23 in this case)
P = the initial deposit ($100)
r = the annual interest rate (5% or 0.05 in decimal form)
n = the number of times interest is compounded per year (monthly in this case, so n = 12)
t = the number of years (10)
Using this formula, we can plug in the given values and solve for A:
15,528.23 = 100(1 + 0.05/12)^(12*10)
Now let's solve the equation step by step:
Step 1: Divide both sides by 100 to isolate the expression with the parentheses.
155.2823 = (1 + 0.05/12)^(120)
Step 2: Take the 120th root of both sides.
(1 + 0.05/12) = (155.2823)^(1/120)
Step 3: Subtract 1 from both sides.
0.05/12 = (155.2823)^(1/120) - 1
Step 4: Multiply both sides by 12 to isolate the numerator of the left side.
0.05 = 12[(155.2823)^(1/120) - 1]
Step 5: Divide both sides by (155.2823)^(1/120) - 1.
0.05 / [ (155.2823)^(1/120) - 1 ] = 12
Now we have the left side of the equation in simplified form. Let's solve for it using a calculator or any other computational tool:
0.05 / [ (155.2823)^(1/120) - 1 ] ≈ 0.005116268752
Thus, the value of the left side of the equation is approximately 0.005116268752.
Now, let's express the amount $15,528.23 in the form of a×10^n, where 'a' is a single digit.
To find 'a', we need to move the decimal point until there is a single non-zero digit to the left of it.
In this case, we can move the decimal point three places to the left by dividing our value by 1000.
0.005116268752 / 1000 = 0.000005116268752
Now, let's find 'n' by counting the number of decimal places we moved the decimal point. In this case, we moved the decimal point three places to the left, so 'n' equals -3.
Therefore, we can express $15,528.23 in the form of a×10^n as approximately 5.116 × 10^(-3).