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Suppose you have $5700 deposited at 5.4% compounded daily. About long will it take your balance to increase to $6900?

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Explanation:

To determine how long it will take for your initial deposit of $5,700 to grow to $6,900 at an annual interest rate of 5.4% compounded daily, you can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A is the future amount ($6,900 in this case).

P is the principal amount ($5,700).

r is the annual interest rate (5.4% or 0.054 as a decimal).

n is the number of times the interest is compounded per year (daily, so n = 365).

t is the number of years we want to find.

Now, let's plug in the values and solve for t:

6,900 = 5,700(1 + 0.054/365)^(365t)

First, let's simplify the fraction inside the parentheses:

6,900 = 5,700(1 + 0.00014794521)^(365t)

Now, calculate the expression inside the parentheses:

1 + 0.00014794521 = 1.00014794521

6,900 = 5,700(1.00014794521)^(365t)

Next, divide both sides by 5,700:

6,900 / 5,700 = (1.00014794521)^(365t)

1.21052631579 = (1.00014794521)^(365t)

Now, take the natural logarithm (ln) of both sides to solve for t:

ln(1.21052631579) = ln((1.00014794521)^(365t))

Using the property of logarithms, we can bring down the exponent:

ln(1.21052631579) = 365t * ln(1.00014794521)

Now, divide by 365ln(1.00014794521) to isolate t:

t = ln(1.21052631579) / (365 * ln(1.00014794521))

Now, calculate the right side of the equation:

t ≈ 6.8257 years

So, it will take approximately 6.8257 years for your balance to increase to $6,900 when compounded daily at an annual interest rate of 5.4%.

User Steve Ritz
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