Question

Anne has opened up a systematic savings account into which she deposits $500 per month compounded monthly at a rate of 7.8%. How long will it take her to reach $500,000? Round your answer to the nearest tenth of a year.

Help, please!! I need a step-by-step tutorial. Or just the formula and how is set up. Thank you <33

Answer 1

Explanation:

The formula for calculating the time it takes to reach a target amount with a constant periodic deposit is given by

Time = ln(Target Amount / (Periodic Deposit x (1 + Interest Rate)^0)) / ln(1 + Interest Rate)

In this case, we have Target Amount = 500,000, Periodic Deposit = 500, and Interest rate = 7.8%. Plugging the values into the formula gives

Time = ln(500,000 / (500 x (1 + 0.078)^0)) / ln(1 + 0.078)

Time = ln(500,000 / 500) / ln(1.078)

Time = ln(1,000) / ln(1.078)

Time = 8.98 years

Rounding to the nearest tenth of a year, the answer is 8.9 years.

Answer 2

To calculate how long it will take for Anne to reach $500,000 in her systematic savings account, we can use the formula for compound interest. After following the steps of the formula and evaluating the equation, we find that it will take Anne approximately 14.3 years to reach her savings goal.

To calculate how long it will take for Anne to reach $500,000 in her systematic savings account, we can use the formula for compound interest:

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

Where:

A = the future value of the account ($500,000)

P = the principal amount deposited each month ($500)

r = the interest rate per period (7.8% or 0.078)

n = the number of times interest is compounded per year (12 times monthly in this case)

t = the number of years

1. Plug in the given values into the formula: $500,000 = $500(1 + 0.078/12)^(12t)
2. Divide both sides of the equation by $500 to isolate the exponential term: 1000 = (1 + 0.078/12)^(12t)
3. Take the logarithm of both sides to solve for t: log(1000) = log((1 + 0.078/12)^(12t))
4. Use logarithmic properties to bring down the exponent: log(1000) = (12t)log(1 + 0.078/12)
5. Divide both sides by (12log(1 + 0.078/12)) to solve for t: t = log(1000) / (12log(1 + 0.078/12))
6. Evaluate the right-hand side using a calculator: t ≈ 14.3 years (rounded to the nearest tenth of a year)

Therefore, it will take Anne approximately 14.3 years to reach $500,000 in her systematic savings account.