Pregunta
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Suppose you wish to retire at the age of 70 with in savings. Determine your monthly payment into an IRA if the APR is compounded monthly and you
begin making payments at 25 years old. Round your answer to the nearest cent, if necessary.

Ask by Olson Medina. in the United States
Jan 24,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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The monthly payment needed is approximately $33.90.

Solución

To determine the monthly payment into an IRA needed to accumulate by the age of 70, we can use the future value of an annuity formula. The formula for the future value of an annuity is:
Where:
  • is the future value of the annuity (the amount you want to have at retirement, which is ).
  • is the monthly payment (the value we want to find).
  • is the monthly interest rate (annual interest rate divided by 12).
  • is the total number of payments (number of years until retirement multiplied by 12).

Step 1: Identify the known values

  • Future Value,
  • Annual Interest Rate,
  • Monthly Interest Rate,
  • Current Age = 25
  • Retirement Age = 70
  • Total Years until Retirement =
  • Total Payments,

Step 2: Calculate and

Now, let’s calculate and :

Step 3: Substitute into the formula and solve for

Rearranging the future value formula to solve for :
Now, let’s calculate , , and then .
I’ll perform these calculations now.
Calculate the value by following steps:
  • step0: Calculate:
  • step1: Multiply the numbers:

    Calculate or simplify the expression .
    Calculate the value by following steps:
  • step0: Calculate:
  • step1: Convert the expressions:
  • step2: Multiply by the reciprocal:
  • step3: Multiply the fractions:
  • step4: Multiply:

    The calculations yield the following results:
  • Monthly Interest Rate,
  • Total Payments,

Step 4: Substitute into the formula to find

Now we can substitute , , and into the rearranged formula for :
Let’s calculate .
Calculate the value by following steps:
  • step0: Calculate:
  • step1: Remove the parentheses:
  • step2: Add the numbers:
  • step3: Convert the expressions:
  • step4: Multiply the numbers:
  • step5: Subtract the numbers:
  • step6: Convert the expressions:
  • step7: Multiply by the reciprocal:
  • step8: Rewrite the expression:
  • step9: Rewrite the expression:
  • step10: Reduce the numbers:
  • step11: Multiply:

    The calculation gives us:

Conclusion

The monthly payment into the IRA needed to accumulate by the age of 70, with an APR of compounded monthly, is approximately $33.90.

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Mind Expander

To achieve your goal of having $80,000 by the age of 70, we need to calculate how much you should contribute monthly starting at age 25 with an APR of 5.5% compounded monthly.
Using the future value of an ordinary annuity formula:
Where:
  • = future value ($80,000)
  • = monthly payment (what we want to find)
  • = monthly interest rate (5.5% / 12 months = 0.00458333)
  • = total number of payments (45 years × 12 months = 540)
Rearranging the formula for :
Now plug in the values:
Calculating :
Now substituting back into the payment formula:
So, the monthly payment you need to make is approximately $87.76.

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