Pregunta
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Marta takes a loan of at an interest rate of , compounded quarterly. Assuming no payments are
made, find the amount owed after 5 years. Round to the nearest dollar.
Type your numerical answer below.

Ask by Clarke Wade. in the United States
Mar 14,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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After 5 years, Marta owes $15,618.

Solución

To find the amount owed after 5 years on a loan with compound interest, we can use the formula for compound interest:
Where:
  • is the amount of money accumulated after n years, including interest.
  • is the principal amount (the initial loan amount).
  • is the annual interest rate (decimal).
  • is the number of times that interest is compounded per year.
  • is the number of years the money is borrowed for.
Given:
  • (since the interest is compounded quarterly)
Now, we can substitute these values into the formula:
Let’s calculate this step by step.
  1. Calculate :
  2. Calculate :
  3. Substitute these values into the formula and calculate .
Now, I will perform these calculations.
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:

    Calculate or simplify the expression .
    Calculate the value by following steps:
  • step0: Calculate:
  • step1: Add the numbers:
  • step2: Convert the expressions:
  • step3: Simplify:
  • step4: Rewrite the expression:
  • step5: Rewrite the expression:
  • step6: Rewrite the expression:
  • step7: Reduce the numbers:
  • step8: Multiply the fractions:

    Now that we have the necessary calculations:
  1. The quarterly interest rate is .
  2. The total number of compounding periods over 5 years is .
Now we can substitute these values into the compound interest formula:
Calculating this gives us:
Rounding to the nearest dollar, the amount owed after 5 years is:

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The Deep Dive

To find the amount owed after 5 years, we can use the formula for compound interest:
Where:
  • is the amount of money accumulated after n years, including interest.
  • is the principal amount (the initial loan amount).
  • is the annual interest rate (decimal).
  • is the number of times that interest is compounded per year.
  • is the number of years the money is invested or borrowed.
For this loan:
  • (since the interest is compounded quarterly)
Plugging these values into the formula gives:
Calculating the values step-by-step:
  1. Calculate
  2. So
  3. That simplifies to
  4. Calculating
  5. Finally,
Rounding to the nearest dollar, the amount owed after 5 years is .

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