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Carl deposited $5,142 into a savings account 20 years ago. The account has an interest rate of 3. 8% and the balance is currently $10,994. 58. How often does the interest compound?

A. Daily


B. Monthly


C. Quarterly


D. Annually

User Thomdask
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1 Answer

5 votes

Answer:

A. Daily

Explanation:

  • To solve this problem, we can use the formula for compound interest, which is given by:

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

  • A is the final amount (aka the current amount),
  • P is the principal (aka the deposit),
  • r is the interest rate (the percentage is always converted to a decimal),
  • n is the number of times the interest is compounded per year,
  • and t is the time in years.

Step 1: Substitute the given values into the formula:

  • We have A = 10994.58, P = 5142, r = 0.038, and t = 20.
  • Plugging in these values for the variables in the compound interest formula gives us: 10994.58 = 5142(1 + 0.038/n)^(20n).

Step 2: Check whether the money is compounded daily, monthly, quarterly, or annually:

Checking if the money is compounded daily:

  • If the money is compounded daily, then n = 365 since there are 365 days in a year we want the number of compounding periods per year.
  • We can check if the money is compounded daily by plugging in 365 for n and seeing if we get 10994.58 on both sides of the equation:

10994.58 = 5142(1 + 0.038/365)^(365 * 20)

10994.58 = 5142(1.00010411)^(7300)

10994.58 < 10994.61436

10994.58 < 10994.61

  • Although our answer is slightly larger than 10994.58, we can still trust our answer (the money is currently $10994.58, but the problem doesn't specify which day of the 365 days the account was checked).
  • When you check for monthly, quarterly, and annually compound, you get answers there are not nearly as close to 10994.58 as 10994.61, further proving that the money is compounded daily. I checked whether the money was compounded monthly, quarterly, and annually to verify that the money was indeed compounded daily and I'll show how I did this below:

Proving the money is not compounded monthly:

  • For money compounded monthly, n = 12 as there are 12 months in a year and we want the number of compounding periods per year.

Thus, we can prove the money is not compounded monthly by plugging in 12 for n and examining how much farther the answer is from 10994.58 than 10994.61 (aka our answer when we did 365 for n to check for daily compounded money):

10994.58 --- 5142(1 + 0.038/12)^(12 * 20)

10994.58 --- 5142(1.003166667)^240)

10994.58 ? 10981.82145

10994.58 > 10981.82

10981.82 is further away from 10994.58 than 10994.61 so the money definitely is not compounded monthly.

Proving the money is not compounded quarterly:

  • For money compounded quarterly, n = 4 as the money is compounded every three months and since there are 12 months in a year, there are four three month periods.

Thus, we can prove the money is not compounded quarterly by plugging in 4 for n and examining how much farther the answer is from 10994.58 than 10994.61 (aka our answer when we did 365 for n to check for daily compounded money):

10994.58 --- 5142(1 + 0.038/4)^(4 * 20)

10994.58 --- 5142(1.0095)^(80)

10994.58 > 10955.64458

10994.58 > 10955.64

10955.64 is further away from 10994.58 than 10994.61 so the money definitely is not compounded monthly.

Proving the money is not compounded annually:

  • For money compounded annually, n = 1 as the money is only compounded once per year.

Thus, we can prove the money is not compounded quarterly by plugging in 4 for n and examining how much farther the answer is from 10994.58 than 10994.61 (aka our answer when we did 365 for n to check for daily compounded money):

10994.58 --- 5142(1 + 0.038/1)^(1 * 20)

10994.58 --- 5142(1.038)^(20)

10994.58 > 10841.24459

10994.58 > 10841.24

10841.24 is further away from 10994.58 than 10994.61 so the money is definitely not compounded annually.

User Arash Afsharpour
by
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