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.