Question
Jaina and Tomas compare their compound interest accounts to see how much they will have in the accounts after three years. They substitute their values shown below into the compound interest formula.
Compound Interest Accounts
Name
Principal
Interest Rate
Number of Years
Compounded
Jaina
$300
7%
3
Once a year
Tomas
$400
4%
3
Once a year
A = P (1 + r) Superscript t
Which pair of equations would correctly calculate their compound interests?
Jaina: A = 400 (1 + 0.07) cubed, Tomas: A = 300 (1 + 0.04) cubed
Jaina: A = 300 (1 + 0.03) Superscript 7, Tomas: A = 400 (1 + 0.03) Superscript 4
Jaina: A = 400 (1 + 0.03) Superscript 7, Tomas: A = 300 (1 + 0.03) Superscript 4
Jaina: A = 300 (1 + 0.07) cubed, Tomas: A = 400 (1 + 0.04) cubed
Compound Interest Accounts
Name
Principal
Interest Rate
Number of Years
Compounded
Jaina
$300
7%
3
Once a year
Tomas
$400
4%
3
Once a year
A = P (1 + r) Superscript t
Which pair of equations would correctly calculate their compound interests?
Jaina: A = 400 (1 + 0.07) cubed, Tomas: A = 300 (1 + 0.04) cubed
Jaina: A = 300 (1 + 0.03) Superscript 7, Tomas: A = 400 (1 + 0.03) Superscript 4
Jaina: A = 400 (1 + 0.03) Superscript 7, Tomas: A = 300 (1 + 0.03) Superscript 4
Jaina: A = 300 (1 + 0.07) cubed, Tomas: A = 400 (1 + 0.04) cubed
Answers
GPT-4o mini
To correctly calculate their compound interests, we will use the compound interest formula:
\[
A = P(1 + r)^t
\]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount).
- \(r\) is the annual interest rate (decimal).
- \(t\) is the time the money is invested for in years.
Now, let's apply this formula to both Jaina and Tomas.
**Jaina's values:**
- Principal \(P = 300\)
- Interest rate \(r = 7\% = 0.07\)
- Number of years \(t = 3\)
Using the formula, Jaina's equation is:
\[
A = 300(1 + 0.07)^3
\]
or
\[
A = 300(1.07)^3
\]
**Tomas's values:**
- Principal \(P = 400\)
- Interest rate \(r = 4\% = 0.04\)
- Number of years \(t = 3\)
Using the formula, Tomas's equation is:
\[
A = 400(1 + 0.04)^3
\]
or
\[
A = 400(1.04)^3
\]
Now, let's see which option matches these calculations:
1. Jaina: \(A = 400(1 + 0.07)^3\), Tomas: \(A = 300(1 + 0.04)^3\) (Incorrect)
2. Jaina: \(A = 300(1 + 0.03)^7\), Tomas: \(A = 400(1 + 0.03)^4\) (Incorrect)
3. Jaina: \(A = 400(1 + 0.03)^7\), Tomas: \(A = 300(1 + 0.03)^4\) (Incorrect)
4. Jaina: \(A = 300(1 + 0.07)^3\), Tomas: \(A = 400(1 + 0.04)^3\) (Correct)
So, the fourth option is the one that correctly calculates their compound interests.
\[
A = P(1 + r)^t
\]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount).
- \(r\) is the annual interest rate (decimal).
- \(t\) is the time the money is invested for in years.
Now, let's apply this formula to both Jaina and Tomas.
**Jaina's values:**
- Principal \(P = 300\)
- Interest rate \(r = 7\% = 0.07\)
- Number of years \(t = 3\)
Using the formula, Jaina's equation is:
\[
A = 300(1 + 0.07)^3
\]
or
\[
A = 300(1.07)^3
\]
**Tomas's values:**
- Principal \(P = 400\)
- Interest rate \(r = 4\% = 0.04\)
- Number of years \(t = 3\)
Using the formula, Tomas's equation is:
\[
A = 400(1 + 0.04)^3
\]
or
\[
A = 400(1.04)^3
\]
Now, let's see which option matches these calculations:
1. Jaina: \(A = 400(1 + 0.07)^3\), Tomas: \(A = 300(1 + 0.04)^3\) (Incorrect)
2. Jaina: \(A = 300(1 + 0.03)^7\), Tomas: \(A = 400(1 + 0.03)^4\) (Incorrect)
3. Jaina: \(A = 400(1 + 0.03)^7\), Tomas: \(A = 300(1 + 0.03)^4\) (Incorrect)
4. Jaina: \(A = 300(1 + 0.07)^3\), Tomas: \(A = 400(1 + 0.04)^3\) (Correct)
So, the fourth option is the one that correctly calculates their compound interests.