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If $5,000,000 is the initial amount placed in an account that coll
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12 Jan 2026, 09:15
12 Jan 2026, 09:15
Expert Reply
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Question Stats:
50% (00:13) correct
50% (00:26) wrong based on 2 sessions
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If $ 5,000,000 is the initial amount placed in an account that collects 7% annual interest, which of the following compounding rates would produce the largest total amount after two years?
◯ compounding annually
◯ compounding quarterly
◯ compounding monthly
◯ compounding daily
◯ All four of these would produce the same total
◯ compounding annually
◯ compounding quarterly
◯ compounding monthly
◯ compounding daily
◯ All four of these would produce the same total
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Re: If $5,000,000 is the initial amount placed in an account that coll
[#permalink]
31 Jan 2026, 13:00
31 Jan 2026, 13:00
Expert Reply
To determine which compounding rate produces the largest amount, we look at the
Compound Interest Formula:
$$
\(A=P\left(1+\frac{r}{n}\right)^{n t}\)
$$
Where:
- $A=$ the final amount (the "Future Value")
- $P=$ the principal $(5,000,000)$
- $r=$ the annual interest rate (0.07)
- $n=$ the number of compounding periods per year
- $t=$ the number of years (2)
The Relationship Between $n$ and $A$
The principle of compounding is that you earn "interest on interest." The more frequently the interest is calculated and added to the principal, the sooner that interest begins to earn its own interest.
In mathematical terms, as the frequency of compounding $(n)$ increases, the total amount $(A)$ also increases. Let's compare the multipliers for each option:
1. Annually $\((n=1):(1+0.07)^2=1.144\)9$
2. Quarterly $\((n=4):\left(1+\frac{0.07}{4}\right)^{4 \times 2}=(1.0175)^8 \approx 1.1488\)$
3. Monthly $\((n=12)$ : $\left(1+\frac{0.07}{12}\right)^{12 \times 2} \approx(1.00583)^{24} \approx 1.1498\)$
4. Daily $\((n=365):\left(1+\frac{0.07}{365}\right)^{365 \times 2} \approx 1.1502\)$
As you can see, the final multiplier grows as the compounding frequency increases.
Final Comparison (Total Amount after 2 years):
- Annually: $\(\$ 5,724,500\)$
- Quarterly: $\(\approx \$ 5,744,410\)$
- Monthly: $\(\approx \$ 5,749,034\)$
- Daily: $\(\approx \$ 5,751,299\)$
Compound Interest Formula:
$$
\(A=P\left(1+\frac{r}{n}\right)^{n t}\)
$$
Where:
- $A=$ the final amount (the "Future Value")
- $P=$ the principal $(5,000,000)$
- $r=$ the annual interest rate (0.07)
- $n=$ the number of compounding periods per year
- $t=$ the number of years (2)
The Relationship Between $n$ and $A$
The principle of compounding is that you earn "interest on interest." The more frequently the interest is calculated and added to the principal, the sooner that interest begins to earn its own interest.
In mathematical terms, as the frequency of compounding $(n)$ increases, the total amount $(A)$ also increases. Let's compare the multipliers for each option:
1. Annually $\((n=1):(1+0.07)^2=1.144\)9$
2. Quarterly $\((n=4):\left(1+\frac{0.07}{4}\right)^{4 \times 2}=(1.0175)^8 \approx 1.1488\)$
3. Monthly $\((n=12)$ : $\left(1+\frac{0.07}{12}\right)^{12 \times 2} \approx(1.00583)^{24} \approx 1.1498\)$
4. Daily $\((n=365):\left(1+\frac{0.07}{365}\right)^{365 \times 2} \approx 1.1502\)$
As you can see, the final multiplier grows as the compounding frequency increases.
Final Comparison (Total Amount after 2 years):
- Annually: $\(\$ 5,724,500\)$
- Quarterly: $\(\approx \$ 5,744,410\)$
- Monthly: $\(\approx \$ 5,749,034\)$
- Daily: $\(\approx \$ 5,751,299\)$
gmatclubot
Re: If $5,000,000 is the initial amount placed in an account that coll [#permalink]
31 Jan 2026, 13:00
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