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$ P were deposited in each of two different accounts, one gives sim
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26 May 2025, 06:57
26 May 2025, 06:57
Expert Reply
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$\(\$ P\)$ were deposited in each of two different accounts, one gives simple interest and in the other amount is compounded annually.
For both accounts the annual rate of interest is $\(R \%\)$. $\((R>0)\)$ (Note: No other transactions were made apart from those mentioned.)
For both accounts the annual rate of interest is $\(R \%\)$. $\((R>0)\)$ (Note: No other transactions were made apart from those mentioned.)
Quantity A |
Quantity B |
Difference between the interests earned in accounts in $\(2^{\text {nd }}\)$ year |
Difference between the interests earned in accounts in $\(1^{\text {st }}\)$ year |
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Re: $ P were deposited in each of two different accounts, one gives sim
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27 May 2025, 05:20
27 May 2025, 05:20
1
Obviously, Quantity A is greater because both the simple and compound account will yield the same interest in the first year because their principal and rates are the same so the difference is zero for B.
Re: $ P were deposited in each of two different accounts, one gives sim
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28 May 2025, 04:00
28 May 2025, 04:00
Expert Reply
We have two accounts where an amount $P$ is deposited:
1. Account 1: Simple Interest at an annual rate of $R \%$.
2. Account 2: Compound Interest (compounded annually) at the same annual rate $R \%$.
We need to compare:
- Quantity A: Difference between the interests earned in the 2nd year for both accounts.
- Quantity B: Difference between the interests earned in the 1st year for both accounts.
Step 1: Understand Simple and Compound Interest
Simple Interest (SI):
- Interest is calculated only on the principal amount $P$ every year.
- Formula for interest in the $n$-th year: $\(\mathrm{SI}=P \times \frac{R}{100}\)$.
Compound Interest (CI):
- Interest is calculated on the principal plus any accumulated interest.
- Formula for amount after $n$ years: $\(A=P\left(1+\frac{R}{100}\right)^n\)$.
- Interest in the $\(n$-th\) year: $\(\mathrm{CI}_n=A_n-A_{n-1}\)$.
Step 2: Calculate Interest for Each Account in the 1st Year
1st Year:
- Simple Interest:
$$
\(\mathrm{SI}_1=P \times \frac{R}{100}\)
$$
- Compound Interest:
$$
\(\mathrm{CI}_1=P\left(1+\frac{R}{100}\right)^1-P=P \times \frac{R}{100}\)
$$
- Difference in 1st Year (Quantity B):
$$
\(\mathrm{CI}_1-\mathrm{SI}_1=\left(P \times \frac{R}{100}\right)-\left(P \times \frac{R}{100}\right)=0\)
$$
Step 3: Calculate Interest for Each Account in the 2nd Year
2nd Year:
- Simple Interest:
$$
\(\mathrm{SI}_2=P \times \frac{R}{100}\)
$$
(Same as the 1st year since Sl is constant per year.)
- Compound Interest:
$$
\(\begin{gathered}
\text { Amount after 1 year }=P\left(1+\frac{R}{100}\right) \\
\mathrm{CI}_2=P\left(1+\frac{R}{100}\right)^2-P\left(1+\frac{R}{100}\right)=P\left(1+\frac{R}{100}\right) \times \frac{R}{100}
\end{gathered}\)
$$
- Difference in 2nd Year (Quantity A):
$$
\(\mathrm{CI}_2-\mathrm{SI}_2=P\left(1+\frac{R}{100}\right) \times \frac{R}{100}-P \times \frac{R}{100}=P \times \frac{R}{100}\left(1+\frac{R}{100}-1\right)=P \times\left(\frac{R}{100}\right)\)
$$
Step 4: Compare Quantity A and Quantity B
- Quantity A (Difference in 2nd Year):
$$
\(P \times\left(\frac{R}{100}\right)^2\)
$$
- Quantity B (Difference in 1st Year):
$$
\(0\)
$$
Since $\(R>0\)$ and $\(P>0, P \times\left(\frac{R}{100}\right)^2>0\)$.
Conclusion
Quantity A > Quantity B
1. Account 1: Simple Interest at an annual rate of $R \%$.
2. Account 2: Compound Interest (compounded annually) at the same annual rate $R \%$.
We need to compare:
- Quantity A: Difference between the interests earned in the 2nd year for both accounts.
- Quantity B: Difference between the interests earned in the 1st year for both accounts.
Step 1: Understand Simple and Compound Interest
Simple Interest (SI):
- Interest is calculated only on the principal amount $P$ every year.
- Formula for interest in the $n$-th year: $\(\mathrm{SI}=P \times \frac{R}{100}\)$.
Compound Interest (CI):
- Interest is calculated on the principal plus any accumulated interest.
- Formula for amount after $n$ years: $\(A=P\left(1+\frac{R}{100}\right)^n\)$.
- Interest in the $\(n$-th\) year: $\(\mathrm{CI}_n=A_n-A_{n-1}\)$.
Step 2: Calculate Interest for Each Account in the 1st Year
1st Year:
- Simple Interest:
$$
\(\mathrm{SI}_1=P \times \frac{R}{100}\)
$$
- Compound Interest:
$$
\(\mathrm{CI}_1=P\left(1+\frac{R}{100}\right)^1-P=P \times \frac{R}{100}\)
$$
- Difference in 1st Year (Quantity B):
$$
\(\mathrm{CI}_1-\mathrm{SI}_1=\left(P \times \frac{R}{100}\right)-\left(P \times \frac{R}{100}\right)=0\)
$$
Step 3: Calculate Interest for Each Account in the 2nd Year
2nd Year:
- Simple Interest:
$$
\(\mathrm{SI}_2=P \times \frac{R}{100}\)
$$
(Same as the 1st year since Sl is constant per year.)
- Compound Interest:
$$
\(\begin{gathered}
\text { Amount after 1 year }=P\left(1+\frac{R}{100}\right) \\
\mathrm{CI}_2=P\left(1+\frac{R}{100}\right)^2-P\left(1+\frac{R}{100}\right)=P\left(1+\frac{R}{100}\right) \times \frac{R}{100}
\end{gathered}\)
$$
- Difference in 2nd Year (Quantity A):
$$
\(\mathrm{CI}_2-\mathrm{SI}_2=P\left(1+\frac{R}{100}\right) \times \frac{R}{100}-P \times \frac{R}{100}=P \times \frac{R}{100}\left(1+\frac{R}{100}-1\right)=P \times\left(\frac{R}{100}\right)\)
$$
Step 4: Compare Quantity A and Quantity B
- Quantity A (Difference in 2nd Year):
$$
\(P \times\left(\frac{R}{100}\right)^2\)
$$
- Quantity B (Difference in 1st Year):
$$
\(0\)
$$
Since $\(R>0\)$ and $\(P>0, P \times\left(\frac{R}{100}\right)^2>0\)$.
Conclusion
Quantity A > Quantity B
