The difference between the C.I and the S.I on Rs 8,400 for two years is Rs 21 at the same rate of intersect per year. Find the rate of interset.

Answer

To solve the problem step by step, we will use the formula for the difference between Compound Interest (C.I) and Simple Interest (S.I):

**Step 1: Understand the formula for the difference between C.I and S.I.**
The difference between Compound Interest and Simple Interest for 2 years can be expressed as:
\[ \text{Difference} = \frac{P \times r}{100} \times \frac{r}{100} \]
where:
- \( P \) = Principal amount
- \( r \) = Rate of interest per annum

**Step 2: Substitute the given values into the formula.**
We know from the problem:
- Principal \( P = 8400 \)
- Difference = 21

Substituting these values into the formula, we have:
\[ 21 = \frac{8400 \times r}{100} \times \frac{r}{100} \]

**Step 3: Simplify the equation.**
This simplifies to:
\[ 21 = \frac{8400 \times r^2}{10000} \]
To eliminate the fraction, multiply both sides by 10000:
\[ 21 \times 10000 = 8400 \times r^2 \]
\[ 210000 = 8400 \times r^2 \]

**Step 4: Solve for \( r^2 \).**
Now, divide both sides by 8400:
\[ r^2 = \frac{210000}{8400} \]
Calculating the right side gives:
\[ r^2 = 25 \]

**Step 5: Find \( r \) by taking the square root.**
To find \( r \), take the square root of both sides:
\[ r = \sqrt{25} \]
Thus,
\[ r = 5 \]

**Step 6: Conclusion.**
The rate of interest is \( 5\% \).

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