The rate of interest in two hanks DNB and HBI are in the ratio of 7: 8. If a person invested some amount in both the banks and received equal amounts from both the banks in two years. The ratio of amount invested in DNB and HBI, respectively is:

Options

A. `15:1`
B. `8:7`
C. `7:8`
D. `108:107`

Answer

To solve the problem, we need to determine the ratio of the amounts invested in DNB and HBI given the interest rates and the condition that the amounts received from both banks after two years are equal.

### Step-by-Step Solution:

1. **Define Variables**:
Let the amount invested in DNB be \( X \) and the amount invested in HBI be \( Y \).

2. **Interest Rates**:
The rates of interest for DNB and HBI are in the ratio of 7:8.
- Let the rate of interest for DNB be \( 7r \) and for HBI be \( 8r \).

3. **Calculate Amounts After 2 Years**:
The formula for the amount \( A \) after \( t \) years at simple interest is given by:
\[ A = P + (P \times r \times t) \]
where \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time in years.

- For DNB:
\[ A_{DNB} = X + (X \times 7r \times 2) = X(1 + 14r) \]
- For HBI:
\[ A_{HBI} = Y + (Y \times 8r \times 2) = Y(1 + 16r) \]

4. **Set Amounts Equal**:
According to the problem, the amounts received from both banks are equal:
\[ A_{DNB} = A_{HBI} \]
Therefore:
\[ X(1 + 14r) = Y(1 + 16r) \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ X + 14Xr = Y + 16Yr \]
This can be rearranged to:
\[ X - Y = 16Yr - 14Xr \]
Or:
\[ X - Y = r(16Y - 14X) \]

6. **Expressing Y in Terms of X**:
From the equation \( X(1 + 14r) = Y(1 + 16r) \), we can express \( Y \) in terms of \( X \):
\[ Y = \frac{X(1 + 14r)}{1 + 16r} \]

7. **Finding the Ratio**:
To find the ratio \( \frac{X}{Y} \):
\[ \frac{X}{Y} = \frac{X(1 + 16r)}{X(1 + 14r)} = \frac{1 + 14r}{1 + 16r} \]
This simplifies to:
\[ \frac{X}{Y} = \frac{8}{7} \]

8. **Final Ratio**:
Thus, the ratio of the amounts invested in DNB and HBI is:
\[ \frac{X}{Y} = \frac{8}{7} \]
Therefore, the ratio of the amount invested in DNB to HBI is \( 8:7 \).

### Conclusion:
The ratio of the amount invested in DNB and HBI is \( 8:7 \).