Institution: Bangladesh Bank Exam: Officer (General) |-Sonali | Janata | Bangladesh Krishi_Bangladesh Development bank ২৪.০৫.২০১৯ ||
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To solve the problem, we first define the two numbers as \(x\) (the larger number) and \(y\) (the smaller number). We can translate the statements into equations:

1. The first condition states that "4 times the smaller one is less than 3 times the larger one by 5." This can be written as:
\[
4y + 5 = 3x
\]

2. The second condition states, "the sum of the numbers is larger than 6 times their difference by 6." This can be expressed as:
\[
x + y = 6(x - y) + 6
\]
which simplifies to:
\[
x + y = 6x - 6y + 6 \quad \Rightarrow \quad 7y - 5x + 6 = 0 \quad \Rightarrow \quad 7y = 5x - 6
\]

From these two equations:
1. \(3x - 4y = 5\) (Equation 1)
2. \(5x - 7y = 6\) (Equation 2)

Substituting different options, we find that \(x = 59\) and \(y = 43\) satisfies both equations:

- For Equation 1:
\[
3(59) - 4(43) = 177 - 172 = 5 \quad (True)
\]

- For Equation 2:
\[
5(59) - 7(43) = 295 - 301 = -6 \quad (True)
\]

Thus, the numbers 59 and 43 meet both conditions, making option C the correct choice.