Solving for the Two-Digit Number
Let the two-digit number be represented as $10x + y$, where $x$ is the digit in the tens place and $y$ is the digit in the units place.
Equation from Sum of Digits
The problem states that the sum of the digits is 12:
$x + y = 12 \quad (1)$
Equation from Reversed Digits
The number formed by reversing the digits is $10y + x$. This new number is 54 greater than the original number:
$ (10y + x) - (10x + y) = 54 $
Simplify the equation:
$ 10y + x - 10x - y = 54 $
$ 9y - 9x = 54 $
Divide by 9:
$ y - x = 6 \quad (2) $
Solving the System of Equations
We now have a system of two linear equations:
- $x + y = 12$
- $-x + y = 6$
Add equation (1) and equation (2):
$ (x + y) + (-x + y) = 12 + 6 $
$ 2y = 18 $
$ y = \frac{18}{2} $
$ y = 9 $
Substitute the value of $y$ back into equation (1):
$ x + 9 = 12 $
$ x = 12 - 9 $
$ x = 3 $
Determining the Original Number
The original number is $10x + y$. Substituting the values $x=3$ and $y=9$:
$ \text{Original Number} = 10(3) + 9 = 30 + 9 = 39 $
Verification
Check if the conditions are met:
- Sum of digits: $3 + 9 = 12$ (Correct).
- Reversed number: $93$.
- Difference: $93 - 39 = 54$ (Correct).
The original number is 39.
Important Questions from Numerical Computation
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If the in-situ density of coal is 1320 kg/m$^3$ and the density of blasted coal is 952 kg/m$^3$, the swell factor is _____________ (rounded off to 3 decimal places)
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The Fourier transform and its inverse transform are respectively defined as $\tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x)e^{i\omega x}dx$ and $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega)e^{-i\omega x}d\omega$. Consider two functions $f$ and $g$. Another function $f * g$ is defined as
$(f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y)g(x - y)dy$
Which of the following relation is/are true?
Note: Tilde ($\sim$) denotes the Fourier transform.
