Solving the Two-Digit Number Problem
Let the two-digit number be represented as $10x + y$, where $x$ is the tens digit and $y$ is the units digit.
Setting Up the Equations
- Condition 1: Sum of digits is 9
This translates to the equation: $x + y = 9 \quad (1)$ - Condition 2: Digits interchange after subtracting 45
The original number is $10x + y$. The number with interchanged digits is $10y + x$. Subtracting 45 from the original number gives the interchanged number: $(10x + y) - 45 = 10y + x$
Solving the Equations
First, simplify the second equation:
$10x + y - 45 = 10y + x$ $10x - x + y - 10y = 45$ $9x - 9y = 45$Divide the entire equation by 9:
$x - y = 5 \quad (2)$Now, we have a system of two linear equations:
- $x + y = 9$
- $x - y = 5$
Add equation (1) and equation (2) to eliminate $y$:
$(x + y) + (x - y) = 9 + 5$ $2x = 14$ $x = \frac{14}{2}$ $x = 7$Substitute the value of $x$ (which is 7) back into equation (1):
$7 + y = 9$ $y = 9 - 7$ $y = 2$Determining the Number
The tens digit ($x$) is 7 and the units digit ($y$) is 2.
The number is $10x + y = 10(7) + 2 = 70 + 2 = 72$.
Verification
- Sum of digits: $7 + 2 = 9$. (Matches Condition 1)
- Subtracting 45: $72 - 45 = 27$. The digits 7 and 2 are interchanged to form 27. (Matches Condition 2)
Therefore, the number is 72.
Important Questions from Numerical Reasoning
- $P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.
$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.
It is known that $PQ$ and $RS$ are consecutive numbers and
$(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.
The value of $Y$ is __________ - Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
If $\oplus \div \odot = 2$, $\oplus \div \triangle = 3$, $\odot + \triangle = 5$, and $\Delta \times \otimes = 10$,
then the value of $(\otimes - \oplus)^2$ is:- The remainder when $98!$ is divided by $101$ is equal to ________
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
