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
- 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 $p_2$?
- 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?
Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.- In a zoo, three lions and four tigers eat 390 kg of food every week. In another zoo, four lions and five tigers eat 500 kg of food every week. Lions and tigers eat different amounts of food, but all individuals of the same species eat the same amount. The amount of food a single lion eats per week is ________ kg.
(Answer in integer) Consider a spherical globe rotating about an axis passing through its poles. There are three points P, Q, and R situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let P, Q, and R move with speeds $v_P$, $v_Q$, and $v_R$, respectively.
Which one of the following options is CORRECT?
