Algebra

Let Ram's present age be and Shyam's present age be . From the problem: - Ten years ago, Ram was twice as old as Shyam: - Ten years from now, the ratio of their ages will be 3:2: Solving the first equation: Substitute this into the second equation: Cross-multiply: Substitute into the equation for : Thus, Ram's present age is .

Step 1: Analyze the system of equations.
We are given the equations: By squaring equation (1), we get: Expanding this: Substitute from equation (2) into this:

Step 2: Use the identity for .
We need to find . Using known algebraic identities, we calculate based on the values and . After solving, we find that .

Step 1: Define the digits.
Let the tens digit of the number be and the ones digit be . Then, the number can be expressed as:
We are given two conditions:
1. The product of the digits is 12, so:
2. If 36 is added to the number, the digits interchange their places, so:

Step 2: Solve the system of equations.
From the second equation:
From the first equation:

Step 3: Solve for and .
From Equation 1:
Substitute this into Equation 2:
Solving this quadratic equation using the quadratic formula:
Thus, or . Since must be a positive digit, we take . Substituting into , we get .

Step 4: Conclusion.
Thus, the number is , and the correct answer is (c).

Step 1: Given the quadratic equation with roots and , by Vieta's formulas, we know:

Step 2: The new quadratic equation will have roots and . Using the fact that the sum and product of the roots of a quadratic equation are given by and respectively, we can derive the quadratic equation.

Step 3: After deriving the equation using the relationships between the roots and coefficients, we find that the correct quadratic equation is:

Let Karan's age be and Deepak's age be . The two given conditions are:

1.

2.

From the second equation, solve for : Substitute this into the first equation: Simplifying: Divide the equation by 2: Now solve this quadratic equation using the quadratic formula: Thus, or . Since age cannot be negative, we take .

Therefore, Deepak is currently 12 years old, and two years ago, he was years old. Thus, the answer is years.