Solving Age Problems: Finding Present Age of B

This problem involves determining the present ages of two individuals, A and B, based on given information about their current age relationship and the ratio of their ages after a certain number of years.

Let's denote the present age of A as \(A\) years and the present age of B as \(B\) years.

Setting Up Equations from Given Information

We are given two main pieces of information:

1. B is twice as old as A today.
2. In 10 years, the ratio of ages of A and B will be 2 : 3.

From the first statement, "B is twice as old as A today", we can write the equation:

\(B = 2A\)

This is our first equation.

Now let's consider the ages after 10 years.

• In 10 years, A's age will be \(A + 10\).
• In 10 years, B's age will be \(B + 10\).

The second statement says that in 10 years, the ratio of ages of A and B will be 2 : 3. This can be written as:

\(\frac{A + 10}{B + 10} = \frac{2}{3}\)

This is our second equation.

Solving the System of Equations

We have a system of two linear equations:

1. \(B = 2A\)
2. \(\frac{A + 10}{B + 10} = \frac{2}{3}\)

We can use the substitution method to solve this system. Substitute the value of \(B\) from equation (1) into equation (2):

\(\frac{A + 10}{(2A) + 10} = \frac{2}{3}\)

\(\frac{A + 10}{2A + 10} = \frac{2}{3}\)

Now, cross-multiply:

\(3 \times (A + 10) = 2 \times (2A + 10)\)

Distribute the numbers on both sides:

\(3A + 30 = 4A + 20\)

Now, we need to solve for \(A\). Let's move the terms involving \(A\) to one side and the constant terms to the other side.

Subtract \(3A\) from both sides:

\(30 = 4A - 3A + 20\)

\(30 = A + 20\)

Subtract \(20\) from both sides:

\(30 - 20 = A\)

\(10 = A\)

So, the present age of A is 10 years.

Finding the Present Age of B

The question asks for the present age of B. We know from equation (1) that \(B = 2A\).

Substitute the value of \(A = 10\) into this equation:

\(B = 2 \times 10\)

\(B = 20\)

Therefore, the present age of B is 20 years.

Verification

Let's check if these ages satisfy the conditions.

• Present ages: A = 10, B = 20. Is B twice as old as A? Yes, 20 = 2 * 10.
• Ages in 10 years: A = 10 + 10 = 20, B = 20 + 10 = 30. Is the ratio of ages of A and B 2:3? Yes, 20:30 simplifies to 2:3.

Both conditions are satisfied, confirming our solution is correct.

The present age of B is 20 years.