Age Sum and Ratio Problem Explained

This problem involves finding the current age of person B, given the sum of the current ages of two people (A and B) and the ratio of their ages five years ago.

Setting Up the Equations

Let's denote the current age of A as $A$ years and the current age of B as $B$ years.

• The sum of their current ages is 46 years. This can be written as the equation:
  $A + B = 46$ (Equation 1)
• Five years ago, A's age was $A - 5$ and B's age was $B - 5$.
• The ratio of their ages five years ago was $4:5$. This gives us the equation:
  $\frac{A - 5}{B - 5} = \frac{4}{5}$ (Equation 2)

Solving for B's Current Age

We need to solve these two equations simultaneously to find the value of $B$.

1. Simplify Equation 2: Cross-multiply Equation 2:
   $5 \times (A - 5) = 4 \times (B - 5)$
   Distribute the numbers:
   $5A - 25 = 4B - 20$
   Rearrange the terms to group variables:
   $5A - 4B = 25 - 20$
   $5A - 4B = 5$ (Equation 3)
2. Express A in terms of B using Equation 1: From Equation 1, we can write:
   $A = 46 - B$
3. Substitute A in Equation 3: Now, substitute the expression for $A$ from the previous step into Equation 3:
   $5 \times (46 - B) - 4B = 5$
   Distribute the 5:
   $230 - 5B - 4B = 5$
   Combine the terms with $B$:
   $230 - 9B = 5$
4. Isolate and solve for B: Subtract 230 from both sides:
   $-9B = 5 - 230$
   $-9B = -225$
   Divide by -9:
   $B = \frac{-225}{-9}$
   $B = 25$

Verification

Let's verify the result.

• If B's current age is 25, then A's current age is $46 - 25 = 21$.
• Five years ago, A's age was $21 - 5 = 16$.
• Five years ago, B's age was $25 - 5 = 20$.
• The ratio of their ages five years ago was $\frac{16}{20}$, which simplifies to $\frac{4}{5}$. This matches the condition given in the problem.

Therefore, B's current age is 25 years.