Step 1: Understanding the Question:
We are given the current ratio of the ages of two people, A and B, and the sum of their current ages. We need to find the ratio of their ages after 4 years.
Step 2: Key Formula or Approach:
Represent the ages using a common multiplier \(x\) based on the given ratio. Use the given sum to find the value of \(x\) and thus their current ages. Then, calculate their future ages and find the new ratio.
Step 3: Detailed Explanation:
Let the current age of A be \(4x\) years and the current age of B be \(5x\) years, based on the ratio 4:5.
The sum of their ages is given as 36 years.
\[ 4x + 5x = 36 \]
\[ 9x = 36 \]
\[ x = \frac{36}{9} = 4 \]
Now, we can find their current ages:
Current age of A = \(4x = 4 \times 4 = 16\) years.
Current age of B = \(5x = 5 \times 4 = 20\) years.
Next, we find their ages after 4 years:
Age of A after 4 years = \(16 + 4 = 20\) years.
Age of B after 4 years = \(20 + 4 = 24\) years.
Finally, we find the ratio of their ages after 4 years:
\[ \text{New Ratio} = \frac{\text{Age of A after 4 years}}{\text{Age of B after 4 years}} = \frac{20}{24} \]
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4.
\[ \text{New Ratio} = \frac{20 \div 4}{24 \div 4} = \frac{5}{6} \]
So, the new ratio is 5:6.
Step 4: Final Answer:
The ratio of their ages after 4 years will be 5:6.