To find the father's age, follow these steps:
1. Define Variables:
- Let the current age of each child be [tex]\( x \)[/tex].
- The current sum of the ages of the two children is [tex]\( x + x = 2x \)[/tex].
2. Current Age Relationship:
- The father's current age is 3 times the sum of the ages of his two children:
[tex]\[
\text{Father's current age} = 3 \times (2x) = 6x
\][/tex]
3. Future Age Relationship:
- Assume that after [tex]\( z \)[/tex] years, the father's age will be twice the sum of the ages of his two children at that future time.
- In [tex]\( z \)[/tex] years, the father's age will be [tex]\( 6x + z \)[/tex].
- In [tex]\( z \)[/tex] years, each of the child's age will be [tex]\( x + z \)[/tex].
- The sum of the children's ages in [tex]\( z \)[/tex] years will be:
[tex]\[
(x + z) + (x + z) = 2x + 2z
\][/tex]
- According to the problem, the father's age at that time will be twice the sum of the children's ages:
[tex]\[
6x + z = 2 \times (2x + 2z)
\][/tex]
4. Set Up the Equation:
- Simplify the equation from the future age relationship:
[tex]\[
6x + z = 4x + 4z
\][/tex]
- Rearrange the terms to isolate [tex]\( z \)[/tex] on one side:
[tex]\[
6x + z - 4x - 4z = 0
\][/tex]
[tex]\[
2x - 3z = 0
\][/tex]
5. Solve for [tex]\( z \)[/tex]:
- From the equation [tex]\( 2x - 3z = 0 \)[/tex], we get:
[tex]\[
z = \frac{2x}{3}
\][/tex]
6. Find Current Ages:
- To find the specific ages, we use the given relationship for the father's current age being 3 times the sum of his children's ages:
[tex]\[
6x = 3 \times (2x)
\][/tex]
- Simplifying this, we observe:
[tex]\[
6x = 6x
\][/tex]
- This equation holds true, which means the variables [tex]\( x \)[/tex] (children's age) and [tex]\( z \)[/tex] can take any value that satisfies the original conditions.
Given the initial equation [tex]\( 6x = 6x \)[/tex] implies any real value, the unique solution provided is that the father's current age [tex]\( = 0 \)[/tex]. This unusual circumstance hints at unique starting conditions; specifying explicit concrete relations showed one acceptable set of values: [tex]\( x = 0 \)[/tex].
Thus, the age of the father is:
[tex]\[
0
\][/tex]
This indicates all ages were initially zero. Since mathematical context arises that fits appropriate constraints, numeric analysis leads to an understanding, asserting father’s age and current sum initially equaled zero.
