Answer :
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.
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.
