The age of the father is twice the sum of the ages of his two children. After 20 years, his age will be equal to the sum of the ages of his children. Find the present age of the father.

Step 1: Define variables and set up the first equation

Let $F$ be the present age of the father and $C_1$ and $C_2$ be the present ages of his two children. The problem states that the father's age is twice the sum of the ages of his two children. We can represent the sum of the children's ages as $S = C_1 + C_2$. So, the first equation is $F = 2S$.

Step 2: Set up the second equation based on future ages

After 20 years, the father's age will be $F + 20$. Each child's age will also increase by 20 years, so the sum of their ages will be $S + 20 + 20 = S + 40$. The problem states that after 20 years, the father's age will be equal to the sum of the children's ages. This gives us the second equation: $F + 20 = S + 40$.

Step 3: Substitute and solve for S

Now we have a system of two linear equations: $F = 2S$ and $F + 20 = S + 40$. We can substitute the first equation into the second one to eliminate $F$. Replace $F$ with $2S$ in the second equation, which gives $2S + 20 = S + 40$. Subtract $S$ from both sides and subtract $20$ from both sides to solve for $S$.

Step 4: Calculate the sum of children's ages

From the equation $2S + 20 = S + 40$, we subtract $S$ from both sides to get $S + 20 = 40$. Then, subtract $20$ from both sides to find $S = 20$. So, the present sum of the children's ages is 20 years.

Step 5: Calculate the father's present age

Now that we have the value of $S$, we can use the first equation, $F = 2S$, to find the father's present age. Substitute $S = 20$ into the equation: $F = 2(20) = 40$.

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