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.
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Step-by-Step Solution
Step 1: Define variables and set up the first equation
Let F be the present age of the father and C1 and C2 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=C1+C2. 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.