How Many Years Until a Father is Three Times As Old as His Son?

How Many Years Until a Father is Three Times As Old as His Son?

When faced with the age relationship problem of a father and son, it can be a fun challenge to solve. Let's explore a detailed exploration into a specific problem where a father is 35 years old, and his son is 7 years old. We'll find out how many years it will take for the father to be three times as old as his son.

Setting Up the Equation

In this problem, we need to define the ages in terms of a variable. Let's denote the number of years from now until the father is three times as old as his son by ( x ).

The ages in ( x ) years will be:

The father's age: ( 35 x ) The son's age: ( 7 x )

Forming the Equation

We need to set up an equation based on the condition that the father's age will be three times the son's age:

[ 35 x 3(7 x) ]

Now let's solve this equation step by step:

Step 1: Expand the Equation

Expand the right side of the equation:

[ 35 x 21 3x ]

Step 2: Rearrange the Equation to Isolate ( x )

Move all terms involving ( x ) to one side and the constant terms to the other side:

[ 35 - 21 3x - x ]

Combine like terms:

[ 14 2x ]

Step 3: Solve for ( x )

Divide both sides by 2 to isolate ( x ):

[ x 7 ]

Therefore, in 7 years, the father will be three times as old as the son.

Verification

To verify the solution, let's calculate the ages in 7 years:

The father's age will be ( 35 7 42 ) years. The son's age will be ( 7 7 14 ) years.

Indeed, 42 is three times 14, confirming that our solution is correct.

Alternative Solutions

There are various ways to solve this problem. Here are a few additional methods:

Algebraic Solution 2

Another approach involves simplifying the equation directly:

[ frac{35}{5x} 3 ]

Solving for ( x ), we find:

[ x 10 ]

Ratio Method

We can use the ratio of their ages. Currently, the father is 7 times older than the son (35:5 or 7:1). After 10 years, the ratio will be 3:1 (45:15 or 35:5 10:7 10).

Therefore, in 10 years, the father's age will be three times the son's age.

Sum of Ages Method

If you find it difficult to solve it quickly, you can use the sum of ages. In 7 years, the sum of their ages will be 50:

Their combined age in 7 years: ( 50 ) The son's age in 7 years: ( 10 ) The father's age in 7 years: ( 40 )

Subtract 7 from 40 to get the correct age for the father after 10 years.

Conclusion

In conclusion, by solving the equation or using alternative methods, we have determined that in 7 years, the father will be three times as old as his son. This problem demonstrates the power of algebraic thinking and provides an interesting real-world application of mathematical concepts.