Five years ago, a man was seven times as old as his son. Five years hence, the father will be three times as old as his son. Find their present ages.

Understand the Problem

The question is an age problem involving a father and son. It provides information about their ages five years ago and five years in the future. The goal is to determine their current ages. We can set up a system of equations to solve for the ages.

Answer

Father is $40$ years old and son is $10$ years old.

Steps to Solve

Define the variables

Let $f$ be the father's current age and $s$ be the son's current age.

Set up the first equation based on the information "Five years ago a man was seven times as old as his son"

Five years ago, the father's age was $f - 5$ and the son's age was $s - 5$. Thus,

$$f - 5 = 7(s - 5)$$

Simplify the first equation

Expanding the equation, we get $f - 5 = 7s - 35$.

Rearranging, we have $f = 7s - 30$

Set up the second equation based on the information "Five years hence, the father will be three times as old as his son"

Five years hence, the father's age will be $f + 5$ and the son's age will be $s + 5$. Thus,

$$f + 5 = 3(s + 5)$$

Simplify the second equation

Expanding the equation, we get $f + 5 = 3s + 15$.

Rearranging, we have $f = 3s + 10$

Solve the system of equations

We have two equations: $f = 7s - 30$ $f = 3s + 10$

Since both equations are solved for $f$, we can set them equal to each other: $7s - 30 = 3s + 10$

Solve for s Subtract $3s$ from both sides: $4s - 30 = 10$ Add $30$ to both sides: $4s = 40$ Divide by $4$: $s = 10$
Solve for f Substitute $s = 10$ into either equation. Let's use $f = 3s + 10$: $f = 3(10) + 10$ $f = 30 + 10$ $f = 40$
State the present ages

The father's current age is 40 and the son's current age is 10.

Father's age: $40$ Son's age: $10$

More Information

The father is currently 40 years old and the son is currently 10 years old. Five years ago, the father was 35 and the son was 5, so the father was indeed seven times as old as his son. Five years from now, the father will be 45 and the son will be 15, so the father will be three times as old as his son.

Tips

A common mistake is to incorrectly set up the equations. For example, confusing "five years ago" with "five years from now" or not properly distributing the multiplication when setting up the equations. Another mistake is to make errors in the algebraic manipulation when solving the system of equations.

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