Solving Age Problems The Father-Daughter Age Puzzle With Linear Equations

by Marta Kowalska 74 views

Hey there, math enthusiasts! Today, we're diving into a classic age problem that we'll solve using a system of linear equations and the substitution method. These types of problems are not only great for sharpening our algebra skills but also show us how math can be applied to everyday scenarios. So, let's jump right into it!

Problem Statement

The sum of the ages of a father and his daughter is 56 years. Ten years ago, the father's age was five times the age his daughter had at that time. What is the current age of each of them?

This problem presents us with a couple of key pieces of information that we can translate into mathematical equations. The first clue is the sum of their current ages, and the second is the relationship between their ages ten years ago. By carefully setting up our equations, we can unravel this age mystery.

Setting up the Equations

To tackle this problem, let's use variables to represent the unknowns. Let:

  • F = the father's current age
  • D = the daughter's current age

From the problem statement, we can form two equations:

  1. The sum of their current ages is 56:

    F + D = 56

  2. Ten years ago, the father's age was five times the daughter's age. To represent their ages ten years ago, we subtract 10 from their current ages:

    F - 10 = 5(D - 10)

Now we have a system of two linear equations with two variables. This is where the substitution method comes into play.

Solving with the Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which we can easily solve.

Let's start by solving the first equation for F:

F = 56 - D

Now, we substitute this expression for F into the second equation:

(56 - D) - 10 = 5(D - 10)

Next, we simplify and solve for D:

46 - D = 5D - 50

Add D to both sides and add 50 to both sides:

46 + 50 = 5D + D

96 = 6D

Divide both sides by 6:

D = 16

So, the daughter's current age is 16 years old.

Now that we have the daughter's age, we can substitute it back into either of our original equations to find the father's age. Let's use the equation F = 56 - D:

F = 56 - 16

F = 40

Therefore, the father's current age is 40 years old.

Verifying the Solution

It's always a good idea to check our solution to make sure it satisfies the conditions of the problem.

  • The sum of their current ages: 40 + 16 = 56 (Correct!)
  • Ten years ago, the father was 30 and the daughter was 6. Was the father's age five times the daughter's age? 30 = 5 * 6 (Correct!)

Our solution checks out, so we can confidently say that the father is currently 40 years old, and the daughter is 16 years old.

Why Linear Equations Matter

You might be wondering,