Problem 22 - Algebra and Trigonometry

Question

A Riddle A father is four times as old as his daughter. In 6 years, he will be three times as old as she is. How old is the daughter now?

Step-by-Step Solution

Answer

The daughter is 12 years old now.

Step 1: Define Variables

Let's begin by defining variables to represent the ages. Let \( f \) be the father's current age, and \( d \) be the daughter's current age.

Step 2: Establish the First Equation

According to the riddle, the father is currently four times as old as his daughter. Thus, we can write the equation: \[ f = 4d \]

Step 3: Establish the Second Equation

The riddle also states that in 6 years, the father will be three times as old as the daughter. In 6 years, the father's age will be \( f + 6 \), and the daughter's age will be \( d + 6 \). This leads to the equation: \[ f + 6 = 3(d + 6) \]

Step 4: Solve the Second Equation for f

Expand the second equation to isolate \( f \): \[ f + 6 = 3d + 18 \] \[ f = 3d + 12 \]

Step 5: Solve the System of Equations

We have two equations: \( f = 4d \) and \( f = 3d + 12 \). Set these equal since both expressions represent \( f \): \[ 4d = 3d + 12 \] Subtract \( 3d \) from both sides to solve for \( d \): \[ d = 12 \]

Step 6: Verify the Solution

Check if \( d = 12 \) satisfies both conditions. Substitute \( d = 12 \) into \( f = 4d \): \( f = 48 \). In 6 years, the daughter's age will be 18 and the father's age will be 54. \( 54 = 3 \times 18 \), which checks out.

Key Concepts

Linear Equations

Linear equations are one of the building blocks of algebra. They express a relationship where one variable is equal to a constant, multiplied by another variable, plus a constant. In simpler terms, they allow us to find unknown values using given information.

A typical linear equation looks like this: \[ ax + b = c \]Where:

• \(a\) is the coefficient or the number that multiplies the unknown variable \(x\),
• \(b\) is a constant added to the product of \(a\) and \(x\),
• \(c\) is what the expression is equal to.

In our exercise, we used the first equation \( f = 4d \) to express the father's age in terms of the daughter's age. Here, \(4\) is the coefficient that indicates the father's age is four times that of the daughter's.

Linear equations are essential because they make it possible to represent real-world situations and solve for unknowns. Once you understand how they work, you unlock the ability to tackle a multitude of problems by describing them mathematically.

System of Equations

A system of equations occurs when we have more than one equation working together to define relationships between variables. By solving these systems, we can determine the values of unknowns that satisfy all equations simultaneously.

In our age problem exercise, we established two equations:

• \( f = 4d \)
• \( f + 6 = 3(d + 6) \)

These two equations together form a system. To solve them:

• We equated both expressions of \(f\) since they must represent the same value. This gave us \( 4d = 3d + 12 \).
• We simplified to find \( d = 12 \), which is the solution that fits both conditions.

Systems of equations are widespread in various mathematical applications because many scenarios involve multiple relationships that must be maintained. Mastering how to solve them is a critical algebraic skill.

Age Problems

Age problems are a common type of word problem in algebra where you determine the ages of individuals based on given relationships. These problems often involve forming equations based on the ages now, in the past, or in the future.

In our exercise, the problem sets up relationships based on the current and future ages of a father and daughter. It uses statements like "the father is four times as old as his daughter now" and "in 6 years, he will be three times as old." From these statements, we derived equations that could be solved systematically.

To tackle age problems effectively:

• Start by defining variables clearly for the ages involved.
• Create equations based on the relationships mentioned in the problem.
• Use algebraic methods like substitution or elimination to solve the equations.
• Check your solutions by substituting back into the context of the problem.

Practicing age problems helps strengthen problem-solving skills essential for more complex algebraic questions and tests your ability to interpret and apply mathematical concepts to real-world scenarios.