Problem 23 - Algebra and Trigonometry

Question

A Riddle A movie star, unwilling to give his age, posed the following riddle to a gossip columnist. "Seven years ago, I was eleven times as old as my daughter. Now I am four times as old as she is." How old is the star?

Step-by-Step Solution

Answer

The movie star is 40 years old.

Step 1: Define the Variables

Let the current age of the movie star be \( S \) and the current age of his daughter be \( D \). We need to find the value of \( S \).

Step 2: Set Up the Equations

According to the riddle, seven years ago, the movie star was eleven times as old as his daughter. This gives us the equation: \( S - 7 = 11(D - 7) \). Currently, the star is four times as old as his daughter, which gives us another equation: \( S = 4D \).

Step 3: Simplify the First Equation

Expand and simplify the first equation: \( S - 7 = 11(D - 7) \) becomes \( S - 7 = 11D - 77 \). Thus, \( S = 11D - 70 \).

Step 4: Substitute and Solve

Substitute the expression for \( S \) from the current age condition \( S = 4D \) into the simplified equation from Step 3: \( 4D = 11D - 70 \).

Step 5: Rearrange and Solve for Daughter's Age

Rearrange the equation \( 4D = 11D - 70 \) to \( 70 = 11D - 4D \), which simplifies to \( 70 = 7D \). Thus, \( D = 10 \).

Step 6: Find the Star's Age

Since we know \( S = 4D \) and \( D = 10 \), substitute \( D = 10 \) into the equation \( S = 4D \) to get \( S = 4 \times 10 = 40 \).

Key Concepts

Variables in Equations

In algebra word problems, variables are used to represent unknown values. This allows us to form equations that we can solve. Here, let’s consider the scenario of our riddle involving a movie star and his daughter. We introduce variables to model the star's and his daughter’s ages.

For the riddle problem, we need a way to express these ages in a way that reflects the conditions given in the statement. So, we set:

• \( S \) as the current age of the movie star.
• \( D \) as the current age of his daughter.

Using variables helps simplify the problem into a form that can be tackled methodically. This sets the stage for using algebraic methods such as equation simplification and substitution to find the values of \( S \) and \( D \).

Equation Simplification

Simplifying equations is a core concept in solving algebraic word problems. It involves making equations easier to solve by reducing their complexity.

In our riddle, we start with the relationship given from seven years ago: \( S - 7 = 11(D - 7) \).
This equation translates seven years in the past, revealing a connection between ages. We expand it:

• Distribute the 11 in \( 11 \times (D - 7) \) to get \( 11D - 77 \).
• Reorganize to isolate \( S \): \( S = 11D - 70 \).

By simplifying equations, each variable becomes more manageable and clear in terms of its relationship to the other variables. This is a stepping stone to applying further algebraic techniques like substitution to solve for unknown values.

Substitution Method

The substitution method is a powerful algebraic tool used to find the values of variables by replacing one variable with an expression derived from another equation. This method is particularly effective when dealing with multiple equations.

For the movie star’s puzzle, once the equations were set up, we had:

• The simplified equation \( S = 11D - 70 \).
• A current age condition \( S = 4D \).

Using substitution, we take the second equation, \( S = 4D \), and substitute \( S \) in the first equation:\[ 4D = 11D - 70 \]This step replaces \( S \) in one equation with its equivalent from another, simplifying the problem to one equation with one unknown, \( D \), allowing us to solve directly for the daughter's age.

Age-related Problems

Age-related problems often leverage basic algebra to solve. They typically involve relationships over time expressed through equations.

In this type of problem, you’re usually given a scenario where ages change over time and are related by conditions, like someone being a multiple of another person’s age in the past or present. For the movie star scenario:

• Seven years ago: The star was eleven times the age of his daughter.
• Current: The star is four times his daughter’s age.

Age-related word problems challenge us to think algebraically about how time affects these relationships. This means setting up equations that capture these changing relationships and solving them using methods such as simplification and substitution. Solving these is a matter of logically expressing relationships and working through the math step-by-step.