Problem 31 - 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 movie star?

Step-by-Step Solution

Answer

The movie star is 40 years old.

Step 1: Define Variables

Let the movie star's current age be \( S \) and the daughter's current age be \( D \). We want to find \( S \).

Step 2: Set Up Equations

From the information given, we have two conditions: seven years ago, the movie star was eleven times as old as his daughter, and now he is four times as old. Therefore, the equations are:1. \( S - 7 = 11(D - 7) \)2. \( S = 4D \)

Step 3: Simplify Equation 1

Substitute the first equation: \( S - 7 = 11D - 77 \) which simplifies to \( S = 11D - 70 \).

Step 4: Solve the System of Equations

We have two equations now:1. \( S = 11D - 70 \)2. \( S = 4D \)Equate these two expressions for \( S \):\[ 11D - 70 = 4D \]Rearrange to find \( D \):\[ 11D - 4D = 70 \]\[ 7D = 70 \]\[ D = 10 \]

Step 5: Find the Movie Star's Age

Substitute \( D = 10 \) into \( S = 4D \):\[ S = 4 \times 10 = 40 \]

Step 6: Check the Solution

Verify the solution by checking the initial condition: seven years ago, the age of the movie star was \( 40 - 7 = 33 \) and the daughter's age was \( 10 - 7 = 3 \). The movie star's age was 11 times his daughter's age: \( 11 \times 3 = 33 \). Therefore, the solution is correct.

Key Concepts

System of Equations

In age-related word problems like this one, we often need a method to find unknown values. A system of equations becomes highly useful here. It allows us to handle multiple pieces of information simultaneously, which in this case revolve around ages and times.

Think of a system of equations as a set of mathematical statements that share common variables. These equations need to be solved together to find one or more unknown values. In our movie star riddle, we identified two situations: today and seven years ago.

• The equation for today's ages relates the movie star’s age to being four times that of his daughter.
• The second equation involves their ages seven years prior, with movie star's age being eleven times the daughter's age.

Both components were solved simultaneously to find the current ages. Through such practice, solving complex life-like problems becomes more structured and manageable.

Algebraic Expressions

Algebraic expressions are fundamental when setting up equations for age word problems. They help translate written descriptions into mathematically solvable expressions.

In our exercise, we made use of algebra to form expressions that correspond to the ages. Let's break it down:

• Define variables: Let the movie star’s age be represented as \( S \), and the daughter’s age as \( D \).
• Use these definitions to create expressions that represent the situations described: \( S - 7 = 11(D - 7) \) and \( S = 4D \).

By representing the problem as algebraic expressions, abstract situations are given clear numerical paths, making it easier to analyze and solve.

Verification of Solutions

Once you've calculated the ages, verification is crucial to ensure the accuracy of your solution. This involves checking your answers against the given conditions of the problem.

For the movie star and daughter scenario, verification ensures that both original conditions hold true:

• The ages calculated today should satisfy the equation \( S = 4D \).
• When you check backwards to 7 years ago, \( S - 7 \) should equal eleven times \( D - 7 \).

Verification process:
- For current ages, \( S = 40 \) and \( D = 10 \): bien checks with today's condition as \( 40 = 4 \times 10 \).
- For past ages: Seven years ago, \( 40 - 7 \) equals \( 33 \), and \( 11 \times 3 = 33 \) satisfies the past condition.
Verifying these solutions helps in avoiding errors and ensuring logical consistency in complex problems.