Problem 61 - Introductory Algebra for College Students

Question

In a film, the actor Charles Coburn played an elderly "uncle" character criticized for marrying a woman when he is 3 times her age. He wittily replies, "Ah, but in 20 years time I shall only be twice her age." How old is the "uncle" and the woman?

Step-by-Step Solution

Answer

The woman is 20 years old, and the uncle is 60 years old.

Step 1: Formulate the equations

Let's denote the uncle's current age as \( X \) and the woman's age as \( Y \). Based on the problem, we have two equations. One is from the statement that the uncle (Charles) is currently three times as old as the woman which gives us \( X = 3Y \). The other one is from the statement that in 20 years, Charles will be twice as old as the woman which can be written as \( X+20 = 2 * (Y+20) \).

Step 2: Substitute the first equation into the second one

Replace X in the second equation with \( 3Y \) (from the first equation), this will result in \( 3Y+20 = 2 * (Y + 20) \).

Step 3: Simplify the equation

Expand and then Simplify the equation. This will result in \( 3Y + 20 = 2Y + 40 \). After rearranging the terms we get the woman's age \( Y = 20 \).

Step 4: Find the uncle's age

Substitute \( Y = 20 \) into the first equation \( X = 3Y \) we get Uncle's age \( X = 60 \).

Key Concepts

algebraic equations

Algebraic equations are mathematical statements that show the equality between two expressions with unknown variables. In these equations, we often use symbols like \( X \) and \( Y \) to represent unknown numbers. The key goal is to manipulate these equations to find the values of the unknowns.

For example, in the exercise, we have two algebraic equations:

• The uncle's age is three times the woman's age: \( X = 3Y \).
• In 20 years, the uncle will be twice the woman's age: \( X + 20 = 2 \times (Y + 20) \).

These equations form the basis of solving for the ages by using algebraic methods to find the values of \( X \) and \( Y \). By understanding and forming these equations correctly, solving age word problems becomes a systematic task.

substitution method

The substitution method is a technique used in algebra to find the values of variables in equations. This method involves solving one equation for one variable and then substituting that expression into another equation.

In our example, we initially have the equation \( X = 3Y \). This tells us that Charles's age \( X \) is three times the woman's age \( Y \). To use substitution, we replace \( X \) in the second equation \( X+20 = 2 \times (Y+20) \) with \( 3Y \):

\[ 3Y + 20 = 2 \times (Y + 20) \]

By substituting \( 3Y \) in place of \( X \), we simplify the problem to a single variable equation in terms of \( Y \), making it easier to solve.

The substitution method is particularly useful in solving systems of equations where you need to reduce the problem to a single variable before finding the solution.

linear equations

Linear equations are equations of the first degree, meaning they have the highest exponent of the variable as one. They form a straight line when graphed, and they are characterized by their simple, direct relationships between variables.

In the given exercise, both equations are linear:

• \( X = 3Y \)
• \( X + 20 = 2 \times (Y + 20) \)

These linear equations correspond to simple proportional relationships between the uncle's age and the woman’s age. The key to solving them is to isolate one variable and then systematically solve for the other.

Linear equations are foundational in algebra and are widely used in many real-world applications, from predicting trends to solving problems involving rates and ratios. Understanding how to manipulate and solve linear equations helps in tackling more complex algebraic problems efficiently.