Problem 69 - Algebra and Trigonometry

Question

In a film, the actor Charles Coburn plays 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 uncle is 60 years old and the woman is 20 years old.

Step 1: Form the First Equation

Let's denote the uncle's current age as \( U \) and the woman's current age as \( W \). The text gives that currently, the uncle is three times her age. This can be represented as an equation: \( U = 3W \)

Step 2: Form the Second Equation

The text also gives that in 20 years, the uncle will be twice as old as the woman. In 20 years, the uncle's age will be \( U + 20 \) and the woman's age will be \( W + 20 \). This gives us equation: \( U + 20 = 2(W + 20) \)

Step 3: Solve the Equations

We can now solve both equations together to find the values of \( U \) and \( W \). Plugging the first equation into the second could be easiest in this case: \( 3W + 20 = 2W + 40 \). Solving for \( W \) shows \( W = 20 \). And substituting \( W = 20 \) into the first equation gives \( U = 60 \)

Step 4: Interpret the Solution

Found the values of \( U \) and \( W \) were 60 and 20 respectively. So, the uncle is 60 years old and the woman is 20 years old. This solution matches with the conditions given in the problem statement.

Key Concepts

Algebraic Equations

Understanding algebraic equations is vital for solving age word problems like the one about Charles Coburn and his movie character's witty retort. An algebraic equation is a mathematical statement that expresses the equality of two algebraic expressions using variables and constants. In the given problem, we start by establishing the equation U = 3W, which directly translates the sentence stating that the uncle is three times the woman's age.

To solve such equations, one can perform arithmetic operations such as addition, subtraction, multiplication, and division on both sides of the equation to isolate the variable. An important thing to keep in mind is that whatever operation is done on one side of the equation must also be done on the other side to maintain the equality.

Systems of Equations

When multiple relationships are presented in a problem, we often need to use systems of equations to find a solution. A system of equations is a set of two or more equations that share common variables, and we're looking for a common solution to all the equations.

In our example, we have a second equation, U + 20 = 2(W + 20), representing the future ages of the characters. Solving the system involves combining these equations using methods such as substitution or elimination. In this case, substituting the value of U from the first equation into the second one allows us to solve for W and consequently find the value of U. Systems of equations can model complex relationships and are useful in fields beyond math, like economics and engineering.

Variables in Algebra

In algebra, variables are symbols—commonly letters—that represent unknown values we aim to find. Variables can stand for anything: numbers, people's ages, quantities of items, or any other measure. The challenge and utility of algebra come from manipulating these variables to model real-world situations and solve problems.

In our age problem, we chose U to represent the uncle's age and W to represent the woman's age. We then use the given information to write down relationships between these variables in the form of equations. Familiarity with variables and their use in algebraic expressions is a foundational skill for all subsequent mathematics. It's important to choose variable representations that make it easier to understand and solve the problem, just like we did in breaking down the ages of the uncle and the woman.