Translating Age Problems into Algebra
Age problems are a classic application of linear equations in introductory algebra. They test your ability to read natural language descriptions regarding time, relationships, and human ages, and accurately translate them into solvable mathematical formulas. The secret to mastering these word problems is understanding how to represent the past, present, and future structurally.
Defining the Variables
The first and most critical step is defining your unknown variable. Usually, we let \( x \) represent the present age of the youngest or primary person in the word problem. Once \( x \) is established, every other person's age is defined relative to \( x \).
- Past Ages: If a problem mentions "5 years ago," you must subtract 5 from the current age expression. E.g., \( (x - 5) \).
- Future Ages: If a problem states "in 3 years" or "3 years from now," you add 3 to the expression. E.g., \( (x + 3) \).
- Multiplicative Relationships: Phrases like "twice as old" or "three times as old" require multiplication. E.g., \( 2x \) or \( 3(x + 5) \).
Setting Up the Linear Equation
Once all ages are represented algebraically, you look for the balancing statement in the word problem to create your equation (using the equal sign). For example, if the problem states: "In 5 years, John will be twice as old as he was 3 years ago," you translate it into the equation: \( x + 5 = 2(x - 3) \).
By inputting this exact linear equation into our calculator above, the mathematical engine will automatically distribute the coefficients, combine like terms, and isolate \( x \) to reveal the person's current chronological age.