Problem 34 - Elementary Algebra

Question

Person A is four times as old as person \(\mathrm{B}\), who is six times as old as person \(\mathrm{C}\), who is twice as old as person D. How old is each person if their combined ages are 189 months?

Step-by-Step Solution

Answer

Answer: Person A is 144 months old, Person B is 36 months old, Person C is 6 months old, and Person D is 3 months old.

Step 1: Representing ages as variables

Let the age of person A be represented as 'a', person B as 'b', person C as 'c', and person D as 'd'. Then we have the following equations based on the given conditions: 1. a = 4b 2. b = 6c 3. c = 2d 4. a + b + c + d = 189

Step 2: Replacing variables in terms of others

From the equations 1, 2 and 3, we can replace each of person A, B, and C's ages in terms of person D's age: 1. a = 4(6c) = 24c (replacing b with 6c based on equation 2) 2. Since c = 2d, a = 24(2d) = 48d (replacing c with 2d based on equation 3) Now, we can rewrite equation 4 as: 48d + 6(2d) + 2d + d = 189

Step 3: Solving for 'd'

Simplify the equation and solve for 'd': 48d + 12d + 2d + d = 189 63d = 189 d = \frac{189}{63} = 3 Now we have the age of person D as 3 months.

Step 4: Finding the ages of other persons A, B, and C

Now, using the value of 'd', we can find the ages of the other persons. c = 2d = 2(3) = 6 months (age of person C) b = 6c = 6(6) = 36 months (age of person B) a = 48d = 48(3) = 144 months (age of person A)

Step 5: Recap the solution

We have the following ages of the persons: Person A: 144 months Person B: 36 months Person C: 6 months Person D: 3 months

Key Concepts

Linear Equations

Linear equations are the foundation of algebra and they're essential for solving age word problems. When we talk about linear equations, we refer to mathematical expressions that represent a straight line when plotted on a graph. These equations have variables that appear to the first power (not squared or cubed) and have constants, which are numbers without variables.

In the context of age word problems, each person’s age can be represented by a linear equation. For instance, if we know that person A is four times as old as person B, this relationship can be expressed as an equation: \( a = 4b \). This simple equation is linear because it describes a direct proportional relation between the age of person A and person B.

To solve age word problems using linear equations, you often set up multiple equations based on the given relationships and then solve for the unknown variables. This leads to an understanding of how the ages are interconnected and allows you to determine the age of each individual.

Variables in Algebra

Variables are symbols used in algebra to represent unknown values. In age word problems like the one provided, we use variables such as 'a', 'b', 'c', and 'd' to represent the ages of persons A, B, C, and D, respectively. Understanding how to choose and use variables is key to setting up and solving algebraic equations.

It's important to define each variable clearly before starting to solve the problem. In our exercise, the variables are defined in relation to each other, encapsulating the information given in the problem statement. This relational approach means that, by finding the value of one variable, such as 'd', we can use substitution to find the values of the other variables. Variables are powerful tools that give us a way to deal with unknown quantities in a logical and systematic manner.

Solving Systems of Equations

Solving systems of equations is about finding the values of the variables that make all the equations true at the same time. When faced with age word problems, we often end up with a system of equations because each stated relationship generates its own equation.

In the exercise, the relationships between people's ages let us write multiple linear equations. To solve them, we apply the substitution method, which is one of the commonly used techniques to deal with systems. This involves expressing all variables in terms of a single one, which reduces the system to a single equation. Once the value of the chosen variable is found, as in 'd = 3' from our exercise, it's substituted back into the other equations to find the ages of the remaining individuals.

Properly solving a system of equations provides coherent and consistent results that align with the context of the original problem, allowing us not only to solve the given task but to check our solutions for possible errors, ensuring that all constraints given in the problem are met.