Problem 52 - College Algebra with Corequisite Support

Question

Three numbers sum up to 147 . The smallest number is half the middle number, which is half the largest number. What are the three numbers?

Step-by-Step Solution

Answer

The numbers are 21, 42, and 84.

Step 1: Define Variables

Let's denote the smallest number as \( x \), the middle number as \( y \), and the largest number as \( z \). According to the problem, the smallest number \( x \) is half the middle number \( y \), and the middle number \( y \) is half the largest number \( z \).

Step 2: Formulate Equations from Problem Statement

From the given information, we can write the following equations: \( x = \frac{y}{2} \) and \( y = \frac{z}{2} \). Additionally, we know that the sum of the three numbers is 147, so \( x + y + z = 147 \).

Step 3: Substitute Expressions

Substitute the expressions for \( x \) and \( y \) from the first two equations into the sum equation. We get: \( \frac{y}{2} + y + z = 147 \) and substitute \( z = 2y \) from \( y = \frac{z}{2} \), giving us \( \frac{y}{2} + y + 2y = 147 \).

Step 4: Simplify and Solve for y

Combine all the terms in \( y \) by adding: \( \frac{y}{2} + y + 2y = 3.5y = 147 \). Solve for \( y \) by dividing both sides by 3.5: \( y = \frac{147}{3.5} = 42 \).

Step 5: Solve for x and z

Now that we know \( y = 42 \), we can find \( x \) and \( z \). Using \( x = \frac{y}{2} \), we have \( x = \frac{42}{2} = 21 \). Using \( z = 2y \), we have \( z = 2(42) = 84 \). Thus, the three numbers are 21, 42, and 84.

Key Concepts

Algebraic Expressions

Algebraic expressions are vital in translating word problems into mathematical equations that we can solve. They represent numbers or quantities in equations with symbols and variables. For instance, when we talk about a number being half of another, in algebraic terms, we might write this as \( x = \frac{y}{2} \). This expression communicates a relationship between two numbers `` and ``. It provides a clear and concise way to represent complex relationships.
Using these expressions, we can simplify problems and explore them more systematically. Through manipulation of equations, we can substitute, rearrange, and calculate solutions step-by-step. This provides an effective strategy for addressing various math challenges.

Key points to remember about algebraic expressions include:

• They allow the representation of relationships between quantities.
• They can be simplified by combining like terms and using algebraic rules.
• They form the basis for more complicated problem-solving techniques in mathematics.

Problem Solving

Problem solving in mathematics involves a systematic approach to finding solutions to given challenges. When tackling a system of equations, this involves understanding the problem, translating it into mathematical language, and using logical steps to arrive at a solution. Let's break down these steps to see how they apply.
First, clearly defining what the problem is asking is crucial. In the provided exercise, we identified relationships between the three numbers. We understood how the smallest related to the middle, and the middle to the largest. This was translated to the equation structure: \( x = \frac{y}{2} \) and \( y = \frac{z}{2} \).

Steps involved in problem solving often include:

• Identifying known quantities and relationships.
• Setting up equations based on these.
• Step-by-step manipulation to isolate variables and solve for unknowns.
• Confirming that solutions satisfy all initial conditions of the problem.

This structured approach makes tackling even complex problems more manageable, thus ensuring a clear, logical pathway to a solution.

Variables

Variables play a central role in mathematical problem solving and are fundamental to forming algebraic expressions. They allow us to represent unknown values in equations, which we can then solve. In our example, we assigned variables \( x, y, z \) to denote the three numbers in question. This assignment helps to express complex ideas in a simpler and more manageable way.

Variables provide the flexibility to manipulate mathematical statements universally. They act as placeholders that can later be ``filled`` with known values once an equation is solved. Here, the relationships between variables are key: \( x = \frac{y}{2} \) and \( y = \frac{z}{2} \) gave us a framework to solve for \( x, y, \) and \( z \).

• Variables can denote unknown quantities whose values are to be found.
• They help in generalizing mathematical ideas and relationships.
• They allow us to form solvable equations, leading us to discover unknown values.

Without variables, solving algebraic expressions in mathematical problems would be considerably more difficult.