Problem 52 - College Algebra

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 define three variables for the numbers: Let the smallest number be \( x \), the middle number be \( y \), and the largest number be \( z \). According to the problem, \( x + y + z = 147 \).

Step 2: Establish Relationships Between Numbers

From the problem, we know the smallest number \( x \) is half the middle number \( y \), so \( x = \frac{y}{2} \). Also, the middle number \( y \) is half the largest number \( z \), so \( y = \frac{z}{2} \).

Step 3: Substitute Relationships Into Equation

Substitute the expressions for \( x \) and \( y \) from Step 2 into the sum equation: \( \frac{y}{2} + y + z = 147 \). Since \( y = \frac{z}{2} \), substitute \( y \) with \( \frac{z}{2} \) in the equation to get: \( \frac{(\frac{z}{2})}{2} + \frac{z}{2} + z = 147 \).

Step 4: Simplify the Equation

Simplify the equation from Step 3: \( \frac{z}{4} + \frac{z}{2} + z = 147 \). Find a common denominator (4) to combine the fractions: \( \frac{z}{4} + \frac{2z}{4} + \frac{4z}{4} = 147 \), which simplifies to \( \frac{7z}{4} = 147 \).

Step 5: Solve for Largest Number

Solve for \( z \) by multiplying both sides by 4: \( 7z = 588 \). Divide by 7: \( z = 84 \).

Step 6: Calculate Middle and Smallest Numbers

Use the relationships \( y = \frac{z}{2} \) and \( x = \frac{y}{2} \) to find \( y \) and \( x \): \( y = \frac{84}{2} = 42 \) and \( x = \frac{42}{2} = 21 \).

Key Concepts

Systems of Equations

In mathematics, solving problems often involves finding values for multiple variables that satisfy a set of equations. These are known as systems of equations. When dealing with systems, each equation provides a different perspective on the relationships involving the variables. In the given problem, we need to find three numbers. We know their sum is 147 and they have specific relationships between each other. This is a perfect example of a system of equations.

The system of equations allows us to combine multiple relationships and conditions into a structured problem-solving approach. These equations can be solved using various methods, such as substitution, elimination, or graphical interpretations. Understanding how to set up and solve systems of equations will pave the way for tackling more complex mathematical problems.

In our case, the first equation is derived from the condition on the sum of the numbers:

• \( x + y + z = 147 \)

It's the interplay and combination of such equations that can lead us efficiently to the solution.

Variable Substitution

Variable substitution is a vital tool in solving systems of equations. It involves expressing one variable in terms of another, thus reducing the number of unknowns in the equations. This technique helps simplify complex problems by breaking them down into more manageable parts.

In our problem, the relationships between the numbers gave us direct expressions for substitution. The smallest number \( x \) is half the middle number \( y \), which we express as:

• \( x = \frac{y}{2} \)

And, the middle number \( y \) being half the largest number \( z \) gives:

• \( y = \frac{z}{2} \)

By substituting these into our sum equation, we reduce the problem from one involving three variables to one that involves just one variable \( z \).

Substitution helps in significantly simplifying the expression for the solution, making it easier to manage and solve. This clean approach ensures no redundant calculations, saving time and reducing errors.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. They are the building blocks of algebraic reasoning. Our problem involves linear equations as the relationships between the variables are proportional and do not involve squares or other higher powers.

Here's how linearity helped solve this problem:

• The equation \( x + y + z = 147 \) is linear.
• The relationships \( x = \frac{y}{2} \) and \( y = \frac{z}{2} \) are also linear, expressing a direct and constant rate of change between the numbers.

By maintaining this linear relationship, we simplify the problem without losing accuracy or complexity. Linear equations are straightforward to manipulate and provide clear, direct answers, which is precisely why they are fundamental in algebra.

They enable us to understand how changes in one variable affect others, and thus offer an efficient way to track and solve problems involving direct proportionality.