Problem 92 - Intermediate Algebra

Question

Translate into mathematical symbols: The weight of an object in ounces is the product of 16 and its weight in pounds.

Step-by-Step Solution

Answer

The equation is \( y = 16x \).

Step 1: Understanding the Problem

We need to convert the given verbal statement into a mathematical equation. The statement involves weights in ounces and pounds.

Step 2: Identifying Variables

Let's assign variables: \( x \) represents the weight in pounds and \( y \) represents the weight in ounces.

Step 3: Interpreting 'The Product Of'

The phrase 'the product of 16 and its weight in pounds' indicates multiplication. Therefore, mathematically, we will express it as \( 16 \, \times \, x \), where \( x \) is the weight in pounds.

Step 4: Formulating the Equation

Now, setting up the equation based on the statement 'the weight of an object in ounces is', we get \( y = 16x \). This equation captures the relationship as described in the statement.

Key Concepts

Translating Verbal Statements into Math Language

Translating verbal statements into mathematical symbols is a crucial skill in algebra. It allows us to represent real-world situations in a form that is easy to manipulate mathematically. In the exercise, we are given a statement about weights in pounds and ounces. The aim is to take this verbal description and create an equation that conveys the same information.
This often involves identifying key phrases that indicate mathematical operations.

• "The product of" suggests multiplication.
• "Is" typically denotes equality.

Understanding these phrases helps convert a verbal statement into a meaningful mathematical expression. With practice, recognizing these patterns becomes easier, allowing for a seamless translation from words to math.

Understanding Mathematical Symbols

Mathematical symbols are the language of forms and operations. In our exercise, symbols such as multiplication (\( \times \)) and equality (=) play vital roles in constructing equations. Multiplication can be symbolized in different ways, but in algebra, we often use a space or parentheses, especially when working with variables.
Let's break it down:

• \( 16 \times x \) means '16 times the weight in pounds', where "\( x \)" is the variable.
• The equals symbol (=) is used to show that two expressions are the same.

These symbols are essential as they form the foundation of any algebraic equation. They not only simplify complex phrases but also allow for powerful solutions when solving mathematical problems.

Assigning Variables for Clarity

Variable assignment is a fundamental concept in mathematics. It helps in presenting a clean and precise version of a problem. Variables are often letters like \( x \), \( y \), and \( z \) that stand for numbers we don't yet know.
In this exercise, we assigned:

• \( x \) to the weight in pounds
• \( y \) to the weight in ounces

This clear assignment makes it easier to write equations and solve them. Without variable assignment, translating complex sentences directly into math could lead to confusion and errors. It allows anyone solving the problem to follow the logical flow of the sequence and understand which quantities are involved.

Crafting the Mathematical Equation

Equation formulation is the ultimate goal when translating verbal descriptions into math problems. We take all elements identified earlier—variables, mathematical operations, and symbols—and weave them into a cohesive statement.
In this specific problem, the equation

• \( y = 16x \)

was formulated from the phrase "the weight of an object in ounces is the product of 16 and its weight in pounds."
The left side of the equation, \( y \), stands for the ounces, while the right side, \( 16x \), shows the mathematical operation that results in ounces from pounds. This equation precisely expresses the relationship and allows for further calculations, such as determining ounces if the pounds are known. It simplifies and solidifies the understanding of the problem's logic.