Problem 28 - Algebra 1

Question

In your collection of 53 stamps, 37 cost less than S.25. Let \(y\) be the number of stamps that cost \(\$ .25\) or more. Which equation models the situation? A. \(53-y=37\) B. \(53+y=37\) C. \(53+37=y\)

Step-by-Step Solution

Answer

The equation that models the situation is \(53-y=37\).

Step 1: Identify given quantities

The problem gives two important pieces of information: a total of 53 stamps in the collection and 37 of them cost less than $0.25.

Step 2: Define variable

The question lets \(y\) to represent the number of stamps that cost $0.25 or more.

Step 3: Translate word problem to equation

We know that the total collection of 53 stamps is made up of stamps that cost less than $0.25 and those that cost $0.25 or more. Therefore, if we subtract the number of stamps that cost $0.25 or more (\(y\)) from the total number of stamps (53), we will get the number of stamps that cost less than $0.25. This gives the equation \(53-y=37\).

Key Concepts

Word Problems

Word problems in math can often seem challenging at first because they mix words with numbers and require translating everyday language into mathematical expressions. In the algebraic equation exercise about stamps, we are given a context where we have a total number of stamps. Our task is to find how many of them are of a certain type, specifically how many cost $0.25 or more.

To solve word problems effectively, start by identifying keywords and numbers provided. For example, here "37 stamps cost less than $0.25" and "total of 53 stamps." Actively understanding what the problem is asking for is crucial: it asks for the number of stamps that cost $0.25 or more.

Follow these tips to handle word problems better:

• Carefully read the problem multiple times.
• Highlight or underline important numbers and keywords.
• Translate words into mathematical expressions step by step.
• Check if your solution makes practical sense within the context of the problem.

Understanding the language used in word problems is a skill you can improve with practice.

Variables

Variables are symbols that represent unknown quantities in mathematical equations and expressions. They are essential in formulating and solving equations. In this exercise, the variable is represented by the letter \(y\).

The variable \(y\) stands for the number of stamps that cost $0.25 or more. Using variables allows us to easily manipulate and move the equation components to find solutions.

Here's how they work in equations:

• Variables act as placeholders for values we want to find.
• They make equations more general and applicable in various scenarios.
• By adjusting variables, we can isolate them to solve for a specific unknown.

In algebra, defining variables accurately is crucial to modeling a problem so that it can be solved.

Mathematical Modeling

Mathematical modeling involves creating representations of real-world situations using mathematical expressions and equations. This method helps solve problems systematically and accurately. In our exercise, the mathematical model aims to represent the relationship between the total number of stamps and the number that cost less than \(0.25.

To create a mathematical model for this problem, we:

• Identified the total number of items involved (53 stamps).
• Recognized the relationship between the two groups (stamps less than \)0.25 and those $0.25 or more).
• Formulated an equation to solve for the unknown quantity \(y\): \(53 - y = 37\).

This kind of modeling is useful because it simplifies complex problems. It breaks them down into manageable mathematical terms that can be solved through algebra.