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Chapter 2: Problem 61

Cliff has \(r\) quarters. Express the value of the quarters in cents.

Short Answer

Expert verified
25r cents

Step by step solution

01

Recognize the value of one quarter

A single quarter has a value of 25 cents.
02

Set up the expression for the total value

If Cliff has \(r\) quarters, then each quarter is worth 25 cents. The total value (in cents) can be expressed by multiplying the number of quarters by the value of a single quarter.
03

Multiply the number of quarters by the value

Multiply \(r\) by 25 to get the total value in cents: \(25r\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Basic Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's like a language that helps us describe relationships and changes.

In this exercise, we use algebra to express the value of Cliff's quarters. Here, the symbol or variable is 'r', representing the number of quarters Cliff has.

The goal is to find an expression for the total value in cents. By following basic algebra rules, specifically focusing on multiplication, we can write a mathematical expression that shows this relationship.
The Concept of Multiplication
Multiplication is one of the foundational operations in mathematics. It allows us to add a number to itself multiple times in a quick and efficient way. For instance, instead of adding 25 four times (25 + 25 + 25 + 25), we can simply multiply 25 by 4 to get 100.

In this exercise, we multiply the value of one quarter (25 cents) by the number of quarters Cliff has (r). This is represented by the expression: \(25r\).

Here, we see how multiplication simplifies our calculations and helps build a concise mathematical expression. It transforms repetitive addition into a single operation, making the problem easier to solve.
Using Variables in Algebra
Variables are symbols in algebra that represent numbers or values that can change. They are very powerful tools because they allow us to write general expressions that can work for any number.

In Cliff's problem, 'r' is a variable representing the number of quarters he has. This means it could be any positive integer, allowing our expression \(25r\) to work whether Cliff has 1, 17, or even 100 quarters.

By using a variable, we can generalize the problem. Instead of solving it over and over for different amounts, we create one single solution that works for any possible number of quarters Cliff might have. This is the strength and beauty of using variables in algebra.

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