Problem 84 - Algebra A Combined Function

Question

Write each algebraic expression described. The value of 7 nickels is \(5 \cdot 7\) cents. Likewise, the value of \(x\) nickels is \(5 x\) cents. If the money box in a drink machine contains \(x\) nickels, \(3 x\) dimes, and \((30 x-1)\) quarters, express their total value in cents as an algebraic expression.

Step-by-Step Solution

Answer

The total value is \(785x - 25\) cents.

Step 1: Calculate the Value of Nickels

The value of one nickel is 5 cents. If we have \(x\) nickels, then the total value of the nickels is \(5x\) cents. This means for \(x\) nickels, the total value is given by the expression \(5x\).

Step 2: Calculate the Value of Dimes

A single dime is worth 10 cents. With \(3x\) dimes, the value is calculated by multiplying the number of dimes by 10. The expression for the value of the dimes is \(10 \cdot 3x = 30x\) cents.

Step 3: Calculate the Value of Quarters

Each quarter is worth 25 cents. We have \((30x - 1)\) quarters, so the expression for their total value is \(25(30x - 1)\) cents. Simplify this to get the expression \(750x - 25\) cents.

Step 4: Add the Values Together

To find the total value in cents, sum up the values of nickels, dimes, and quarters. The total value expression is: \(5x + 30x + 750x - 25\).

Step 5: Simplify the Expression

Combine the like terms in the expression: \(5x + 30x + 750x - 25 = (5 + 30 + 750)x - 25 = 785x - 25\). Thus, the total value in cents is given by the simplified expression \(785x - 25\).

Key Concepts

Understanding the Value of Coins

When working with coins, it's essential to know the value of each type. Nickels, dimes, and quarters each have a different worth, which needs to be considered when calculating the total value of several coins.

• A nickel is worth 5 cents.
• A dime is worth 10 cents.
• A quarter is worth 25 cents.

Knowing these values allows you to create algebraic expressions representing the total value of varying amounts of these coins. For example, if you have a certain number of nickels, you multiply the value of one nickel (5 cents) by the number of nickels to find their total value.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means combining all like terms to make the expression as neat as possible. This process helps you see the relationship between terms more clearly and makes it easier to work with the expression in calculations or equations.
Here is a simple guide for simplifying expressions:

• Identify like terms: These are terms that have the same variable raised to the same power.
• Add or subtract the coefficients of the like terms.
• Rewrite the expression with the simplified terms.

For instance, in the expression \(5x + 30x + 750x - 25\), the terms \(5x\), \(30x\), and \(750x\) are like terms, meaning they all have the same variable \(x\). By adding their coefficients, you simplify the expression to \(785x - 25\), where \(785x\) represents the total simplified value, and \(-25\) remains as a constant term.

Step-by-Step Solution to Evaluate Expression

Solving algebraic problems often requires you to follow a structured approach. Let's walk through the example step-by-step.First, calculate the value of each type of coin:

• Nickels: For \(x\) nickels, the total value is \(5x\) cents.
• Dimes: With \(3x\) dimes, the value becomes \(30x\) cents (because each dime is worth 10 cents).
• Quarters: If there are \((30x - 1)\) quarters, the value is \(25(30x - 1)\) cents, which simplifies to \(750x - 25\).

Next, you sum these individual values to form one expression: \(5x + 30x + 750x - 25\).
Finally, simplify the expression by combining like terms, resulting in \((5 + 30 + 750)x - 25 = 785x - 25\).
This detailed process ensures you accurately represent the total value using algebraic expressions.