Problem 34 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation. Let x represent the number. Use mental math to solve the equation. Then check your solution. The product of a number and 25 is 100.

Step-by-Step Solution

Answer

The number 'x' that makes the sentence 'The product of a number and 25 is 100' true is 4.

Step 1: Write The Given Sentence as an Equation

The sentence 'The product of a number and 25 is 100.' can be written as the equation \(25x = 100\), where x represents the unknown number.

Step 2: Solve The Equation Using Mental Math

To solve for x, we divide both sides of the equation by 25: \(x = 100 / 25\). This simplifies to \(x = 4\).

Step 3: Check the Solution

To ensure the solution is correct, substitute \(x = 4\) back into the original equation \(25x = 100\). The left-hand side (LHS) of the equation then becomes \(25 * 4 = 100\), which is exactly equal to right-hand side (RHS) (100). Thus, confirming the solution is correct.

Key Concepts

Product in Equations

In mathematics, the term _product_ refers to the result obtained when multiplying two or more numbers together. In algebra, we often work with equations that include products. When you see a phrase like "the product of a number and 25," it signals that you'll be multiplying the unknown number by 25. To form an equation from this description, you use a multiplication operation. Suppose we let the unknown number be represented by the variable \( x \). The sentence "the product of a number and 25 is 100" translates into the algebraic equation:

• \( 25x = 100 \)

This means 25 times some number \( x \) equals 100. Identifying the product in such equations is crucial as it sets the foundation for finding the value of the variable through solving the equation.

Variables in Algebra

Variables are symbols, often letters, that represent unknown values in mathematical expressions and equations. They serve as placeholders for numbers that can vary or that we need to find. In this case, the variable \( x \) has been used to represent an unknown number in the equation \( 25x = 100 \).Understanding variables is a fundamental concept in algebra, enabling us to formulate and manipulate equations. Here are some key points about variables:

• Variables can be any letter, but \( x \) is commonly used.
• They allow us to generalize mathematical relationships.
• You can solve equations to find the specific value that a variable represents.

By using \( x \) in our equation, we can systematically solve for the number by finding out what value makes the equation true. Solving for \( x \) involves operations that isolate the variable, enabling us to determine its numerical value.

Checking Solutions in Equations

After finding a solution to an equation, it's vital to check that the solution is correct. This step ensures that our mathematical operations were accurate and that our understanding of the problem is sound. To verify a solution, substitute the value back into the original equation and see if both sides are equivalent.For the equation \( 25x = 100 \), we determined \( x = 4 \) by dividing both sides by 25:

• \( x = 100/25 = 4 \)

To check, we substitute \( x = 4 \) back into the equation:

• Multiply 25 by 4: \( 25 \times 4 = 100 \).
• The original equation is \( 25x = 100 \), so the left side matches the right side, confirming \( 100 = 100 \).

This step reassures us that \( x = 4 \) is indeed the correct solution. Checking is an essential habit in solving equations to prevent errors and ensure understanding.