Problem 8 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ 15 a, 5 a $$

Step-by-Step Solution

Answer

Answer: The other factor is 3.

Step 1: Rewrite the problem statement in terms of multiplication

We need to understand the problem in terms of multiplication. The product of two factors is given as \(15a\), and one of the factors is \(5a\). We can express this as: $$ 15a = 5a \times (\text{other factor}) $$

Step 2: Divide the product by the given factor

To isolate the other factor, we need to divide the product (\(15a\)) by the given factor (\(5a\)). When we do this, we will obtain the other factor. Using the equation from step 1, this can be written as: $$ \text{other factor} = \frac{15a}{5a} $$

Step 3: Simplify the expression

Now, we need to simplify the expression \(\frac{15a}{5a}\) by canceling out the common terms and reducing the coefficients: $$ \text{other factor} = \frac{15}{5} \cdot \frac{a}{a} $$ Since \(\frac{a}{a} = 1\), and \(\frac{15}{5} = 3\), we can simplify it further as: $$ \text{other factor} = 3 \cdot 1 $$

Step 4: Write the final answer

The other factor in the multiplication that results in the product \(15a\) is \(3\). So, $$ 15a = 5a \times 3 $$

Key Concepts

Factorization

Factorization is a fundamental concept in algebra that involves breaking down complex expressions into their simplest components, called factors. When you factor an expression, you are essentially reversing multiplication to find out what numbers or expressions were multiplied together to get the original expression.

• In our exercise, we start with a product, \(15a\), and one known factor, \(5a\).
• Our goal is to determine the missing factor that, when multiplied with \(5a\), results in \(15a\).

The process involves looking for the greatest common factor (GCF) shared between the two terms, the product and the given factor. In this case, both terms share the factor \(a\), which can be used to simplify the expression. Recognizing common factors simplifies the algebraic problem, making it easier to handle further calculations.

Multiplication

Multiplication in algebra involves combining expressions or numbers to form a product. It's essential to understand both the direct multiplication of simple constants and variables, as well as dealing with expressions.

• In our exercise, the equation \(15a = 5a \times (\text{other factor})\) shows how numbers and variables can combine to create a product.
• The aim is to rewrite the product as a multiplication involving all known factors and to determine any unknown ones.

When we multiply expressions such as \(5a\) and the unknown factor, we apply the distributive property if necessary to manage expressions involving sums or differences. Here, however, multiplication is straightforward as we are dealing with single terms. Multiplying and later dividing terms is common in revealing the more hidden components of multiplication in terms of factorization.

Simplifying Expressions

Simplifying expressions is a crucial process in algebra that makes working with expressions more manageable and interpretable. It involves reducing the expression to its simplest form, usually by canceling out terms or combining like terms.

• In our exercise, once we have \(\frac{15a}{5a}\), simplifying involves canceling common terms.
• This process reduces the fraction \(\frac{15}{5} \cdot \frac{a}{a}\), simplifying to \(3 \cdot 1\), revealing the factor as \(3\).

The key to simplification is recognizing terms that are identical and can cancel out, such as \(a/a\) which equals 1. This indicates that these terms are essentially the same, reducing the complexity of the expression. It’s about making expressions as streamlined as possible for ease of understanding and further use in equations. Simplifying reduces the potential for errors and enhances the clarity of the mathematical process.