Problem 41 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ \frac{18 a}{5} \cdot \frac{15}{6} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{18a}{5} \cdot \frac{15}{6} \) is \( 9a \)

Step 1: Multiplying the numerators

Multiply the numerators of the two fractions, which are 18a and 15. This gives a result of \( 18a \times 15 = 270a \)

Step 2: Multiplying the denominators

Multiply the denominators of the two fractions, which are 5 and 6. This gives a value of \( 5 \times 6 = 30 \)

Step 3: Write the resulting fraction

The result of the multiplication should be written as a new fraction with the result from Step 1 as the numerator and the result from Step 2 as the denominator. Therefore, the simplified expression will be \( \frac{270a}{30} \)

Step 4: Simplify the fraction

When 270a is divided by 30, we get 9a. Therefore, the final simplified expression of the given is \( 9a \)

Key Concepts

Multiplying Fractions

Multiplying fractions is an essential skill in algebra that allows you to combine two fractions into a single fraction. The process is quite straightforward. To multiply two fractions, you simply multiply their numerators (the top numbers) to get the new numerator, and multiply their denominators (the bottom numbers) to get the new denominator.

For example, consider the multiplication of \( \frac{18a}{5} \) and \( \frac{15}{6} \). Here, you'd multiply \( 18a \) by \( 15 \) to get the numerator of the new fraction, and \( 5 \) by \( 6 \) to get the denominator, resulting in \( \frac{270a}{30} \) before simplification. Remember to always express your final answer in simplest form by reducing the fraction to its lowest terms, if possible.

Simplifying Fractions

Simplifying fractions is another important concept in algebra, where you reduce the fraction to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

In the context of our example, to simplify the fraction \( \frac{270a}{30} \), we notice that both numbers are divisible by 30. Therefore, when you divide the numerator and the denominator by 30, the fraction simplifies to \( \frac{9a}{1} \), or simply \( 9a \). It's crucial to simplify your fractions as much as possible to achieve the most straightforward expression of the solution.

Algebraic Numerators and Denominators

In algebra, numerators and denominators can contain variables, as well as numbers. When you have algebraic expressions in the numerator or the denominator, the approach to multiplication or simplification is the same as with numerical fractions.

However, you need to pay attention to any like terms and factors that can be simplified or canceled out. In the product of \( \frac{18a}{5} \) and \( \frac{15}{6} \), for example, the variable 'a' is part of the numerator and should be treated as a factor when simplifying. It is important not to ignore the variable component in algebraic fractions, as it is essential in simplifying the expression correctly.

Elementary Algebra

Elementary algebra is the backbone of all higher-level mathematics and includes the fundamental operations of solving equations, multiplying and simplifying algebraic expressions, like we practiced with fractions.

A good grasp of elementary algebra is essential for understanding and performing operations with algebraic fractions. This includes knowing how to multiply binomials, factor polynomials, and simplify expressions with variables. As in our example, simplifying the fraction \( \frac{270a}{30} \) to \( 9a \) involves elementary algebraic techniques like recognizing factors and understanding variable manipulation. These skills are foundational for progressing through more advanced mathematical concepts.