Question

Multiply and reduce. Do some by calculator. $$\frac{x+y}{10} \cdot \frac{a x}{3(x+y)}$$

Short answer

\(\frac{ax}{30}\)

Step-by-step solution

Simplify Common Factors

Identify and cancel out the common factors in the numerator and the denominator. In this case, the term \(x+y\) is present in both the numerator and the denominator, so it can be canceled out.

Multiply the Numerators

Multiply the numerators of the fractions. This is done by multiplying \(\frac{1}{10}\) and \(ax\) together.

Multiply the Denominators

Multiply the denominators of the fractions. Since the common factor \(x+y\) has been canceled, the only term left in the denominator is 3.

Express the Simplified Result

After canceling and multiplying, write down the simplified result of the fractions.

Key concepts

Common Factors in Algebra

Identifying common factors is a crucial step in simplifying algebraic expressions. A common factor is a term or number that is a factor of two or more terms or numbers. In the context of algebraic fractions, finding and canceling common factors can vastly reduce the complexity of an expression. For our exercise, the term (x+y) appears in both the numerator and the denominator, meaning it can be canceled out, simplifying the expression.

If you spot a common factor in both the top and bottom of a fraction, you can divide each by that factor. This method is essentially applying the property that states a/a = 1, so they cancel each other out. Remember, though, that you can only cancel factors — not terms that are separated by addition or subtraction — which is why finding common factors is so powerful in algebraic fraction reduction.

Multiplication of Fractions

Multiplication of fractions is quite straightforward, yet it can intimidate students when variables are involved. The key rule to remember is to multiply the numerators (top numbers) together and the denominators (bottom numbers) together. In our example, once the common factor of (x+y) is removed, you continue by multiplying the numerators: 1/10 by ax, which is just ax/10.

This multiplication doesn't alter the nature of the factors; it combines them. With numbers, this is a simple arithmetic operation, but with algebra, it combines variables and constants into a single term or polynomial. This is why understanding multiplication with fractions is essential, even more so when simplifying algebraic expressions.

Fraction Reduction Techniques

There are several techniques to reduce fractions, making them simpler to work with, especially when handling algebraic fractions. After cancelling common factors, as shown in the previous sections, we sometimes need to take additional steps to simplify the expression further. In the given example, once the common factor is canceled, we are left to multiply the numerically expressed fractions. The reduction process may involve factoring algebraic expressions, finding the greatest common divisor (GCD), or simplifying complex fractions to a lower term.

These reduction techniques help in presenting expressions in their simplest form, enabling further algebraic operations or making it easier to evaluate. Always look for opportunities to reduce fractions; it will not only clean up your workings but also improve your understanding of the relationships between different algebraic quantities.