Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ \frac{a+b}{b} \div \frac{a-b}{a b}=\frac{a(a+b)}{a-b} $$

Step-by-Step Solution

Answer

The real number property that justifies the statement is the Commutative Property of Addition, which allowed us to rewrite the expression in a simpler form and see the relationship more clearly: \( \frac{a(a+b)}{b(a-b)} \).

Step 1: Convert division to multiplication

To simplify this expression, we'll convert the division operation to multiplication by using the reciprocal of the second fraction. The reciprocal of a fraction is found by swapping the numerator and the denominator. So, the given equation can be rewritten as: $$ \frac{a+b}{b} \cdot \frac{ab}{a-b} $$ Step 2: Multiply Fractions

Step 2: Multiply Fractions

Now, we'll multiply the fractions. To multiply fractions, we'll simply multiply the numerators and then the denominators separately. Therefore, $$ \left(\frac{a+b}{b}\right) \cdot \left(\frac{ab}{a-b}\right) = \frac{(a+b)(a b)}{b(a-b)} $$ Step 3: Factor Out "a" From the Numerator

Step 3: Factor "a" from the numerator

We can factor out "a" from the numerator to make it easier to see any relationships between the numbers. Factoring out "a" gives us: $$ \frac{a(a+b)}{b(a-b)} $$ Step 4: Identify the Real Number Property

Step 4: Identify the real number property

The real number property that justifies the statement is the Commutative Property of Addition. The Commutative Property of Addition states that for any real numbers a and b, if a + b exists, then a + b = b + a. In our problem, this property allowed us to rewrite the expression in a simpler form where we could see the relationship more clearly. Therefore, the real number property that justifies the statement is the Commutative Property of Addition.

Key Concepts

Commutative Property of Addition

One of the foundational rules in mathematics is the commutative property of addition, which asserts that the order in which two numbers are added does not affect the sum. In other words, for any pair of real numbers, call them a and b, flipping their order will not change the result:
\( a + b = b + a \).
This property is essential for simplifying expressions and solving equations efficiently. It provides flexibility in the approach to addition. For instance, when looking at the expression \( \frac{a+b}{b} \), applying the commutative property allows for the understanding that \( a+b \) can also be represented as \( b+a \), which makes no difference in the calculation of the sum.

Multiplying Fractions

Multiplying fractions might seem tricky at first, but it follows a simple set of steps. When you multiply two fractions, you multiply the numerators (the top numbers) together and then multiply the denominators (the bottom numbers) together. Therefore:
\( \frac{m}{n} \times \frac{o}{p} = \frac{m \times o}{n \times p} \).
This straightforward procedure is the same no matter how complicated the fractions might appear. Let's take \( \frac{a+b}{b} \times \frac{ab}{a-b} \) as an example. By following the rule, we multiply \( a+b \) by \( ab \), and then \( b \) by \( a-b \), simplifying our expression to \( \frac{(a+b)(ab)}{b(a-b)} \).

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is a critical concept in arithmetic, particularly when dividing fractions. The reciprocal is simply the inverted form of the original fraction, where the numerator becomes the denominator and vice versa. For fraction \( \frac{c}{d} \), its reciprocal would be \( \frac{d}{c} \).
In the context of division by a fraction, as seen in the original exercise, the reciprocal is used to convert the division into multiplication. Therefore, dividing by \( \frac{a-b}{ab} \) is the same as multiplying by its reciprocal \( \frac{ab}{a-b} \). This technique makes the division operation more manageable and allows one to apply the established rules of multiplying fractions.

Factoring Algebraic Expressions

Factoring is a powerful algebraic tool used to simplify expressions and solve equations. The process involves breaking down a complex expression into a product of simpler factors. It can help to reveal common terms and reduce fractions to their simplest form. When you factor an expression such as \( (a+b)ab \), you're searching for common elements that can be taken out. For instance, if 'a' appears in each term, it can be factored out, giving us \( a \times (a+b) \).
Factoring often paves the way to further simplification, making it easier to understand the underlying structure of algebraic expressions and solve mathematical problems more efficiently.