Problem 26 - Algebra 2

Question

Name the property illustrated by each equation. $$ \left(1 \frac{2}{7}\right)\left(\frac{7}{9}\right)=1 $$

Step-by-Step Solution

Answer

The equation illustrates the multiplicative inverse property.

Step 1: Identify the Given Equation

The given equation is \( \left(1 \frac{2}{7}\right) \left(\frac{7}{9}\right) = 1 \). Our goal is to determine what property of numbers this equation illustrates.

Step 2: Rewrite the Mixed Number

A mixed number like \(1 \frac{2}{7}\) can be expressed as an improper fraction. \(1 \frac{2}{7} = \frac{9}{7}\). Therefore, the equation can be rewritten as \(\left(\frac{9}{7}\right)\left(\frac{7}{9}\right) = 1\).

Step 3: Simplify the Multiplication

Multiply the two fractions: \(\frac{9}{7} \times \frac{7}{9}\). The numerators (9 and 7) and the denominators (7 and 9) multiply to give: \(\frac{9 \times 7}{7 \times 9}\). This simplifies because each term cancels: \(\frac{63}{63} = 1\).

Step 4: Recognize the Inverse Property

The equation demonstrates that when a number is multiplied by its reciprocal, the result is 1. This illustrates the multiplicative inverse property, which states that any number multiplied by its reciprocal equals 1.

Key Concepts

Reciprocal Exploration

In mathematics, the term 'reciprocal' refers to a specific kind of relationship between two numbers. When we talk about the reciprocal, we are referring to two numbers that multiply together to give a product of 1. This is also known as the multiplicative inverse. Imagine a number, say "a", and its reciprocal will be "1/a". When "a" is multiplied by "1/a", the result is always 1.

Finding the reciprocal of a fraction is straightforward:

• Flip the numerator and the denominator. If you have a fraction \( \frac{3}{4} \), its reciprocal is \( \frac{4}{3} \).
• For a whole number, treat it as if it were over 1. For example, the reciprocal of 5 is \( \frac{1}{5} \).
• Negative numbers have reciprocals too. The reciprocal of \( -2/3 \) is \( -3/2 \).

Reciprocals are crucial in division and solving equations.

They also help simplify complex expressions and work hand-in-hand with the multiplicative inverse property.

Understanding Improper Fractions

An improper fraction is one of those quirky numbers which might seem upside down. It's one where the numerator (the number on top) is greater than or equal to the denominator (the number on the bottom). For example, \( \frac{9}{7} \) is improper because 9 is bigger than 7.

Improper fractions often come from converting mixed numbers. Mixed numbers contain a whole part and a fraction part, such as \(1 \frac{2}{7}\). Turning a mixed number into an improper fraction involves:

• Multiplying the whole number by the denominator (1 x 7 = 7).
• Adding this result to the original numerator (7 + 2 = 9).
• Placing the total as the new numerator over the original denominator, so \(1 \frac{2}{7} = \frac{9}{7}\).

This form is particularly useful in calculations involving multiplication or division with other fractions as it often simplifies the math process.

The Technique of Simplification

Simplification is the process of making an expression easier to manage or solve. It's like tidying up your room—removing what's unnecessary and organizing what remains. In fractions, simplification means reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1.

For example, if we start with \( \frac{18}{24} \), to simplify:

• Find the greatest common factor of 18 and 24, which is 6.
• Divide the numerator and the denominator by this factor \( \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \).

Simplifying fractions is a key skill because it allows for clearer understanding of numerical relationships and easier computation. In complex equations, like our multiplied fractions \( \frac{9}{7} \times \frac{7}{9} \), simplification helps us quickly recognize that such multiplication leads to the neat result of 1, thanks to the cancellation of equal factors in the numerator and denominator.