Problem 64 - Algebra A Combined Function

Question

Find each multiplicative inverse or reciprocal. $$ \frac{7}{6} $$

Step-by-Step Solution

Answer

The multiplicative inverse of \( \frac{7}{6} \) is \( \frac{6}{7} \).

Step 1: Understanding the Concept

The multiplicative inverse (or reciprocal) of a fraction is essentially flipping the fraction upside down. This means the numerator becomes the denominator, and the denominator becomes the numerator.

Step 2: Apply the Concept

Start with the given fraction \( \frac{7}{6} \). To find its reciprocal, swap the numerator and the denominator.

Step 3: Swap Numerator and Denominator

The fraction \( \frac{7}{6} \) becomes \( \frac{6}{7} \) after swapping the numerator and denominator.

Step 4: Validate the Reciprocal

Multiply the original fraction by its reciprocal to check if the result is 1: \( \frac{7}{6} \times \frac{6}{7} = \frac{42}{42} = 1 \). This confirms that \( \frac{6}{7} \) is indeed the multiplicative inverse of \( \frac{7}{6} \).

Key Concepts

Fractions

A fraction is a way of representing numbers that are not whole. It shows how many parts of a whole you have. Fractions are written with two numbers separated by a line, such as \( \frac{1}{2} \). In this example, the number above the line is called the numerator, and the number below the line is the denominator. Fractions are particularly useful when dividing things into equal parts. You can think of fractions as pieces of a pizza. If you have a pizza cut into 6 pieces, and you eat 2, you have eaten \( \frac{2}{6} \) of the pizza. This can also be simplified to \( \frac{1}{3} \) because 2 out of 6 pieces is the same as 1 out of 3 equal sections. Understanding fractions is essential for grasping the concept of reciprocals and multiplicative inverses.

Reciprocal

The reciprocal, or multiplicative inverse, of a number is another number that, when multiplied with the original number, results in 1. For fractions, finding the reciprocal is straightforward: you simply swap the numerator and the denominator. This turns the fraction upside down.
For example, the reciprocal of \( \frac{7}{6} \) is \( \frac{6}{7} \). When you multiply these two fractions together, \( \frac{7}{6} \times \frac{6}{7} \), you get \( \frac{42}{42} \), which simplifies to 1.
This property is what makes them multiplicative inverses. Reciprocals are important in many areas of mathematics, including solving equations and simplifying complex problems.

Numerator and Denominator

The numerator and denominator are the two key parts of a fraction. The numerator is the top number, and it represents how many parts of the whole we are looking at. The denominator is the bottom number, showing how many equal parts the whole is divided into.

• Numerator: If you have \( \frac{3}{5} \), the numerator is 3. This tells us we have 3 parts.
• Denominator: With the same fraction, the denominator is 5, meaning the whole is divided into 5 equal parts.

When finding the reciprocal of a fraction, we swap these two parts. This means the numerator becomes the new denominator and vice versa. In the fraction \( \frac{7}{6} \), finding the reciprocal results in \( \frac{6}{7} \), effectively flipping the roles of the numerator and denominator. Understanding these terms is crucial because they determine the fraction's value and its reciprocal.