Problem 22 - Prealgebra

Question

Find the multiplicative inverse of each number. $$-3 \frac{2}{9}$$

Step-by-Step Solution

Answer

The multiplicative inverse of $$-3 \frac{2}{9}$$ is $$-\frac{9}{29}$$.

Step 1: Convert the Mixed Number to an Improper Fraction

To find the multiplicative inverse of $$-3 \frac{2}{9}$$, first convert the mixed number to an improper fraction. Multiply the whole number part by the denominator and add the numerator. This gives $$-3 \frac{2}{9} = -\frac{(3 \times 9) + 2}{9} = -\frac{27 + 2}{9} = -\frac{29}{9}.$$

Step 2: Find the Multiplicative Inverse

The multiplicative inverse of a fraction \(-\frac{29}{9}\) is found by flipping the numerator and the denominator. Therefore, the inverse is \(-\frac{9}{29}\) (note that if either the numerator or denominator are negative, the inverse will also be negative).

Key Concepts

Improper Fractions

Improper fractions are a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means the fraction represents a value that is equal to or greater than one whole.
A classic example of an improper fraction is when you have things grouped together, like 29/9. In this fraction, 29 is much larger than 9, clearly indicating a value greater than one.

• To convert a mixed number into an improper fraction involves a simple process: multiply the whole number by the denominator, add the numerator, and keep the same denominator.
• For instance, with -3 2/9, you multiply 3 (the whole number) by 9 (the denominator) getting 27, add 2 (the numerator) resulting in 29. So, -3 2/9 becomes -29/9.

Conversions like these are so common in math because improper fractions are often easier to work with, especially when performing arithmetic operations like multiplication and division.

Mixed Numbers

Mixed numbers are numbers that combine a whole number and a fraction, like 3 2/9. They're often used because they make values easier to understand at a quick glance.
Mixed numbers are frequently converted into improper fractions to facilitate mathematical operations, especially those involving multiplication or division.

• To convert a mixed number back from an improper fraction, divide the numerator by the denominator to find the whole number.
• Use the remainder as the new numerator over the original denominator. For example, -29/9 becomes -3 (because 29 divided by 9 is 3 with a remainder of 2) and the fraction 2/9, resulting in -3 2/9.

On paper, mixed numbers appear simpler but are not ideal for calculations. That's why mathematics often leans towards improper fractions in equations.

Fraction Inversion

Fraction inversion, often referred to as finding the multiplicative inverse, involves flipping a fraction. Doing this changes the fraction to its reciprocal.
It's like flipping the positions of the numerator and denominator. The term 'multiplicative inverse' is used because when you multiply a fraction by its inverse, the result is 1.

• Consider the improper fraction -29/9; its multiplicative inverse is -9/29 because you swap 29 and 9.
• Make sure to keep track of negative signs, as they affect the inversion process but are straightforward to manage: if the original fraction is negative, so is its inverse.

Fraction inversion is particularly useful for division problems because dividing by a fraction is the same as multiplying by its inverse.