Question

Find the reciprocal of the number. $$-2 \frac{1}{5}$$

Short answer

The reciprocal of -2 1/5 is -5/11.

Step-by-step solution

Convert the mixed number to an improper fraction

To convert -2 1/5 to an improper fraction, multiply the whole number by the denominator of the fraction, add the numerator, and then write the result as the numerator over the original denominator. This gives: \[ -2 \frac{1}{5}= -\frac{(2 \times 5 + 1)}{5}=-\frac{11}{5} \].

Find the reciprocal of the number

The reciprocal of a number is found by swapping the numerator and the denominator. So, the reciprocal of -11/5 is: \[ -\frac{5}{11} \].

Key concepts

Improper Fraction

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. Unlike a proper fraction, where the numerator is less than the denominator, an improper fraction represents a value that is equal to or greater than one.

For instance, when working with mixed numbers such as \( -2 \frac{1}{5} \), you encounter a whole number alongside a proper fraction. To solve certain problems, like finding reciprocals, you need to convert the mixed number into an improper fraction. You do this by multiplying the whole number part by the denominator of the fraction and adding the result to the numerator, which then becomes the new numerator of the improper fraction, while the denominator remains unchanged.

In our example, converting \( -2 \frac{1}{5} \) to an improper fraction involves the following steps:

• Multiply the whole number (2) by the denominator of the fraction (5), which equals 10.
• Add the numerator (1) to this result, resulting in 11.
• Place the sum (11) over the original denominator (5), producing the improper fraction \( -\frac{11}{5} \).

Mixed Numbers

Mixed numbers are another form of representing numbers composed of a whole number and a proper fraction combined. They are particularly handy when dealing with quantities that aren't exact wholes, providing a clearer picture of the value beyond the whole number part.

For example, the mixed number \( -2 \frac{1}{5} \) represents the whole number \( -2 \) and the fraction \( \frac{1}{5} \) together. Mixed numbers can be particularly intuitive for human understanding because they clearly separate the whole part and the fractional part of a value.

When the need arises to perform arithmetic operations with mixed numbers, such as finding a reciprocal, they must be converted into improper fractions. This is because operations with mixed numbers, as in original form, can be complex and lead to inaccuracies if not handled correctly.

Numerators and Denominators

Every fraction consists of two parts, a numerator and a denominator. The numerator, located above the fraction bar, signifies how many parts of the whole are being considered, while the denominator, below the fraction bar, denotes the total number of equal parts the whole is divided into.

In the context of reciprocals, understanding the numerator and denominator is crucial. The reciprocal of any fraction is found by inverting these two components. For the fraction \( \frac{11}{5} \) the numerator is 11, and the denominator is 5. To find the reciprocal, we swap these numbers, resulting in the reciprocal \( -\frac{5}{11} \), where the numerator (5) is now in the denominator's position and the original denominator (11) has become the numerator.

It is important to preserve the sign of the original number when finding reciprocals; if the original fraction is negative, so too will be its reciprocal. Therefore, the reciprocal maintains the magnitude but inversely reflects the ratio originally established between the numerator and the denominator.