Finding the Reciprocal of a Mixed Number

The question asks us to find the reciprocal of the mixed number \(2\frac{3}{5}\). Understanding how to work with reciprocals and mixed numbers is important for this problem.

What is a Reciprocal?

The reciprocal of a number is simply \(1\) divided by that number. Another way to think about it is finding a number that, when multiplied by the original number, gives a result of \(1\).

• For a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are non-zero, the reciprocal is \(\frac{b}{a}\). We just flip the numerator and the denominator.
• For a whole number \(n\), we can think of it as the fraction \(\frac{n}{1}\). Its reciprocal is \(\frac{1}{n}\).

Converting the Mixed Number to an Improper Fraction

Before finding the reciprocal of a mixed number, it's easiest to convert it into an improper fraction. A mixed number like \(a\frac{b}{c}\) means \(a + \frac{b}{c}\). To convert it to an improper fraction, we use the formula \(\frac{(a \times c) + b}{c}\).

Let's convert \(2\frac{3}{5}\) to an improper fraction:

$$2\frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}$$

So, the mixed number \(2\frac{3}{5}\) is equivalent to the improper fraction \(\frac{13}{5}\).

Calculating the Reciprocal

Now that we have the number as a fraction \(\frac{13}{5}\), we can easily find its reciprocal. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

The reciprocal of \(\frac{13}{5}\) is found by flipping the numerator and the denominator:

$$\text{Reciprocal of } \frac{13}{5} = \frac{5}{13}$$

Verifying the Reciprocal

To verify that \(\frac{5}{13}\) is indeed the reciprocal of \(2\frac{3}{5}\) (or \(\frac{13}{5}\)), we can multiply the original number by its reciprocal. The result should be \(1\).

$$\frac{13}{5} \times \frac{5}{13} = \frac{13 \times 5}{5 \times 13} = \frac{65}{65} = 1$$

Since the product is \(1\), our calculation for the reciprocal is correct.

Final Answer for Reciprocal

The reciprocal of \(2\frac{3}{5}\) is \(\frac{5}{13}\).

Comparing this with the given options, we see that option 1 matches our result.

Revision Table: Understanding Reciprocals

Additional Information on Reciprocals and Fractions

• The number \(0\) does not have a reciprocal, because division by zero is undefined.
• The reciprocal of \(1\) is \(1\), and the reciprocal of \(-1\) is \(-1\).
• Converting mixed numbers to improper fractions is a fundamental step in many fraction operations, including multiplication, division, and finding reciprocals.
• Understanding reciprocals is essential for fraction division, as dividing by a fraction is the same as multiplying by its reciprocal. For example, \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\).