What Is a Reciprocal in Math?
A reciprocal is what you get when you flip a fraction. In other words, you switch the numerator and denominator.
For example, the reciprocal of \large \frac{2}{3} is \large \frac{3}{2} .
Reciprocals are very important in math, especially when you divide fractions. Once you understand this idea, many fraction problems become much easier.
How to Find the Reciprocal
Here is the basic rule:
- Flip the numerator and denominator.
That’s it. However, you do need to be careful with whole numbers and mixed numbers. We’ll look at those below.
Examples of Reciprocals
Example 1: Proper Fraction
Find the reciprocal of \large \frac{4}{7}
Flip the fraction:
\large \frac{4}{7}\rightarrow\frac{7}{4}So, the reciprocal of \large \frac{4}{7} is \large \frac{7}{4} .
Example 2: Improper Fraction
Find the reciprocal of \large \frac{9}{5}
\large \frac{9}{5}\rightarrow\frac{5}{9}So, the reciprocal of \large \frac{9}{5} is \large \frac{5}{9} .
Example 3: Whole Number
Find the reciprocal of \large 6
First, write the whole number as a fraction:
\large 6=\frac{6}{1}Now flip it:
\large \frac{6}{1}\rightarrow\frac{1}{6}So, the reciprocal of 6 is \large \frac{1}{6} .
Example 4: Mixed Number
Find the reciprocal of \large 2\frac{1}{3}
First, rewrite the mixed number as an improper fraction:
\large 2\frac{1}{3}=\frac{7}{3}Now flip it:
\large \frac{7}{3}\rightarrow\frac{3}{7}So, the reciprocal of \large 2\frac{1}{3} is \large \frac{3}{7} .
Why Reciprocals Matter
Reciprocals are especially important when dividing fractions.
Remember this rule: when you divide by a fraction, you multiply by its reciprocal.
For example:
\large \frac{1}{2}\div\frac{3}{4}=\frac{1}{2}\times\frac{4}{3}This is why learning reciprocals is such an important fraction skill.
Reciprocal vs Opposite: What’s the Difference?
Students often confuse a reciprocal with an opposite, but these are not the same thing.
The opposite of a number changes the sign.
The reciprocal of a number flips the fraction.
For example, for \large \frac{2}{3} :
Opposite: \large -\frac{2}{3}
Reciprocal: \large \frac{3}{2}
These are completely different ideas, so make sure not to mix them up.
Practice Problems: Find the Reciprocal
Try these on your own first, then check the answers below.
Practice Problems
- Find the reciprocal of \large \frac{5}{8}
- Find the reciprocal of \large \frac{11}{3}
- Find the reciprocal of \large 9
- Find the reciprocal of \large 1\frac{2}{5}
- Find the reciprocal of \large \frac{7}{10}
Answers
1. \large \frac{8}{5}
2. \large \frac{3}{11}
3. \large \frac{1}{9}
4. First write \large 1\frac{2}{5} as \large \frac{7}{5} , so the reciprocal is \large \frac{5}{7}
5. \large \frac{10}{7}
More Practice with Reciprocals
Now let’s do a few more examples together.
Example 1
Find the reciprocal of \large \frac{3}{11}
\large \frac{3}{11}\rightarrow\frac{11}{3}Example 2
Find the reciprocal of \large 12
\large 12=\frac{12}{1}\rightarrow\frac{1}{12}Example 3
Find the reciprocal of \large 3\frac{1}{4}
\large 3\frac{1}{4}=\frac{13}{4}\rightarrow\frac{4}{13}Watch John Explain Reciprocals in Action
Even though this lesson is on complex fractions, John uses reciprocals as part of the process. This is a great way to see why reciprocals matter and how they are used in real fraction problems.
Common Mistakes with Reciprocals
- Confusing reciprocal with opposite. Flipping a fraction is not the same as changing its sign.
- Forgetting to write a whole number as a fraction over 1 first.
- Flipping a mixed number before converting it to an improper fraction.
- Thinking every number has a reciprocal of itself. That is only true for 1 and -1.
- Trying to use reciprocals when multiplying fractions. Reciprocals are mainly used when dividing fractions.
Need More Help with Fractions?
If you want more help with fractions, check out John’s full courses. They include clear instruction, practice problems, and step-by-step solution videos.
👉 Explore the Pre-Algebra Course
👉 Explore the Foundations Math Course