# Reciprocals

Reciprocal of the fraction is fraction that is turned "upside down", i.e. reciprocal of the fraction ${\color{green}{{\frac{{a}}{{b}}}}}$ is ${\color{red}{{\frac{{b}}{{a}}}}}$.

There is very nice fact about reciprocals.

Fact. Product of fraction and its reciprocal always equals 1.

Indeed, $\frac{{a}}{{b}}\cdot\frac{{b}}{{a}}=\frac{{{a}{b}}}{{{a}{b}}}={1}$.

If we take fraction $\frac{{3}}{{4}}$ then its reciprocal is $\frac{{4}}{{3}}$. Now, reciprocal of $\frac{{4}}{{3}}$ is $\frac{{3}}{{4}}$, i.e. initial fraction.

Fact. Reciprocal of reciprocal of the number ${a}$ is number ${a}$.

Example 1. Find reciprocal of $\frac{{5}}{{7}}$.

We just turn fraction "upside down": $\frac{{7}}{{5}}$.

Answer: $\frac{{7}}{{5}}={1}\frac{{2}}{{5}}$.

Next example.

Example 2. Find reciprocal of 4.

Recall that each integer can be represented as fraction: ${4}=\frac{{4}}{{1}}$.

Now turn fraction "upside down": $\frac{{1}}{{4}}$.

Answer: $\frac{{1}}{{4}}$.

Next example.

Example 3. Find reciprocal of $-{2}\frac{{1}}{{7}}$.

Convert mixed number to improper fraction: $-{2}\frac{{1}}{{7}}=-\frac{{15}}{{7}}$.

Now turn fraction "upside down": $-\frac{{7}}{{15}}$.

Answer: $-\frac{{7}}{{15}}$.

Now, do a couple of exercises.

Exercise 1. Find reciprocal of $\frac{{7}}{{11}}$.

Answer: $\frac{{11}}{{7}}={1}\frac{{4}}{{7}}$.

Next exercise.

Exercise 2. Find reciprocal of -5.

Answer: $-\frac{{1}}{{5}}$.

Next exercise.

Exercise 3. Find reciprocal of $\frac{{1}}{{4}}$.

Next exercise.

Exercise 4. Find reciprocal of ${2}\frac{{8}}{{9}}$.

Answer: $\frac{{9}}{{26}}$.

Next exercise.

Exercise 5. Find reciprocal of $\frac{{1}}{{\frac{{5}}{{8}}}}$.

Answer: $\frac{{5}}{{8}}$. Hint: reciprocal of $\frac{{1}}{{a}}$ is ${a}$. Here, ${a}$ is $\frac{{5}}{{8}}$.

Next exercise.

Exercise 6. Find reciprocal of reciprocal of -3.

Answer: -3. Hint: reciprocal of -3 is $-\frac{{1}}{{3}}$, reciprocal of $-\frac{{1}}{{3}}$ is again -3.