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