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.