Reciprocals

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

There is very nice fact about reciprocals.

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

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

If we take fraction 34\frac{{3}}{{4}} then its reciprocal is 43\frac{{4}}{{3}}. Now, reciprocal of 43\frac{{4}}{{3}} is 34\frac{{3}}{{4}}, i.e. initial fraction.

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

Example 1. Find reciprocal of 57\frac{{5}}{{7}}.

We just turn fraction "upside down": 75\frac{{7}}{{5}}.

Answer: 75=125\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=41{4}=\frac{{4}}{{1}}.

Now turn fraction "upside down": 14\frac{{1}}{{4}}.

Answer: 14\frac{{1}}{{4}}.

Next example.

Example 3. Find reciprocal of 217-{2}\frac{{1}}{{7}}.

Convert mixed number to improper fraction: 217=157-{2}\frac{{1}}{{7}}=-\frac{{15}}{{7}}.

Now turn fraction "upside down": 715-\frac{{7}}{{15}}.

Answer: 715-\frac{{7}}{{15}}.

Now, do a couple of exercises.

Exercise 1. Find reciprocal of 711\frac{{7}}{{11}}.

Answer: 117=147\frac{{11}}{{7}}={1}\frac{{4}}{{7}}.

Next exercise.

Exercise 2. Find reciprocal of -5.

Answer: 15-\frac{{1}}{{5}}.

Next exercise.

Exercise 3. Find reciprocal of 14\frac{{1}}{{4}}.

Answer: 4.

Next exercise.

Exercise 4. Find reciprocal of 289{2}\frac{{8}}{{9}}.

Answer: 926\frac{{9}}{{26}}.

Next exercise.

Exercise 5. Find reciprocal of 158\frac{{1}}{{\frac{{5}}{{8}}}}.

Answer: 58\frac{{5}}{{8}}. Hint: reciprocal of 1a\frac{{1}}{{a}} is a{a}. Here, a{a} is 58\frac{{5}}{{8}}.

Next exercise.

Exercise 6. Find reciprocal of reciprocal of -3.

Answer: -3. Hint: reciprocal of -3 is 13-\frac{{1}}{{3}}, reciprocal of 13-\frac{{1}}{{3}} is again -3.