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.