Reciprocals

Reciprocal of the fraction is fraction that is turned "upside down", i.e. reciprocal of the fraction `color(green)(a/b)` is `color(red)(b/a)`.

There is very nice fact about reciprocals.

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

Indeed, `a/b*b/a=(ab)/(ab)=1`.

If we take fraction `3/4` then its reciprocal is `4/3`. Now, reciprocal of `4/3` is `3/4`, i.e. initial fraction.

Fact. Reciprocal of reciprocal of the number `a` is number `a`.

Example 1. Find reciprocal of `5/7`.

We just turn fraction "upside down": `7/5`.

Answer: `7/5=1 2/5`.

Next example.

Example 2. Find reciprocal of 4.

Recall that each integer can be represented as fraction: `4=4/1`.

Now turn fraction "upside down": `1/4`.

Answer: `1/4`.

Next example.

Example 3. Find reciprocal of `-2 1/7`.

Convert mixed number to improper fraction: `-2 1/7=-15/7`.

Now turn fraction "upside down": `-7/15`.

Answer: `-7/15`.

Now, do a couple of exercises.

Exercise 1. Find reciprocal of `7/11`.

Answer: `11/7=1 4/7`.

Next exercise.

Exercise 2. Find reciprocal of -5.

Answer: `-1/5`.

Next exercise.

Exercise 3. Find reciprocal of `1/4`.

Answer: 4.

Next exercise.

Exercise 4. Find reciprocal of `2 8/9`.

Answer: `9/26`.

Next exercise.

Exercise 5. Find reciprocal of `1/(5/8)`.

Answer: `5/8`. Hint: reciprocal of `1/a` is `a`. Here, `a` is `5/8`.

Next exercise.

Exercise 6. Find reciprocal of reciprocal of -3.

Answer: -3. Hint: reciprocal of -3 is `-1/3`, reciprocal of `-1/3` is again -3.