# 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.