# Improper Fraction

**Let you have a fraction a ⁄ b, then how can you know if it's an improper fraction?**

It's easy! Look at "a", then look at "b". What is greater?

If "a" is greater than "b" (a > b), or "a" equals "b" (a = b), it means this time you have an improper fraction.

##### Example:

4 ⁄ 3 (4 > 3) 10 ⁄ 7 (10 > 7) 15 ⁄ 15 (15 = 15)

### It's interesting!

An improper fraction as a number has its own place on the number scale.

Every positive improper fraction is always greater than "1", or it equals "1".

(7 ⁄ 3 > 1; 7 ⁄ 7 = 1)

But every negative improper fraction is always less than "-1", or it equals "-1".

(-7 ⁄ 3 < -1; -7 ⁄ 7 = -1)

### And what about "0", "1", "-1"?

You can write these numbers by the following way:

0 = 0 ⁄ 12 – proper fraction (0 < 12)

1 = 13 ⁄ 13 – improper fraction (13 = 13)

-1 = 8 ⁄ -8 – improper fraction (8 > -8)

Ok, now some words about addition, subtraction, multiplication, division, raising to a power.

### How can you do it with improper fractions?

There are special rules for improper fractions, but you have to do multiplication, division, raising to a power with improper fractions the same way as with proper fractions.

### Multiplication

a ⁄ b × c ⁄ d = a × c ⁄ b × d ; b ≠ 0, d ≠ 0

##### Example:

9 ⁄ 5 × 7 ⁄ 2 = 9 × 7 ⁄ 5 × 2 = 63 ⁄ 10, (63 > 10) – improper fraction.

##### Notice!

Doing multiplication with improper fractions you always get an improper fraction.

### Division

a ⁄ b ÷ c ⁄ d = a ⁄ b × d ⁄ c = a × d ⁄ b × c, c ≠ 0, d ≠ 0, b ≠ 0

##### Example:

5 ⁄ 3 ÷ 10 ⁄ 7 = 5 ⁄ 3 × 7 ⁄ 10 = 5 × 7 ⁄ 3 × 10 = 35 ⁄ 30, (35 > 30) - improper fraction.

17 ⁄ 17 ÷ 10 ⁄ 1 = 17 ⁄ 17 × 1 ⁄ 10 = 17 × 1 ⁄ 17×10 = 17 ⁄ 170 , (17 < 170) - proper fraction.

##### Notice!

When you divide improper fractions, you may get proper fraction or improper fraction.

### Rasing to a power

(a ⁄ b)^{n} = a^{n} ⁄ b^{n}, b ≠ 0

##### Example:

(5 ⁄ 2)^{2} = 5^{2} ⁄ 2^{2} = 25 ⁄ 4, (25 > 4) - improper fraction.

(10 ⁄ 3)^{4} = 10^{4} ⁄ 3^{4} = 10000 ⁄ 81, (10000 > 81) - improper fraction.

(7 ⁄ 7)^{10} = 7^{10} ⁄ 7^{10}, (7^{10} = 7^{10}) - improper fraction.

##### Notice!

When you do raising to a power with improper fraction, you always get only improper fraction.

### Addition and Subtraction

You may do addition and subtraction with improper fractions like with proper ones. But, you'd better do it first making improper fractions as mixed numbers.

##### Example:

7 ⁄ 2 + 5 ⁄ 3 = (7 × 3 + 5 × 2) ⁄ 2 × 3 = (21 + 10) ⁄ 6 = 31 ⁄ 6

**OR**

7 ⁄ 2 + 5 ⁄ 3 = 3 + 1 ⁄ 2 + 1 + 2 ⁄ 3 = 3 + 1 + 1 ⁄ 2 + 2 ⁄ 3 = 4 + 7 ⁄ 6 = 4 + 1 + 1 ⁄ 6 =

5 + 1 ⁄ 6

##### Notice!

Every time you may do a mixed number from improper fraction and never the same with proper fraction!

Hope, now you see - "It is never late to learn".

#### Note

⁄ - fraction slash

= - is equal to; equals

≠ - is not equal to; does not equal

< - is less than

> - is greater than

× - multiplied by

÷ - divided by

+ - plus; add

- - minus; subtract