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# Subtracting fractions

Content

## What is the subtraction of fractions?

The subtraction of fractions is one of the basic operations that allows to obtain the difference between two fractions in an equivalent fraction, which is known as "difference" or "Subtraction".

### Symbol or sign of subtraction of fractions

Subtraction of fractions is represented by the symbol of a dash or intermediate line "-" which is known as "minus."

## How do we subtract fractions?

To obtain the numerical value as a fraction, you must first identify if the subtraction of fractions has the same denominator or different denominator, therefore, there are two procedures:

### 1) Subtraction of fractions with the same denominator

Subtraction of fractions with the same denominator or also known as subtraction of homogeneous fractions is the most simplified and simple procedure, since the subtraction procedure is based on subtracting the numerators and the denominator remains the same.

3 / 4
-
1 / 4
=
2 / 4
3/4
1/4
2/4

Example:

5 / 3
-
4 / 3
=
5 - 4 /
3
=
1 / 3
8 / 2
-
6 / 2
=
8 - 6 /
2
=
2 /
2
9 / 6
-
2 / 6
=
9 - 2 /
6
=
7 /
6
5 / 3
-
2 / 3
=
5 - 2 /
3
=
3 /
3

From the previous examples you can simplify 2/2 = 1 and 3/3 = 1.

Exercise:

A)
5 / 3
-
3 / 3
= ?
B)
9 / 2
-
5 / 2
= ?
C)
6 / 5
-
4 / 5
= ?
D)
6 / 8
-
2 / 8
= ?

### 2) Subtraction of fractions with different denominator

For the subtraction of fractions with different denominator or also known as subtraction of heterogeneous fractions, it is recommended to know how to obtain the least common multiple (lcm), since we can simplify the equations.

3 / 4
-
1 / 2
=
1 / 4
3/4
1/2
1/4

Two different methods are considered for the subtraction of fractions with different denominator, in this case the first method corresponds to the direct form since we cannot obtain a least common multiple of the denominator and the second method corresponds to obtaining the least common multiple.

Note: It is recommended to work with previously simplified fractions.

First Method: The first method can be solved in two ways.
A) Method of the Division of the denominators by the numbered: It consists of looking for the common denominator of the fractions to be subtracted. For example:
1 / 2
-
1 / 4
• 1.- Multiply the denominators of the fractions 2 x 4 = 8.
1
/
2
-
1
/
4
=
/
8
• 2.- The common denominator is divided by the denominator of the first fraction: 8/2 = 4.
1
/
2
-
1 / 4
=
/
8
• 3.- The result of the division is multiplied by the numerator of the same fraction: 4 x 1
1
/
2
-
1 / 4
=
/ 8
• 4.- Once it is divided and multiplied, the result is placed in the numerator with the sign of the fraction, in this case the fraction is positive but it is worth putting the sign.
1 / 2
-
1 / 4
=
4
/
8
• 5.- The same procedure is performed with the other fraction considering the sign and the subtraction is performed with the resulting numerators.
1 / 2
-
1 / 4
=
4 - 2 /
8
=
2 /
8
=
1 /
4
B) Cross multiplication method: It consists of looking for the common denominator of the fractions to be subtracted. For example:
2 / 3
-
3 / 5
• 1.- Multiply the denominators of the fractions 3 x 5 = 15.
2
/
3
-
3
/
5
=
/
15
• 2.- The numerator of the first fraction is multiplied by the denominator of the second fraction: 2 x 5 = 10. The result is placed in the numerator with the sign of the first fraction.
2
/
3
-
3
/
5
=
10
/ 15
• 3.- The denominator of the first fraction is multiplied by the numerator of the second fraction: 3 x 3 = 9. The result is placed in the numerator with the sign of the second fraction.
2
/
3
-
3
/
5
=
10 - 9 /
15
• 4.- Subtraction is performed with the numerators that resulted.
2 / 3
-
3 / 5
=
10 - 9 /
15
=
1
/
15
Second Method: It consists in obtaining the least common multiple of the denominators, it is enough to identify the largest multiple between them to perform the subtraction of fractions. For example: Para restar fracciones con múltiplos en el denominador, se lleva a cabo el siguiente procedimiento tomando de ejemplo la resta:
1 / 2
-
2 / 6
• 1.- Identify the least common multiple of the fractions to be subtracted, the denominator 6 is a multiple of 2, with the number 6 being the least common multiple.
1
/
2
-
2
/
6
• 2.- The least common multiple is divided by the denominator of the first fraction: 6/2 = 3.
1
/
2
-
2
/
6
=

/
6
• 3.- The result of the division is multiplied by the numerator of the same fraction: 3x1 = 3.
1
/
2
-
2
/
6
=

/
6
• 4.- Once it is divided and multiplied, the result is placed in the numerator with the sign of the fraction, in this case the fraction is positive but it is worth putting the sign.
1
/
2
-
2
/
6
=
3
/
6
• 5.- The same procedure is performed with the other fraction considering the sign and the subtraction is performed with the numerators that resulted.
1
/
2
-
2
/
6
=
3 - 2
/
6
=
1
/
6
• Note: It is recommended to learn this method, since it allows to simplify the equation in simpler fractions.

Examples:

3 / 2
-
4 / 3
=
9 - 8 /
6
=
1 / 6
5 / 2
-
3 / 4
=
10 - 3 /
4
=
7 /
4
7 / 2
-
4 / 8
=
28 - 4 /
8
=
24 /
8
8 / 5
-
2 / 3
=
24 - 10 /
15
=
14 /
15

From the previous examples it can be simplified 24/8 = 3.

Exercise:

A)
7 / 2
-
5 / 3
= ?
B)
3 / 2
-
5 / 4
= ?
C)
3 / 4
-
3 / 5
= ?
D)
6 / 6
-
2 / 2
= ?

## Subtraction of mixed fractions

In the subtraction of mixed fractions, it is necessary that the whole part be expressed as a fraction with the same denominator as in the fractional part that accompanies it. For example, to perform the following mixed subtraction:

4
2 / 5
- 3
4 / 8
1.- The integer part is multiplied by the denominator of the fraction that accompanies it.

4 x 5 = 20
3 x 8 = 24

2.- The result of the multiplication is added with the numerator of the accompanying fraction.
20 + 2 /
5
-
24 + 4 /
8
3.- Once the mixed fractions are converted, the subtraction can be performed.
22 /
5
-
28 /
8
=
176 - 140 /
40
=
36 /
40