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Math18

Adding of fractions

Content

What is the sum or adding of fractions?

The sum or adding of fractions is one of the basic operations that allows two or more fractions to be combined in an equivalent number, known as the "Sum" or "Result of the Sum".

Symbol or sign of the adding of fractions

The sum of fractions is represented by the symbol of a cross "+" which is known as “plus”.


How do we add fractions?

To obtain the numerical value in the form of fractions, you must first identify whether the sum of fractions has the same denominator or a different denominator, therefore, there are two procedures:

1) Sum of fractions with the same denominator

The sum of fractions with the same denominator or also known as sum of homogeneous fractions is the most simplified and simple procedure, since the summation process is based on adding the numerators and the denominator remains the same.

1 / 4
+
2 / 4
=
3 / 4
1/4
2/4
3/4

Example:

2 / 3
+
4 / 3
=
2 + 4 /
3
=
6 / 3
5 / 2
+
6 / 2
=
5 + 6 /
2
=
11 /
2
5 / 6
+
4 / 6
=
5 + 4 /
6
=
9 /
6
8 / 3
+
2 / 3
=
8 + 2 /
3
=
10 /
3

From the previous examples you can simplify 6/3 = 2 and 9/6 = 3/2.

Exercise:

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

2) Sum of fractions with different denominator

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

1 / 4
+
1 / 2
=
3 / 4
1/4
1/2
3/4

Two different methods are considered for the sum of fractions with different denominator, in this case, the first method corresponds to the direct form since we cannot obtain a minimum 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 added. For example:
      1 / 2
      +
      3 / 5
    • 1.- To do this, multiply the denominators of the fractions 2 x 5 = 10.
      1
      /

      2

      +
      3
      /

      5

      =
        /
      10
    • 2.- The common denominator is divided by the denominator of the first fraction: 10/2 = 5.
      1
      /

      2

      +
      3 / 5
      =
        /

      10

    • 3.- The result of the division is multiplied by the numerator of the same fraction: 5 x 1.

      1

      /
      2
      +
      3 / 5
      =
        / 10
    • 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 too much to put the sign.
      1 / 2
      +
      3 / 5
      =
      5
      / 10
    • 5.- The same procedure is performed with the other fraction and the sum is made with the resulting numerators.
      1 / 2
      +
      3 / 5
      =
      5 + 6 /
      10
      =
      11 /
      10
      B) Cross multiplication method: It consists of looking for the common denominator of the fractions to be added. For example:
      1 / 3
      +
      3 / 5
    • 1.- Multiply the denominators of the fractions 3 x 5 = 15.
      1
      /

      3

      +
      3
      /

      5

      =
        /
      15
    • 2.- The numerator of the first fraction is multiplied by the denominator of the second fraction: 1 x 5 = 5. The result is placed in the numerator with the sign of the fraction.

      1

      /
      3
      +
      3
      /

      5

      =
      5
      / 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 fraction.
      1
      /

      3

      +

      3

      /
      5
      =
      5 + 9 /
      15
    • 4.- The sum is made with the numerators that resulted.
      1 / 3
      +
      3 / 5
      =
      5 + 9 /
      15
      =
      14 /
      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 make the sum of fractions. To add fractions with multiples in the denominator, the following procedure is carried out taking as an example the sum:
      1 / 2
      +
      4 / 6
    • 1.- Identify the least common multiple of the fractions to be added, the denominator 6 is a multiple of 2, with the number 6 being the least common multiple.
      1
      /

      2

      +
      4
      /
      6
    • 2.- The least common multiple is divided by the denominator of the first fraction: 6/2.
      1
      /

      2

      +
      4
      /
      6
      =
       
      /

      6

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

      1

      /
      2
      +
      4
      /
      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 too much to put the sign.
      1
      /
      2
      +
      4
      /
      6
      =
      3
      /
      6
    • 5.- The same procedure is done with the other fraction and the sum is done with the numerators that resulted.
      1
      /
      2
      +
      4
      /
      6
      =
      3 + 4
      /
      6
      =
      7
      /
      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
=
17 / 6
3 / 4
+
5 / 2
=
3 + 10 /
4
=
13 /
4
4 / 8
+
7 / 2
=
4 + 28 /
8
=
32 /
8
8 / 5
+
2 / 3
=
24 + 10 /
15
=
34 /
15

From the previous examples you can simplify 32/8 = 4.

Exercise:

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

Adding of three or more fractions

The procedure is similar to adding two fractions, first you must identify if they have a different denominator. If the denominators are equal, we can do the sum by adding the numerators, which corresponds to the method of “Sum of fractions with the same denominator”. If the denominators are different, then the least common multiple of the denominators must be obtained which corresponds to the method of “Sum of fractions with different denominator”

Addition of three or more fractions with the same denominator

Having the same denominator simplifies the procedure since the denominator passes the same and the numerator must be added.

2 / 3
+
1 / 3
+
5
/ 3
=
2 + 1 + 5
/
3
=
8
/
3

Addition of three or more fractions with different denominator

Having three or more fractions with a different denominator, it is recommended to use method 2 of “sum of fractions with a different denominator” to simplify the equation and obtain a correct result. Example:

2 / 3
+
1 / 4
+
5
/ 12
    1.- Identify the least common multiple of the fractions to be added, denominator 12 is a multiple of 3 and 4, with number 12 being the greatest common denominator.
    2
    /

    3

    +
    1
    /

    4

    +
    5
    /

    12

    =
      /
    12
    2.- The least common multiple is divided by the denominator of the first fraction: 12/3 = 4.
    2
    /

    3

    +
    1 /
    4
    +
    5
    /
    12
    =
      /

    12

    3.- The result of the division is multiplied by the numerator of the same fraction: 4x2 = 8.

    2

    /
    3
    +
    1 /
    4
    +
    5
    /
    12
    =
      /
    12
    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.
    2
    /
    3
    +
    1 /
    4
    +
    5
    /
    12
    =

    8

    /
    12
    5.- The same procedure is done with the other fractions and the sum is done with the numerators that resulted.
    2
    /
    3
    +
    1 /
    4
    +
    5
    /
    12
    =
    8 + 3 + 5
    /
    12
    =
    16
    /
    12
    =
    4
    /
    3

Examples:

3 / 2
+
4 / 2
+
8 /
2
=
15 /
2
3 / 4
+
5 / 4
+
10 /
4
=
18 /
4
2 / 3
+
4 / 2
+
4 /
6
=
4 + 12 + 4 /
6
=
20 /
6
5 / 4
+
4 / 8
+
3 /
2
=
10 + 4 + 12 /
8
=
26 /
8

From the previous examples you can simplify 18/4 = 9/2, 20/6 = 10/3 and 26/8 = 13/4.

Exercise:

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

Adding of mixed fractions

In the sum 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 sum:

4
2 / 5
+ 3
4 / 8
    1.- The whole part is multiplied by the denominator of the fraction.

    4 x 5 = 20
    3 x 8 = 24

    2.- The result of the multiplication is added with the numerator of the fraction.
    20 + 2 /
    5
    +
    24 + 4 /
    8
    3.- Once the mixed fractions are converted, the sum can be made.
    22 /
    5
    +
    28 /
    8
    =
    176 + 140 /
    40
    =
    316 /
    40