Addition and subtraction of fractions with different denominators

To add or subtract two fractions with different denominators, you must first bring them to a common denominator, and then calculate the sum or difference of the fractions with the resulting identical denominators.

Reducing fractions to the lowest common denominator

To reduce a fraction to a new denominator, both the numerator and the denominator of the fraction must be multiplied by the same factor. Fractions are usually reduced to the least common denominator. It is equal to the least common multiple of the denominators of the given fractions.

You can reduce several fractions to their lowest common denominator as follows:

1) find the least common multiple of the denominators of these fractions, this will be the least common denominator of these fractions;

2) divide the least common denominator by the denominators of the fractions to find the corresponding multiplier for each fraction;

3) multiply the numerator and denominator of each fraction by the factor found for it.

Examples of reducing fractions to the lowest common denominator

Example 1.
Reduce a fraction to its lowest common denominator .

Solution.
First, let's find the least common multiple of the denominators of these fractions. To do this, we factorize the numbers 5, 15, 100 into prime factors:
5 = 5;
15 = 3*5;
100 = 2*2*5*5.
We choose the number 100 and add to its factorization the factors from the factorization of the numbers 5 and 15, which are not yet in the factorization: 2*2*5*5*3=300. Therefore, the least common multiple is 300.
Now we divide the number 300 by the denominator of each fraction to find the corresponding additional factor for it:
300:5=60 => ;
300:15=20 => ;
300:100=3 => .
Answer: .

Example 2.
Reduce a fraction to its lowest common denominator .
Solution.
First, let's find the least common multiple of the denominators of these fractions. To do this, we factorize the numbers 12, 60, 80 into prime factors:
12 = 2*2*3;
60 = 2*2*3*5;
80 = 2*2*2*2*5.
We choose the number 80 and add to its expansion the factors from the expansions of the numbers 12 and 15, which are not yet in the expansion: 2*2*2*2*5*3=240. Therefore, the least common multiple is 240.
Now we divide the number 240 by the denominator of each fraction to find the corresponding additional factor for it:
240:12=20 => ;
240:60=4 => ;
240:80=3 => .

Answer:

Example 3.
Reduce a fraction to its lowest common denominator .
Solution.
First, let's find the least common multiple of the denominators of these fractions. To do this, we factorize the numbers 13, 8, 5 into prime factors:
13 = 13;
8 = 2*2*2;
5 = 5.
We choose the number 13 and add to its expansion the factors from the expansions of the numbers 8 and 5, which are not yet in the expansion: 13*2*2*2*5=520. Therefore, the least common multiple is 520.
Now we divide the number 520 by the denominator of each fraction to find the corresponding additional factor for it:
520:13=40 => ;
520:8=65 => ;
520:5=104 => .
Answer:

Examples of adding fractions with different denominators

Example 4.
Calculate the sum .
Solution.
To find the sum of these fractions, you must first bring them to a common denominator, and then calculate the sum of the fractions with the resulting identical denominators.
Let's find the least common denominator of these fractions:
5 = 5;
25 = 5*5.
Therefore, the least common denominator is 5*5 = 25.
.

Answer: .

Example 5.
Calculate the sum .
Solution.
To find the sum of these fractions, you must first bring them to a common denominator, and then calculate the sum of the fractions with the resulting identical denominators.
24 = 2*2*2*3;
16 = 2*2*2*2.
Therefore, the least common denominator is 2*2*2*3*2=48.
.
Answer: .

Example 6.
Calculate the sum .
Solution.
Let's find the least common denominator of fractions:
12 = 2*2*3;
20 = 2*2*5.
Therefore, the least common denominator is 2*2*3*5=60.
.
Answer: .

Let's look at subtraction of fractions with different denominators using examples.

Examples of subtracting fractions with different denominators

Example 7.
Calculate the difference .
Solution.
To find the difference between these fractions, you must first bring them to a common denominator, and then calculate the difference between the fractions with the resulting identical denominators.
.
Answer: .

Example 8.
Calculate the difference .
Solution.
Let's find the least common denominator of fractions:
9 = 3*3;
6 = 2*3.
Therefore, the least common denominator is 3*3*2=18.
.
Answer: .