Fraction addition with unlike denominators**
In mathematics, a fraction is a number that represents a part of a whole. It consists of a numerator and a denominator, which are integers. The numerator represents the number of equal parts of a whole, while the denominator is the total number of parts that make up that whole. For example, in the fraction 3/8, the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator.
When adding fractions with unlike denominators, the first step is to find a common denominator. This involved multiplying the denominators of all of the fractions involved by the product of the denominators of each fraction. The denominators are then combined to form a new lower denominator lcm that is a multiple of each individual denominator. The numerators are also multiplied by the appropriate factors to preserve the value of the fraction as a whole. This is the simplest way to ensure that the fractions have a common denominator.
Examples using the common denominator method
To add fractions using the common denominator method, perform the following steps:
Example 1:
Given the fractions 3/4 and 2/3, find the least common denominator.
The denominators 4 and 3 have a product of 12 as their lcm.
Rewrite the fractions with the common denominator 12:
3/4 = (3*3)/(4*3) = 9/122/3 = (2*4)/(3*4) = 8/12
Now add the fractions:
(9/12) + (8/12) = (9+8)/12 = 17/12
The sum, 17/12, is the final answer.
Example 2:
Given the fractions 5/6 and 1/2, find the least common denominator.
The denominators 6 and 2 have a product of 12 as their lcm.
Rewrite the fractions with the common denominator 12:
5/6 = (5*2)/(6*2) = 10/121/2 = (1*6)/(2*6) = 6/12
Now add the fractions:
(10/12) + (6/12) = (10+6)/12 = 16/12
The sum, 16/12, simplifies to 4/3 or 1 1/3.
Another method: LCM
While the common denominator method is straightforward, another method is used to find the least common multiple (LCM) of the denominators, then add or subtract the numerators as one would an integer. Using the LCM can be more efficient and is more likely to result in a fraction in simplified form.
Example 3:
继续计算
Consider the fractions 1/4 and 2/8.
Determine the LCM:
The denominators 4 of 2, LCM (4, L): LCM(4, 8) = 8
Rewrite the fractions:
1/4 = (1*8)/(4*8) = 8/322/8 = (2*4)/(8*4) = 1/2
Now add or subtract the numerators. Since the LCM has already been determined (8), just add or subtract the numerators that are multiplied by this common denominator.
(8/32) + (1/2) = (8 + 32)/32 = 40/32
Simplify the answer:
- Reduce (40, 32): 40/32 =
5/4
The final answer is .25 or 5/4., when fractions have unlike denominators, the common denominator method is often the easiest and most direct approach. However, for larger numbers in practical applications, the LCM method can be more efficient. Always choose the approach that is most appropriate for the given problem.
