How is adding and subtracting mixed numbers similar to adding and subtracting fractions? How is adding and subtracting mixed numbers different than adding and subtracting fractions?
Give examples to explain your answer.
Adding and subtracting mixed numbers and fractions are similar in that they both involve finding a common denominator and manipulating the numerators to perform the operation. The basic concept of adding or subtracting parts of a whole remains the same.
For example, let's add the fraction 3/4 and the mixed number 1 1/2:
Adding mixed numbers:
1 1/2 + 3/4
First, we convert the mixed number to an improper fraction:
1 1/2 = (2 * 1) + 1/2 = 2/2 + 1/2 = 3/2
Next, we find a common denominator for the fractions:
3/2 + 3/4
We multiply the numerator and denominator of the first fraction by 2 to get a common denominator of 4:
(3/2) * (2/2) + 3/4 = 6/4 + 3/4
Now that we have a common denominator, we can add the numerators:
6/4 + 3/4 = 9/4
Finally, we simplify the fraction if necessary:
9/4 = 2 1/4
Adding fractions:
3/4 + 1/2
Since the denominators are different, we find the least common denominator (LCD) of 4:
3/4 + 1/2 = 3/4 + (1*2)/(2*2) = 3/4 + 2/4
Now that we have a common denominator, we add the numerators:
3/4 + 2/4 = 5/4
Lastly, we simplify the fraction if needed:
5/4 = 1 1/4
The process of adding and subtracting mixed numbers and fractions is different in terms of handling the whole number part of the mixed number. When adding or subtracting mixed numbers, we perform the operations on the whole number and the fractional part separately. However, when adding or subtracting fractions, there is no whole number part to consider.
Let's subtract the fraction 3/4 from the mixed number 2 1/2:
Subtracting mixed numbers:
2 1/2 - 3/4
First, we convert the mixed number to an improper fraction:
2 1/2 = (2 * 2) + 1/2 = 4/2 + 1/2 = 5/2
Next, we find a common denominator for the fractions:
5/2 - 3/4
We multiply the numerator and denominator of the first fraction by 2 to get a common denominator of 4:
(5/2) * (2/2) - 3/4 = 10/4 - 3/4
Now that we have a common denominator, we subtract the numerators:
10/4 - 3/4 = 7/4
Finally, we simplify the fraction if necessary:
7/4 = 1 3/4
Subtracting fractions:
2 1/2 - 3/4
Again, we convert the mixed number to an improper fraction:
2 1/2 = (2 * 2) + 1/2 = 4/2 + 1/2 = 5/2
Finding a common denominator:
5/2 - 3/4
We multiply the numerator and denominator of the first fraction by 2 to get a common denominator of 4:
(5/2) * (2/2) - 3/4 = 10/4 - 3/4
Now that we have a common denominator, we subtract the numerators:
10/4 - 3/4 = 7/4
Lastly, we simplify the fraction if needed:
7/4 = 1 3/4
In both cases, we get the same result of 1 3/4. Both the process and the outcome are the same regardless of whether we are adding or subtracting mixed numbers or fractions.