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Magazine Chapter 4: Problem 27
Add or subtract the mixed fractions, as indicated, by using vertical format. Express your answer as a mixed fraction. $$1 \frac{3}{8}+1 \frac{1}{4}$$
Short Answer
Expert verified
The sum is \(2 \frac{5}{8}\).
Step by step solution
01
Set up the fractions vertically
Place the mixed fractions in a vertical format to make it easier to add them. Align the whole parts and the fractional parts like this:\[\begin{array}{c} 1 \frac{3}{8} \+1 \frac{1}{4} \ \end{array}\]
02
Find a common denominator for the fractions
The fractions \(\frac{3}{8}\) and \(\frac{1}{4}\) need a common denominator. The least common denominator between 8 and 4 is 8. Rewrite \(\frac{1}{4}\) as \(\frac{2}{8}\) by multiplying the numerator and denominator by 2.
03
Add the fractions
Now that both fractions have the same denominator, we can add them:\[\frac{3}{8} + \frac{2}{8} = \frac{5}{8}\]
04
Add the whole numbers
Add the whole number parts of the mixed numbers:\[1 + 1 = 2\]
05
Combine the whole number and fractional parts
Bring together the sum of the whole numbers and the fractional parts to form a single mixed number:\[2 \frac{5}{8}\]
06
Verify your solution
Ensure that the fraction part of \(\frac{5}{8}\) is less than 1 and cannot be simplified further. The final answer, \(2 \frac{5}{8}\), is correctly expressed as a mixed fraction.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Least Common Denominator
In fraction addition, finding a common denominator is crucial. Specifically, the **least common denominator** allows us to add fractions with different denominators without altering their values. For example, when adding \(\frac{3}{8}\) and \(\frac{1}{4}\), we need a common denominator to combine these fractions. The **least common denominator** between 8 and 4 is 8. This means both fractions can be rewritten using 8 as the denominator.
- **\(\frac{3}{8}\)** remains unchanged.
- Convert **\(\frac{1}{4}\)** to have the least common denominator by multiplying both the numerator and denominator by 2, resulting in **\(\frac{2}{8}\)**.
Vertical Format
Using a **vertical format** to arrange fractions and mixed numbers streamlines the process of addition, making it visually more straightforward. To do this:
- Firstly, align all whole numbers in the mix directly above each other, vertically.
- Secondly, place the fractional parts under each other, again ensuring alignment of numerators and denominators.
Fraction Addition
After finding a common denominator, **fraction addition** becomes a straightforward task. Once fractions are aligned vertically and denominators match, you simply add the numerators together. In our example:
- Add \(\frac{3}{8} + \frac{2}{8} = \frac{5}{8}\)
- The denominator remains 8 because the fractions must have equivalent denominators before addition.
Mixed Number
A **mixed number** consists of a whole number and a fraction, representing a sum of these parts. For example, \(1 \frac{3}{8}\) consists of the whole number 1 and the fraction \(\frac{3}{8}\).
- To convert results into mixed numbers, add the whole parts together.
- Then, add the related fractions after finding a least common denominator.
