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Chapter 1: Problem 52

Add. Do not use the number line except as a check. \(-\frac{4}{13}+\frac{1}{2}\)

Short Answer

Expert verified
\frac{5}{26}

Step by step solution

01

- Find a Common Denominator

To add fractions, first find a common denominator. The denominators are 13 and 2. The least common multiple (LCM) of 13 and 2 is 26.
02

- Convert Fractions to Common Denominator

Convert each fraction to an equivalent fraction with the denominator of 26. For \(-\frac{4}{13}\): Multiply numerator and denominator by 2 to get \(-\frac{8}{26}\).For \(\frac{1}{2}\): Multiply numerator and denominator by 13 to get \(\frac{13}{26}\).
03

- Add the Fractions

Now, add the numerators of the equivalent fractions:\(-\frac{8}{26} + \frac{13}{26} = \frac{-8 + 13}{26} = \frac{5}{26}\).
04

- Simplify the Result

In this case, \(\frac{5}{26}\) is already in its simplest form, so no further simplification is needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
To add fractions, we need a common denominator. A common denominator is a shared multiple of the denominators of both fractions.
This allows us to rewrite the fractions with the same bottom number, making addition possible.
In our exercise, the denominators are 13 and 2. Using a common denominator helps in aligning the fractions on the same scale before performing arithmetic operations.
Least Common Multiple (LCM)
The least common multiple (LCM) is the smallest number that is a multiple of both denominators.
To find it, you list the multiples of each denominator and choose the smallest one they have in common.
For the denominators 13 and 2, the LCM is 26. This common multiple facilitates converting the fractions so that their denominators are identical, essential for adding or subtracting them.
Fraction Simplification
Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1.
In this exercise, after adding \(-\frac{8}{26} \) and \(\frac{13}{26} \), we get \(\frac{5}{26} \).
We check the factors of 5 and 26 to confirm that there are no common factors: 5 is a prime number, and 26 = 2 脳 13, so the fraction cannot be simplified further. Simplification ensures our solution is as straightforward as possible.

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