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Chapter 7: Problem 92

Perform the operations. Then simplify, if possible. \(\frac{13}{24 w}+\frac{17}{24 w}\)

Short Answer

Expert verified
\(\frac{5}{4w}\)

Step by step solution

01

Combine the Fractions

To combine the fractions \(\frac{13}{24w}\) and \(\frac{17}{24w}\), we note that they already have the same denominator, \(24w\). Therefore, we can directly add the numerators, \(13 + 17\).
02

Add the Numerators

The addition of the numerators \(13 + 17\) results in \(30\). Thus, we have a new fraction \(\frac{30}{24w}\).
03

Simplify the Fraction

To simplify \(\frac{30}{24w}\), we need to find the greatest common divisor (GCD) of 30 and 24, which is 6. Divide both the numerator and the denominator by 6: \(\frac{30\div6}{24\div6} = \frac{5}{4}\). Therefore, the simplified form of the fraction is \(\frac{5}{4w}\).

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Key Concepts

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

Fraction Addition
When you want to add fractions, you must ensure that the denominators are the same. Once the denominators match, you can simply add the numerators. Let's say you want to add
  • \( \frac{13}{24w} \) and
  • \( \frac{17}{24w} \).
Each has the same denominator \( 24w \). This means you can add the two parts just like regular numbers:
  • Combine the numerators: \( 13 + 17 = 30 \).
So, the resulting fraction is \( \frac{30}{24w} \).
If fractions have different denominators, you would have to adjust the fractions to make their denominators the same before adding. This process often involves finding a common multiple of the denominators.
But lucky for us, in this problem, the denominators were already matched up, making it a straight shot to the answer!
Simplifying Fractions
Once you have added fractions, you often need to simplify. Simplifying means converting the fraction to its smallest form. This involves making both the numerator and the denominator as small as possible while keeping their ratio the same.
In our example, we have the fraction \( \frac{30}{24w} \). To simplify it, look for the biggest number that divides both the numerator and the denominator evenly. This leads us to our next topic: finding the greatest common divisor (GCD).
After finding the GCD, divide both the numerator and denominator by this number. If doing it in this example:
  • Divide 30 by 6 to get 5.
  • Divide 24 by 6 to get 4.
      • So, \( \frac{30}{24w} \) simplifies to \( \frac{5}{4w} \). Simplifying makes fractions easier to read and work with, especially in larger equations or when comparing fractions.
Greatest Common Divisor
The greatest common divisor (GCD) is crucial when simplifying fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder. Finding the GCD helps to reduce a fraction to its simplest form.
To find the GCD, you can:
  • List the factors of each number. For example, for 30, the factors are 1, 2, 3, 5, 6, 10, 15, and 30.
  • For 24, the factors are 1, 2, 3, 4, 6, 8, 12, and 24.
  • Identify the largest number that appears in both lists, which in this case is 6.
So, the GCD of 30 and 24 is 6.
Once you have the GCD, use it to simplify fractions easily, making calculations more straightforward and results cleaner. Using the GCD ensures that you can always simplify fractions accurately, regardless of their size or complexity.

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