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Chapter 7: Problem 41

Perform the indicated operation. Simplify, if possible. $$ \frac{3 a}{4 a-20}+\frac{9 a}{6 a-30} $$

Short Answer

Expert verified
\( \frac{9a}{4(a-5)} \)

Step by step solution

01

Factor the denominators

Identify the common factors in the denominators and factor them out. For the first denominator, \[ 4a - 20 = 4(a - 5) \]For the second denominator,\[ 6a - 30 = 6(a - 5) \]
02

Find the common denominator

Determine the least common denominator (LCD) of the factored forms. The LCD of \( 4(a - 5) \) and \( 6(a - 5) \) is \( 12(a - 5) \).
03

Rewrite each fraction

Rewrite each fraction with the common denominator. For \( \frac{3a}{4(a - 5)} \), multiply the numerator and the denominator by 3: \[ \frac{3a}{4(a - 5)} = \frac{3a \cdot 3}{4(a-5) \cdot 3} = \frac{9a}{12(a-5)} \] For \( \frac{9a}{6(a - 5)} \), multiply the numerator and the denominator by 2: \[ \frac{9a}{6(a-5)} = \frac{9a \cdot 2}{6(a-5) \cdot 2} = \frac{18a}{12(a-5)} \]
04

Add the fractions

Add the fractions keeping the common denominator: \[ \frac{9a}{12(a-5)} + \frac{18a}{12(a-5)} = \frac{9a + 18a}{12(a-5)} = \frac{27a}{12(a-5)} \]
05

Simplify the result

Simplify the fraction by reducing it to its lowest terms. Both the numerator and the denominator can be divided by 3: \[ \frac{27a}{12(a-5)} = \frac{27 \div 3 \cdot a}{12 \div 3 \cdot (a-5)} = \frac{9a}{4(a-5)} \]

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Key Concepts

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

Finding common denominators
To add fractions, we need a common denominator. This means the bottom part of both fractions should be the same. In our problem, the denominators are different: \(4a - 20\) and \(6a - 30\).
First, we factor each denominator. Factoring simplifies the problem and helps in finding the least common denominator (LCD).
For \(4a - 20\), we divide by 4: \[4a - 20 = 4(a - 5)\]
For \(6a - 30\), we divide by 6: \[6a - 30 = 6(a - 5)\]
You can see now that both factored denominators share a common part, \(a - 5\). To find the LCD, we take the highest factor from each unique part. Hence, the LCD for our fractions is \(12(a-5)\).
Factoring denominators
Factoring helps us break down complex expressions into simpler ones. This makes it easier to find common denominators.
Let's look at our denominators again: \(4a - 20\) and \(6a - 30\). Both expressions can be broken down as follows:
  • \(4a - 20 = 4(a - 5)\)
  • \(6a - 30 = 6(a - 5)\)
Factoring means we find components that multiply to give the original expression. Always look for the greatest common factor (GCF) which, in these cases, are 4 and 6. This helps to simplify the fractions and leads us to the common denominator we need for addition or subtraction.
Simplifying fractions
After finding the common denominator and rewriting both fractions, the next step is to combine and simplify them.
From the exercise: \frac{9a}{12(a-5)} + \frac{18a}{12(a-5)}\, we add the numerators: \[ \frac{9a + 18a}{12(a-5)} = \frac{27a}{12(a-5)} \]Now, we simplify by reducing to the lowest terms. Both the numerator (27a) and the denominator (12(a-5)) have a common factor of 3:
We divide both by 3:
\[ \frac{27a}{12(a-5)} = \frac{27 \textdiv 3 \times a}{12 \textdiv 3 \times (a-5)} = \frac{9a}{4(a-5)} \]This is our fully simplified result. Remember, simplifying makes fractions easier to understand and work with.

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Most popular questions from this chapter

To prepare for Chapter \(8,\) review solving inequalities \((\text {Section } 2.6)\) Solve. $$ 2(x-1) \geq 4(x+1) $$

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