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Magazine Chapter 4: Problem 17
$$ \frac{5 a}{11}+\frac{4 a}{9} $$
Short Answer
Expert verified
The simplified expression is \( \frac{89a}{99} \).
Step by step solution
01
Identify the Problem
We need to simplify the expression \( \frac{5a}{11} + \frac{4a}{9} \). This is an addition of two algebraic fractions.
02
Find a Common Denominator
To add fractions, we first need a common denominator. The denominators are 11 and 9. The least common multiple (LCM) of 11 and 9 is 99.
03
Rewrite Each Fraction
Now, we rewrite each fraction with the common denominator 99.\[\frac{5a}{11} = \frac{5a \times 9}{11 \times 9} = \frac{45a}{99}\] \[\frac{4a}{9} = \frac{4a \times 11}{9 \times 11} = \frac{44a}{99}\]
04
Add the Fractions
We can now add the fractions since they have a common denominator.\[ \frac{45a}{99} + \frac{44a}{99} = \frac{45a + 44a}{99} = \frac{89a}{99} \]
05
Simplify If Necessary
Check if \( \frac{89a}{99} \) can be simplified. Since 89 is a prime number and does not divide evenly into 99, this fraction is already in its simplest form.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominator
When dealing with fractions, it鈥檚 important to have a common denominator before you can perform addition or subtraction. This means both fractions need to have the same denominator.
- The denominator is the number located at the bottom of a fraction, indicating into how many equal parts the whole is divided.
- When denominators are different, as they are in the fractions \( \frac{5a}{11} \) and \( \frac{4a}{9} \), you need to find a common denominator to combine them.
Addition of Fractions
Once a common denominator is established, adding fractions becomes straightforward. You simply add the numerators while keeping the denominator the same.
- The numerator is the top number of a fraction, indicating how many parts you have.
- In the expression \(\frac{5a}{11} + \frac{4a}{9}\), once converted to fractions with the common denominator of 99, we have \(\frac{45a}{99}\) and \(\frac{44a}{99}\).
Least Common Multiple (LCM)
The least common multiple (LCM) of two numbers is the smallest multiple that is evenly divisible by both numbers. It plays a crucial role in finding a common denominator when adding or subtracting fractions.
- To find the LCM of two numbers, such as 11 and 9, you list the multiples of each number until you find the smallest multiple common to both lists.
- For example, the multiples of 11 are 11, 22, 33, ..., and the multiples of 9 are 9, 18, 27, 36, ..., and so on. The smallest number common to both lists is 99.
