Write $\frac{r+2}{5}+\frac{4 r}{9}$ as a single fraction in its simplest form.
Answer (2)
Find the least common denominator (LCD) of 5 and 9, which is 45.
Rewrite each fraction with the common denominator: 5 r + 2 = 45 9 ( r + 2 ) and 9 4 r = 45 5 ( 4 r ) .
Add the fractions: 45 9 ( r + 2 ) + 5 ( 4 r ) = 45 9 r + 18 + 20 r .
Simplify the numerator: 45 29 r + 18 . The final answer is 45 29 r + 18 .
Explanation
Understanding the Problem We are asked to write the expression 5 r + 2 + 9 4 r as a single fraction in its simplest form. This involves finding a common denominator, combining the fractions, and simplifying the resulting expression.
Finding the Common Denominator To add two fractions, we need to find a common denominator. The least common denominator (LCD) of 5 and 9 is the least common multiple (LCM) of 5 and 9. Since 5 and 9 are coprime (they have no common factors other than 1), their LCM is simply their product: 5 × 9 = 45 .
Rewriting Fractions with Common Denominator Now, we rewrite each fraction with the common denominator 45. To do this, we multiply the numerator and denominator of each fraction by the appropriate factor: 5 r + 2 = 9 ( 5 ) 9 ( r + 2 ) = 45 9 ( r + 2 ) 9 4 r = 5 ( 9 ) 5 ( 4 r ) = 45 5 ( 4 r )
Adding the Fractions Now that both fractions have the same denominator, we can add them: 45 9 ( r + 2 ) + 45 5 ( 4 r ) = 45 9 ( r + 2 ) + 5 ( 4 r )
Expanding the Numerator Next, we expand the numerator: 9 ( r + 2 ) + 5 ( 4 r ) = 9 r + 18 + 20 r
Combining Like Terms Now, we combine like terms in the numerator: 9 r + 18 + 20 r = 29 r + 18
Writing as a Single Fraction So, the expression becomes: 45 29 r + 18
Checking for Simplification Finally, we check if the fraction can be simplified further. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 45 are 1, 3, 5, 9, 15, and 45. The greatest common divisor of 18 and 45 is 9. However, 29 is a prime number, and it does not share any common factors with 45 (other than 1). Therefore, the fraction is already in its simplest form.
Thus, the simplified expression is: 45 29 r + 18
Final Answer The expression 5 r + 2 + 9 4 r written as a single fraction in its simplest form is 45 29 r + 18 .
Examples
In real-world applications, combining fractions is essential in various fields, such as engineering and physics. For instance, when calculating the total resistance in a parallel circuit, you often need to add fractions representing individual resistances. Simplifying these fractions into a single term makes further calculations easier and more efficient. Similarly, in mixture problems, combining fractional parts of different components helps determine the overall composition of the mixture. Understanding how to manipulate and simplify fractional expressions is a fundamental skill that enhances problem-solving capabilities in these practical scenarios.
To add 5 r + 2 + 9 4 r , we first find a common denominator, which is 45. We rewrite each fraction as 45 9 ( r + 2 ) and 45 20 r and combine them to get 45 29 r + 18 . Since this fraction has no further simplification, the final answer is 45 29 r + 18 .
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