Write $\frac{x+5}{13}+\frac{x+2}{26}$ as a fraction in its simplest form.
Answer (2)
Find a common denominator: Rewrite the fractions with a common denominator of 26.
Combine the numerators: Add the numerators of the fractions.
Simplify the numerator: Expand and combine like terms in the numerator.
Write the simplified fraction: 26 3 ( x + 4 )
Explanation
Problem Analysis We are given the expression 13 x + 5 + 26 x + 2 and we want to write it as a single fraction in its simplest form.
Finding a Common Denominator To add these two fractions, we need to find a common denominator. The least common multiple of 13 and 26 is 26. So, we will rewrite the first fraction with a denominator of 26.
Rewriting the First Fraction To rewrite 13 x + 5 with a denominator of 26, we multiply both the numerator and the denominator by 2: 13 x + 5 = 2 ( 13 ) 2 ( x + 5 ) = 26 2 ( x + 5 )
Combining the Fractions Now we can rewrite the original expression as: 26 2 ( x + 5 ) + 26 x + 2 Since the fractions now have a common denominator, we can add the numerators: 26 2 ( x + 5 ) + ( x + 2 )
Simplifying the Numerator Next, we expand and simplify the numerator: 26 2 x + 10 + x + 2 = 26 3 x + 12
Final Simplification Now we check if the fraction can be simplified further. We can factor out a 3 from the numerator: 26 3 x + 12 = 26 3 ( x + 4 ) Since 3 and 26 have no common factors other than 1, the fraction is in its simplest form.
Final Answer Therefore, the simplified form of the given expression is 26 3 ( x + 4 ) .
Examples
In real-world applications, simplifying algebraic expressions like this can be useful in various scenarios. For example, if you are calculating the total cost of two different items with prices expressed as fractions involving a variable, simplifying the expression allows you to easily determine the total cost for any value of the variable. Suppose the cost of one item is 13 x + 5 dollars and the cost of another item is 26 x + 2 dollars, where x represents a discount value. By simplifying the expression to 26 3 ( x + 4 ) , you can quickly calculate the total cost for any given discount value x .
To simplify 13 x + 5 + 26 x + 2 , we find a common denominator of 26 and rewrite the first fraction. After combining and simplifying, we get 26 3 ( x + 4 ) as the final result.
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