Write $\frac{x+5}{13}+\frac{x+2}{26}$ as a fraction in its simplest form.
Answer (2)
To simplify 13 x + 5 + 26 x + 2 , first find a common denominator and rewrite the fractions. Combining the fractions yields 26 3 ( x + 4 ) , which is the simplest form of the expression. Thus, the final answer is 26 3 ( x + 4 ) .
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Find a common denominator: Rewrite 13 x + 5 as 26 2 ( x + 5 ) = 26 2 x + 10 .
Add the fractions: 26 2 x + 10 + 26 x + 2 = 26 3 x + 12 .
Factor the numerator: 26 3 x + 12 = 26 3 ( x + 4 ) .
The simplified fraction is 26 3 ( x + 4 ) .
Explanation
Problem Analysis We want to simplify the expression 13 x + 5 + 26 x + 2 into a single fraction in its simplest form.
Finding 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 convert the first fraction 13 x + 5 to have 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 + 10
Adding the Fractions Now we can add the two fractions: 26 2 x + 10 + 26 x + 2 = 26 ( 2 x + 10 ) + ( x + 2 )
Simplifying the Numerator Next, we simplify the numerator by combining like terms: 26 2 x + 10 + x + 2 = 26 3 x + 12
Factoring the Numerator Now we factor the numerator to see if we can simplify the fraction further: 26 3 x + 12 = 26 3 ( x + 4 )
Final Simplification Since 3 and 26 have no common factors, the fraction is already in its simplest form. Therefore, the simplified expression is: 26 3 ( x + 4 )
Examples
Fractions are used in everyday life, such as when calculating proportions in recipes or determining discounts while shopping. Simplifying algebraic fractions helps in solving equations and understanding relationships between variables, which is crucial in fields like engineering and economics. For instance, if you're mixing ingredients for a chemical solution, you might need to simplify a complex fraction to determine the correct proportions of each component. This skill ensures accuracy and efficiency in various practical applications.
