Write $\frac{x+5}{13}+\frac{x+2}{26}$ as a fraction in its simplest form.
Answer (1)
Find a common denominator: Rewrite the fractions with a common denominator of 26.
Add the fractions: Combine the numerators over the common denominator.
Simplify the numerator: Expand and combine like terms in the numerator.
Factor and simplify: Factor the numerator and simplify the fraction to its simplest form: 26 3 ( x + 4 ) .
Explanation
Understanding the problem We are given the expression 13 x + 5 + 26 x + 2 and we want to write it as a 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.
Rewriting the first fraction We rewrite the first fraction with the common denominator 26: 13 x + 5 = 2 ( 13 ) 2 ( x + 5 ) = 26 2 ( x + 5 ) .
Adding the fractions Now we can add the two fractions: 26 2 ( x + 5 ) + 26 x + 2 = 26 2 ( x + 5 ) + ( x + 2 ) .
Simplifying the numerator Next, we simplify the numerator: 2 ( x + 5 ) + ( x + 2 ) = 2 x + 10 + x + 2 = 3 x + 12 .
Writing the resulting fraction So the fraction becomes: 26 3 x + 12 .
Factoring the numerator We can factor the numerator: 3 x + 12 = 3 ( x + 4 ) .
Simplifying the fraction Thus, the fraction is: 26 3 ( x + 4 ) . Since 3 and 26 have no common factors, this is the simplest form.
Examples
In real-world scenarios, combining fractions like this can be useful when you're mixing ingredients in a recipe or calculating the total amount of resources needed for a project. For instance, if you need 13 x + 5 of a cup of flour from one bag and 26 x + 2 of a cup of flour from another bag, this calculation helps you determine the total amount of flour you'll have. Simplifying the expression allows for easier measurement and planning.
