Write $\frac{5 c}{3}+\frac{c+1}{2}$ as a single fraction in its simplest form.
Answer (2)
Find a common denominator for the fractions, which is 6.
Rewrite each fraction with the common denominator: 3 5 c = 6 10 c and 2 c + 1 = 6 3 c + 3 .
Add the fractions: 6 10 c + 6 3 c + 3 = 6 10 c + 3 c + 3 .
Simplify the numerator to get the final answer: 6 13 c + 3 .
Explanation
Understanding the Problem We are asked to add two fractions, 3 5 c and 2 c + 1 , and express the result as a single fraction in its simplest form. To do this, we need to find a common denominator for the two fractions.
Finding a Common Denominator The least common multiple (LCM) of the denominators 3 and 2 is 6. So, we will rewrite each fraction with a denominator of 6.
Rewriting the Fractions To rewrite 3 5 c with a denominator of 6, we multiply both the numerator and the denominator by 2: 3 5 c = 3 × 2 5 c × 2 = 6 10 c . To rewrite 2 c + 1 with a denominator of 6, we multiply both the numerator and the denominator by 3: 2 c + 1 = 2 × 3 ( c + 1 ) × 3 = 6 3 c + 3 .
Adding the Fractions Now that both fractions have the same denominator, we can add them: 6 10 c + 6 3 c + 3 = 6 10 c + ( 3 c + 3 ) .
Simplifying the Numerator Combine like terms in the numerator: 6 10 c + 3 c + 3 = 6 13 c + 3 .
Final Answer The fraction 6 13 c + 3 is now in its simplest form because there are no common factors between the numerator ( 13 c + 3 ) and the denominator (6). Therefore, the sum of the two fractions is 6 13 c + 3 .
Examples
Fractions are a fundamental concept in mathematics and are used in various real-life situations. For example, when you're cooking and need to combine different amounts of ingredients, you often have to add fractions. Suppose you need 3 5 c cups of flour and 2 c + 1 cups of sugar for a recipe. Combining these two amounts requires adding the fractions, just like in the problem we solved. Knowing how to add fractions allows you to accurately measure and combine ingredients to get the desired result in your recipe.
To combine 3 5 c and 2 c + 1 , we find a common denominator of 6, rewrite each fraction as 6 10 c and 6 3 c + 3 , and then add them to get 6 13 c + 3 as the final answer. The fraction is simplified and in its simplest form.
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