Write $\frac{2}{3}+\frac{5}{12 n}$ as a single fraction in its simplest form.
Answer (1)
Find a common denominator for the fractions, which is 12 n .
Rewrite 3 2 with the common denominator: 3 2 = 12 n 8 n .
Add the fractions: 12 n 8 n + 12 n 5 = 12 n 8 n + 5 .
The simplified fraction is 12 n 8 n + 5 .
Explanation
Understanding the Problem We are given the expression 3 2 + 12 n 5 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 3 and 12 n is 12 n .
Rewriting the First Fraction We rewrite the first fraction with the common denominator 12 n : 3 2 = 3 × 4 n 2 × 4 n = 12 n 8 n .
Adding the Fractions Now we can add the two fractions: 12 n 8 n + 12 n 5 = 12 n 8 n + 5 .
Checking for Common Factors and Simplifying We need to check if the numerator and the denominator have any common factors. In general, 8 n + 5 and 12 n do not have any common factors, so the fraction is already in its simplest form. Therefore, the simplified fraction is 12 n 8 n + 5 .
Final Answer Thus, the expression 3 2 + 12 n 5 can be written as a single fraction in its simplest form as 12 n 8 n + 5 .
Examples
Fractions are a fundamental part of everyday life. For example, if you are baking a cake and a recipe calls for 3 2 cup of flour and 12 n 5 cup of sugar (where 'n' depends on the number of servings), combining these ingredients requires adding the fractions. Expressing the sum as a single fraction, 12 n 8 n + 5 , helps you measure the total amount of dry ingredients needed more efficiently. This skill is also useful in managing finances, calculating proportions, and understanding various mathematical concepts.
