Question #1

Perform the indicated operation for the given rational expressions:

[tex]\[ \frac{5}{5n} + \frac{2}{6m} \][/tex]

Write the answer in simplest form.

A. [tex]\(\frac{7}{11mn}\)[/tex]

B. [tex]\(\frac{7}{5n + 6m}\)[/tex]

C. [tex]\(\frac{3m + n}{3mn}\)[/tex]

D. [tex]\(\frac{6m + 2n}{mn}\)[/tex]



Answer :

To solve the given problem, we need to perform the addition of two rational expressions and write the result in its simplest form.

The given rational expressions are:
[tex]\[ \frac{5}{5 n} + \frac{2}{6 m} \][/tex]

### Step 1: Simplify the individual fractions

First, we simplify each fraction separately:

1. Simplify [tex]\(\frac{5}{5 n}\)[/tex]:
[tex]\[ \frac{5}{5 n} = \frac{5 \div 5}{5 n \div 5} = \frac{1}{n} \][/tex]

2. Simplify [tex]\(\frac{2}{6 m}\)[/tex]:
[tex]\[ \frac{2}{6 m} = \frac{2 \div 2}{6 m \div 2} = \frac{1}{3 m} \][/tex]

### Step 2: Find a common denominator

To add the fractions, we need a common denominator. The denominators are [tex]\(n\)[/tex] and [tex]\(3m\)[/tex]. The least common multiple (LCM) of [tex]\(n\)[/tex] and [tex]\(3m\)[/tex] is [tex]\(3mn\)[/tex].

### Step 3: Adjust the numerators to the common denominator

Convert each fraction to have the common denominator [tex]\(3mn\)[/tex].

1. Convert [tex]\(\frac{1}{n}\)[/tex]:
[tex]\[ \frac{1}{n} = \frac{1 \cdot 3m}{n \cdot 3m} = \frac{3m}{3mn} \][/tex]

2. Convert [tex]\(\frac{1}{3 m}\)[/tex]:
[tex]\[ \frac{1}{3 m} = \frac{1 \cdot n}{3m \cdot n} = \frac{n}{3mn} \][/tex]

### Step 4: Add the fractions

Now we can add the fractions as they have the same denominator:
[tex]\[ \frac{3m}{3mn} + \frac{n}{3mn} = \frac{3m + n}{3mn} \][/tex]

### Step 5: Simplify if possible

The fraction [tex]\(\frac{3m + n}{3mn}\)[/tex] is already in its simplest form. No further simplification is needed.

### Conclusion

The simplified form of the given rational expressions [tex]\(\frac{5}{5 n} + \frac{2}{6 m}\)[/tex] is:
[tex]\[ \frac{3 m + n}{3 m n} \][/tex]

Among the given options:
- [tex]\(\frac{7}{11 m n}\)[/tex]
- [tex]\(\frac{7}{5 n + 6 m}\)[/tex]
- [tex]\(\frac{3 m + n}{3 m n}\)[/tex]
- [tex]\(\frac{6 m + 2 n}{m n}\)[/tex]

The correct answer is:
[tex]\[ \frac{3 m + n}{3 m n} \][/tex]