Answer :
Certainly! Let's solve the problem of adding the fractions [tex]\( \frac{32}{5} + \frac{11}{6} \)[/tex] and ensure our result is in its simplest form.
### Step 1: Find a Common Denominator
To add fractions, we need a common denominator. The denominators here are 5 and 6. The least common multiple (LCM) of 5 and 6 is 30, so we will use 30 as the common denominator.
### Step 2: Convert Each Fraction
Next, we will convert each fraction to have the common denominator of 30.
1. Convert [tex]\( \frac{32}{5} \)[/tex]:
- Multiply both the numerator and the denominator by the same number to get the new fraction with 30 as the denominator.
- Since [tex]\( 5 \times 6 = 30 \)[/tex], multiply the numerator by 6:
[tex]\[
\frac{32}{5} = \frac{32 \times 6}{5 \times 6} = \frac{192}{30}
\][/tex]
2. Convert [tex]\( \frac{11}{6} \)[/tex]:
- Similarly, multiply both the numerator and the denominator to change the denominator to 30.
- Since [tex]\( 6 \times 5 = 30 \)[/tex], multiply the numerator by 5:
[tex]\[
\frac{11}{6} = \frac{11 \times 5}{6 \times 5} = \frac{55}{30}
\][/tex]
### Step 3: Add the Converted Fractions
Now that both fractions have the same denominator, simply add the numerators:
[tex]\[
\frac{192}{30} + \frac{55}{30} = \frac{192 + 55}{30} = \frac{247}{30}
\][/tex]
### Step 4: Simplify if Possible
Check if the fraction [tex]\( \frac{247}{30} \)[/tex] can be simplified. Since 247 and 30 have no common factors other than 1, the fraction is in its simplest form.
Thus, the final answer is:
[tex]\[
\frac{247}{30}
\][/tex]
This fraction represents the sum of [tex]\( \frac{32}{5} \)[/tex] and [tex]\( \frac{11}{6} \)[/tex] in its simplest form.
### Step 1: Find a Common Denominator
To add fractions, we need a common denominator. The denominators here are 5 and 6. The least common multiple (LCM) of 5 and 6 is 30, so we will use 30 as the common denominator.
### Step 2: Convert Each Fraction
Next, we will convert each fraction to have the common denominator of 30.
1. Convert [tex]\( \frac{32}{5} \)[/tex]:
- Multiply both the numerator and the denominator by the same number to get the new fraction with 30 as the denominator.
- Since [tex]\( 5 \times 6 = 30 \)[/tex], multiply the numerator by 6:
[tex]\[
\frac{32}{5} = \frac{32 \times 6}{5 \times 6} = \frac{192}{30}
\][/tex]
2. Convert [tex]\( \frac{11}{6} \)[/tex]:
- Similarly, multiply both the numerator and the denominator to change the denominator to 30.
- Since [tex]\( 6 \times 5 = 30 \)[/tex], multiply the numerator by 5:
[tex]\[
\frac{11}{6} = \frac{11 \times 5}{6 \times 5} = \frac{55}{30}
\][/tex]
### Step 3: Add the Converted Fractions
Now that both fractions have the same denominator, simply add the numerators:
[tex]\[
\frac{192}{30} + \frac{55}{30} = \frac{192 + 55}{30} = \frac{247}{30}
\][/tex]
### Step 4: Simplify if Possible
Check if the fraction [tex]\( \frac{247}{30} \)[/tex] can be simplified. Since 247 and 30 have no common factors other than 1, the fraction is in its simplest form.
Thus, the final answer is:
[tex]\[
\frac{247}{30}
\][/tex]
This fraction represents the sum of [tex]\( \frac{32}{5} \)[/tex] and [tex]\( \frac{11}{6} \)[/tex] in its simplest form.
