Sure, let's work through this problem step-by-step.
### Step 1: Understand the Fractions
We have two fractions:
[tex]\[ \frac{1}{5} \][/tex]
and
[tex]\[ 1 \frac{2}{3} = 1 + \frac{2}{3} \][/tex]
First, we'll convert the mixed number [tex]\(1 \frac{2}{3}\)[/tex] into an improper fraction.
### Step 2: Convert the Mixed Number to an Improper Fraction
A mixed number like [tex]\(1 \frac{2}{3}\)[/tex] can be converted to an improper fraction as follows:
[tex]\[ 1 \frac{2}{3} = 1 + \frac{2}{3} \][/tex]
To add these, we first convert the whole number part to a fraction:
[tex]\[ 1 = \frac{3}{3} \][/tex]
Now we can add the two fractions:
[tex]\[ \frac{3}{3} + \frac{2}{3} = \frac{3 + 2}{3} = \frac{5}{3} \][/tex]
So:
[tex]\[ 1 \frac{2}{3} = \frac{5}{3} \][/tex]
### Step 3: Find a Common Denominator
To add [tex]\(\frac{1}{5}\)[/tex] and [tex]\(\frac{5}{3}\)[/tex], we need a common denominator.
The denominators are 5 and 3. The least common multiple of 5 and 3 is 15.
### Step 4: Convert Each Fraction to Have the Common Denominator
Now, we convert both fractions to have a denominator of 15:
For [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[ \frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15} \][/tex]
For [tex]\(\frac{5}{3}\)[/tex]:
[tex]\[ \frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15} \][/tex]
### Step 5: Add the Fractions
Now, we add the two like fractions:
[tex]\[ \frac{3}{15} + \frac{25}{15} = \frac{3 + 25}{15} = \frac{28}{15} \][/tex]
### Step 6: Simplify the Fraction (if possible)
In this case, the fraction [tex]\(\frac{28}{15}\)[/tex] is already in its simplest form because 28 and 15 have no common factors other than 1.
### Final Answer:
So, the sum of the fractions [tex]\(\frac{1}{5}\)[/tex] and [tex]\(1 \frac{2}{3}\)[/tex] is:
[tex]\[ \frac{28}{15} \][/tex]
