Certainly! Let's solve the expression [tex]\(3 \frac{5}{6} - 2 \frac{3}{5} + \frac{3}{5}\)[/tex] step-by-step.
Step 1: Convert the mixed numbers to improper fractions.
1. [tex]\(3 \frac{5}{6}\)[/tex]:
[tex]\[
3 \frac{5}{6} = 3 + \frac{5}{6} = \frac{18}{6} + \frac{5}{6} = \frac{23}{6}
\][/tex]
2. [tex]\(2 \frac{3}{5}\)[/tex]:
[tex]\[
2 \frac{3}{5} = 2 + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{13}{5}
\][/tex]
3. [tex]\(\frac{3}{5}\)[/tex] is already an improper fraction.
Step 2: Find a common denominator for the fractions.
The denominators are 6 and 5. The least common multiple (LCM) of 6 and 5 is 30.
Step 3: Convert all fractions to have the common denominator of 30.
1. [tex]\(\frac{23}{6}\)[/tex]:
[tex]\[
\frac{23}{6} = \frac{23 \times 5}{6 \times 5} = \frac{115}{30}
\][/tex]
2. [tex]\(\frac{13}{5}\)[/tex]:
[tex]\[
\frac{13}{5} = \frac{13 \times 6}{5 \times 6} = \frac{78}{30}
\][/tex]
3. [tex]\(\frac{3}{5}\)[/tex]:
[tex]\[
\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}
\][/tex]
Now we have:
[tex]\[
3 \frac{5}{6} = \frac{115}{30}
\][/tex]
[tex]\[
2 \frac{3}{5} = \frac{78}{30}
\][/tex]
[tex]\[
\frac{3}{5} = \frac{18}{30}
\][/tex]
Step 4: Perform the calculations.
Subtraction followed by addition:
[tex]\[
\frac{115}{30} - \frac{78}{30} + \frac{18}{30}
\][/tex]
First, subtract:
[tex]\[
\frac{115}{30} - \frac{78}{30} = \frac{115 - 78}{30} = \frac{37}{30}
\][/tex]
Next, add:
[tex]\[
\frac{37}{30} + \frac{18}{30} = \frac{37 + 18}{30} = \frac{55}{30}
\][/tex]
Step 5: Simplify or convert back to a mixed number (if necessary).
To convert [tex]\(\frac{55}{30}\)[/tex] back to a mixed number, divide the numerator by the denominator:
[tex]\[
55 \div 30 = 1 \text{ (remainder } 25\text{)}
\][/tex]
So:
[tex]\[
\frac{55}{30} = 1 \frac{25}{30}
\][/tex]
Final answer:
The result of [tex]\(3 \frac{5}{6} - 2 \frac{3}{5} + \frac{3}{5}\)[/tex] is [tex]\(1 \frac{25}{30}\)[/tex].
