Sure, let's work out the sum of [tex]\(1 \frac{3}{5}\)[/tex] and [tex]\(2 \frac{1}{4}\)[/tex] step by step.
### Step 1: Convert the mixed numbers to improper fractions
A mixed number consists of a whole number and a fraction. To add mixed numbers, we first convert each mixed number to an improper fraction.
First Mixed Number: [tex]\(1 \frac{3}{5}\)[/tex]
[tex]\[
1 \frac{3}{5} = 1 + \frac{3}{5} = \frac{5}{5} + \frac{3}{5} = \frac{5 + 3}{5} = \frac{8}{5}
\][/tex]
Second Mixed Number: [tex]\(2 \frac{1}{4}\)[/tex]
[tex]\[
2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{8 + 1}{4} = \frac{9}{4}
\][/tex]
Now we have the improper fractions [tex]\( \frac{8}{5} \)[/tex] and [tex]\( \frac{9}{4} \)[/tex].
### Step 2: Find a common denominator
To add these fractions, we need a common denominator. The denominators are 5 and 4. The common denominator is the least common multiple (LCM) of these two numbers.
The LCM of 5 and 4 is 20.
### Step 3: Convert each fraction to have the common denominator
Convert [tex]\( \frac{8}{5} \)[/tex] to a denominator of 20:
[tex]\[
\frac{8}{5} = \frac{8 \times 4}{5 \times 4} = \frac{32}{20}
\][/tex]
Convert [tex]\( \frac{9}{4} \)[/tex] to a denominator of 20:
[tex]\[
\frac{9}{4} = \frac{9 \times 5}{4 \times 5} = \frac{45}{20}
\][/tex]
### Step 4: Add the fractions
Now that both fractions have the same denominator, we can add them directly:
[tex]\[
\frac{32}{20} + \frac{45}{20} = \frac{32 + 45}{20} = \frac{77}{20}
\][/tex]
### Step 5: Convert the improper fraction back to a mixed number
To convert [tex]\( \frac{77}{20} \)[/tex] back to a mixed number, we divide 77 by 20. This will give us the whole number part and the fractional part.
[tex]\[
77 \div 20 = 3 \quad \text{remainder} \quad 17
\][/tex]
This can be written as:
[tex]\[
\frac{77}{20} = 3 \frac{17}{20}
\][/tex]
### Final Answer
Thus, the sum of [tex]\(1 \frac{3}{5}\)[/tex] and [tex]\(2 \frac{1}{4}\)[/tex] is:
[tex]\[
3 \frac{17}{20}
\][/tex]
