Answer :
We want to add the two mixed numbers:
[tex]$$1\frac{3}{5} \quad \text{and} \quad 3\frac{1}{4}.$$[/tex]
Step 1. Convert each mixed number to an improper fraction.
For the first number, [tex]$1\frac{3}{5}$[/tex]:
- Multiply the whole number by the denominator:
[tex]$$1 \times 5 = 5.$$[/tex]
- Add the numerator:
[tex]$$5 + 3 = 8.$$[/tex]
- So,
[tex]$$1\frac{3}{5} = \frac{8}{5}.$$[/tex]
For the second number, [tex]$3\frac{1}{4}$[/tex]:
- Multiply the whole number by the denominator:
[tex]$$3 \times 4 = 12.$$[/tex]
- Add the numerator:
[tex]$$12 + 1 = 13.$$[/tex]
- So,
[tex]$$3\frac{1}{4} = \frac{13}{4}.$$[/tex]
Step 2. Find a common denominator and convert both fractions.
The denominators are [tex]$5$[/tex] and [tex]$4$[/tex], and a common denominator is [tex]$20$[/tex].
Convert [tex]$\frac{8}{5}$[/tex] to a fraction with denominator [tex]$20$[/tex]:
[tex]$$\frac{8}{5} = \frac{8 \times 4}{5 \times 4} = \frac{32}{20}.$$[/tex]
Convert [tex]$\frac{13}{4}$[/tex] to a fraction with denominator [tex]$20$[/tex]:
[tex]$$\frac{13}{4} = \frac{13 \times 5}{4 \times 5} = \frac{65}{20}.$$[/tex]
Step 3. Add the fractions.
Now add the fractions:
[tex]$$\frac{32}{20} + \frac{65}{20} = \frac{32+65}{20} = \frac{97}{20}.$$[/tex]
Step 4. Convert the improper fraction to a mixed number.
Divide [tex]$97$[/tex] by [tex]$20$[/tex] to find the whole number part:
[tex]$$97 \div 20 = 4 \quad \text{with a remainder of} \quad 97 - 4 \times 20 = 17.$$[/tex]
Thus, we have:
[tex]$$\frac{97}{20} = 4\frac{17}{20}.$$[/tex]
Therefore, the final answer is:
[tex]$$\boxed{4\frac{17}{20}}.$$[/tex]
[tex]$$1\frac{3}{5} \quad \text{and} \quad 3\frac{1}{4}.$$[/tex]
Step 1. Convert each mixed number to an improper fraction.
For the first number, [tex]$1\frac{3}{5}$[/tex]:
- Multiply the whole number by the denominator:
[tex]$$1 \times 5 = 5.$$[/tex]
- Add the numerator:
[tex]$$5 + 3 = 8.$$[/tex]
- So,
[tex]$$1\frac{3}{5} = \frac{8}{5}.$$[/tex]
For the second number, [tex]$3\frac{1}{4}$[/tex]:
- Multiply the whole number by the denominator:
[tex]$$3 \times 4 = 12.$$[/tex]
- Add the numerator:
[tex]$$12 + 1 = 13.$$[/tex]
- So,
[tex]$$3\frac{1}{4} = \frac{13}{4}.$$[/tex]
Step 2. Find a common denominator and convert both fractions.
The denominators are [tex]$5$[/tex] and [tex]$4$[/tex], and a common denominator is [tex]$20$[/tex].
Convert [tex]$\frac{8}{5}$[/tex] to a fraction with denominator [tex]$20$[/tex]:
[tex]$$\frac{8}{5} = \frac{8 \times 4}{5 \times 4} = \frac{32}{20}.$$[/tex]
Convert [tex]$\frac{13}{4}$[/tex] to a fraction with denominator [tex]$20$[/tex]:
[tex]$$\frac{13}{4} = \frac{13 \times 5}{4 \times 5} = \frac{65}{20}.$$[/tex]
Step 3. Add the fractions.
Now add the fractions:
[tex]$$\frac{32}{20} + \frac{65}{20} = \frac{32+65}{20} = \frac{97}{20}.$$[/tex]
Step 4. Convert the improper fraction to a mixed number.
Divide [tex]$97$[/tex] by [tex]$20$[/tex] to find the whole number part:
[tex]$$97 \div 20 = 4 \quad \text{with a remainder of} \quad 97 - 4 \times 20 = 17.$$[/tex]
Thus, we have:
[tex]$$\frac{97}{20} = 4\frac{17}{20}.$$[/tex]
Therefore, the final answer is:
[tex]$$\boxed{4\frac{17}{20}}.$$[/tex]
