Let's walk through each mathematical sentence step by step to arrive at the answers and shade the respective models appropriately.
### 1. [tex]\(\frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \)[/tex]
When you add [tex]\(\frac{1}{4}\)[/tex] three times, you get:
[tex]\[\frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}\][/tex]
So, you need to shade three-quarters of the model:
[tex]\[ \square \square \square \boxed \][/tex]
The result is [tex]\(\boxed{0.75}\)[/tex].
### 2. [tex]\(2 \frac{1}{3} + \frac{2}{3} = \)[/tex]
First, convert [tex]\(2 \frac{1}{3}\)[/tex] to an improper fraction:
[tex]\[2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\][/tex]
Now, add [tex]\(\frac{7}{3}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[\frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3\][/tex]
This gives us exactly 3.
[tex]\[ \square = 1 \,\, \square = 1 \,\, \square = 1 \][/tex]
[tex]\[\boxed{3.0}\][/tex]
So, the result is [tex]\(\boxed{3.0}\)[/tex].
### 3. [tex]\(1 \frac{1}{5} + 2 \frac{2}{5} = \)[/tex]
First, convert [tex]\(1 \frac{1}{5}\)[/tex] and [tex]\(2 \frac{2}{5}\)[/tex] to improper fractions:
[tex]\[1 \frac{1}{5} = 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5}\][/tex]
[tex]\[2 \frac{2}{5} = 2 + \frac{2}{5} = \frac{10}{5} + \frac{2}{5} = \frac{12}{5}\][/tex]
Now, add [tex]\(\frac{6}{5}\)[/tex] and [tex]\(\frac{12}{5}\)[/tex]:
[tex]\[\frac{6}{5} + \frac{12}{5} = \frac{18}{5} = 3 \frac{3}{5} = 3.6\][/tex]
[tex]\[ \boxed{3.6} \][/tex]
So, the result is [tex]\(\boxed{3.6}\)[/tex].
### 4. [tex]\(3 - \frac{1}{3} = \)[/tex]
To subtract [tex]\(\frac{1}{3}\)[/tex] from 3, it helps to convert 3 to a fraction with a common denominator:
[tex]\[3 = \frac{9}{3}\][/tex]
Now, subtract:
[tex]\[\frac{9}{3} - \frac{1}{3} = \frac{8}{3} = 2 \frac{2}{3}\][/tex]
[tex]\[ \boxed{2}\,\,\square\,\,\boxed{\frac{2}{3}}\][/tex]
So, the result is [tex]\(\boxed{2.67}\)[/tex].
### 5. [tex]\(5 \frac{3}{4} - 3 \frac{1}{8} = \)[/tex]
First, convert [tex]\(5 \frac{3}{4}\)[/tex] and [tex]\(3 \frac{1}{8}\)[/tex] to improper fractions:
[tex]\[5 \frac{3}{4} = 5 + \frac{3}{4} = \frac{20}{4} + \frac{3}{4} = \frac{23}{4}\][/tex]
[tex]\[3 \frac{1}{8} = 3 + \frac{1}{8} = \frac{24}{8} + \frac{1}{8} = \frac{25}{8}\][/tex]
To subtract, we need a common denominator, which in this case would be 8:
[tex]\[\frac{23}{4} = \frac{23 \times 2}{4 \times 2} = \frac{46}{8}\][/tex]
Now, subtract:
[tex]\[\frac{46}{8} - \frac{25}{8} = \frac{21}{8} = 2 \frac{5}{8} \][/tex]
[tex]\[ \boxed{2}\,\,\square\,\,\boxed{\frac{5}{8}} \][/tex]
So, the result is [tex]\(\boxed{2.625}\)[/tex].
