Answer :
Let's solve the expressions given that [tex]\( x = 1 \frac{1}{3} \)[/tex].
First, convert [tex]\( x = 1 \frac{1}{3} \)[/tex] into an improper fraction:
[tex]\[ x = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \][/tex]
Now, let's evaluate each expression.
### Expression 1
[tex]\[ 4 \frac{1}{2} + x + 3 \frac{5}{6} \][/tex]
Convert each mixed number to an improper fraction:
- [tex]\( 4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} \)[/tex]
- [tex]\( 3 \frac{5}{6} = 3 + \frac{5}{6} = \frac{18}{6} + \frac{5}{6} = \frac{23}{6} \)[/tex]
Now, add these fractions together. First, find a common denominator (which is 6 here):
[tex]\[ \frac{9}{2} = \frac{27}{6}, \quad \frac{4}{3} = \frac{8}{6} \][/tex]
Add them all together:
[tex]\[ \frac{27}{6} + \frac{8}{6} + \frac{23}{6} = \frac{27 + 8 + 23}{6} = \frac{58}{6} \][/tex]
Convert [tex]\(\frac{58}{6}\)[/tex] into a mixed number:
[tex]\[ \frac{58}{6} = 9 \frac{4}{6} = 9 \frac{2}{3} \][/tex]
So, the value is:
[tex]\[ \text{D. } 9 \frac{2}{3} \][/tex]
### Expression 2
[tex]\[ \frac{2}{5} + \left(x - \frac{1}{4}\right) \][/tex]
First, calculate [tex]\( x - \frac{1}{4} \)[/tex]:
[tex]\[ x = \frac{4}{3} \quad \text{(converted earlier)} \][/tex]
Find a common denominator for [tex]\( \frac{4}{3} \)[/tex] and [tex]\( \frac{1}{4} \)[/tex] (which is 12):
[tex]\[ \frac{4}{3} = \frac{16}{12}, \quad \frac{1}{4} = \frac{3}{12} \][/tex]
Subtract:
[tex]\[ \frac{16}{12} - \frac{3}{12} = \frac{13}{12} \][/tex]
Now add [tex]\( \frac{2}{5} \)[/tex], finding a common denominator (which is 60):
[tex]\[ \frac{2}{5} = \frac{24}{60}, \quad \frac{13}{12} = \frac{65}{60} \][/tex]
Add:
[tex]\[ \frac{24}{60} + \frac{65}{60} = \frac{89}{60} \][/tex]
Convert [tex]\(\frac{89}{60}\)[/tex] into a mixed number:
[tex]\[ \frac{89}{60} = 1 \frac{29}{60} \][/tex]
So, the value is:
[tex]\[ \text{C. } 1 \frac{29}{60} \][/tex]
These are the results for each expression given [tex]\( x = 1 \frac{1}{3} \)[/tex].
First, convert [tex]\( x = 1 \frac{1}{3} \)[/tex] into an improper fraction:
[tex]\[ x = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \][/tex]
Now, let's evaluate each expression.
### Expression 1
[tex]\[ 4 \frac{1}{2} + x + 3 \frac{5}{6} \][/tex]
Convert each mixed number to an improper fraction:
- [tex]\( 4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} \)[/tex]
- [tex]\( 3 \frac{5}{6} = 3 + \frac{5}{6} = \frac{18}{6} + \frac{5}{6} = \frac{23}{6} \)[/tex]
Now, add these fractions together. First, find a common denominator (which is 6 here):
[tex]\[ \frac{9}{2} = \frac{27}{6}, \quad \frac{4}{3} = \frac{8}{6} \][/tex]
Add them all together:
[tex]\[ \frac{27}{6} + \frac{8}{6} + \frac{23}{6} = \frac{27 + 8 + 23}{6} = \frac{58}{6} \][/tex]
Convert [tex]\(\frac{58}{6}\)[/tex] into a mixed number:
[tex]\[ \frac{58}{6} = 9 \frac{4}{6} = 9 \frac{2}{3} \][/tex]
So, the value is:
[tex]\[ \text{D. } 9 \frac{2}{3} \][/tex]
### Expression 2
[tex]\[ \frac{2}{5} + \left(x - \frac{1}{4}\right) \][/tex]
First, calculate [tex]\( x - \frac{1}{4} \)[/tex]:
[tex]\[ x = \frac{4}{3} \quad \text{(converted earlier)} \][/tex]
Find a common denominator for [tex]\( \frac{4}{3} \)[/tex] and [tex]\( \frac{1}{4} \)[/tex] (which is 12):
[tex]\[ \frac{4}{3} = \frac{16}{12}, \quad \frac{1}{4} = \frac{3}{12} \][/tex]
Subtract:
[tex]\[ \frac{16}{12} - \frac{3}{12} = \frac{13}{12} \][/tex]
Now add [tex]\( \frac{2}{5} \)[/tex], finding a common denominator (which is 60):
[tex]\[ \frac{2}{5} = \frac{24}{60}, \quad \frac{13}{12} = \frac{65}{60} \][/tex]
Add:
[tex]\[ \frac{24}{60} + \frac{65}{60} = \frac{89}{60} \][/tex]
Convert [tex]\(\frac{89}{60}\)[/tex] into a mixed number:
[tex]\[ \frac{89}{60} = 1 \frac{29}{60} \][/tex]
So, the value is:
[tex]\[ \text{C. } 1 \frac{29}{60} \][/tex]
These are the results for each expression given [tex]\( x = 1 \frac{1}{3} \)[/tex].
