To solve the expression ( 6 \frac{2}{25} + 7 \frac{23}{25} - 8 \frac{1}{30} ), we should first convert the mixed numbers to improper fractions.
Convert ( 6 \frac{2}{25} ):
[
6 \frac{2}{25} = \frac{6 \times 25 + 2}{25} = \frac{150 + 2}{25} = \frac{152}{25}
]
Convert ( 7 \frac{23}{25} ):
[
7 \frac{23}{25} = \frac{7 \times 25 + 23}{25} = \frac{175 + 23}{25} = \frac{198}{25}
]
Convert ( 8 \frac{1}{30} ):
[
8 \frac{1}{30} = \frac{8 \times 30 + 1}{30} = \frac{240 + 1}{30} = \frac{241}{30}
]
Now, we can rewrite the expression using the improper fractions:
[
\frac{152}{25} + \frac{198}{25} - \frac{241}{30}
]
Next, we combine the first two fractions, since they have a common denominator:
[
\frac{152 + 198}{25} = \frac{350}{25} = 14
]
Now we need to subtract ( \frac{241}{30} ) from ( 14 ):
First, we convert ( 14 ) to a fraction with a denominator of 30:
[
14 = \frac{14 \times 30}{30} = \frac{420}{30}
]
So now our expression becomes:
[
\frac{420}{30} - \frac{241}{30} = \frac{420 - 241}{30} = \frac{179}{30}
]
Thus, the final answer is:
[
\frac{179}{30}
]
If needed, we can convert it back to a mixed number:
[
\frac{179}{30} = 5 \frac{29}{30}
]
So the final result is:
[
5 \frac{29}{30}
]
