To solve this problem, we need to find the total fraction of the annual income that the family spends on various items by summing up the fractions provided for housing, food and clothing, general expenses, and entertainment.
Here are the fractions given:
- Housing: [tex]\( \frac{1}{10} \)[/tex]
- Food and clothing: [tex]\( \frac{1}{4} \)[/tex]
- General expenses: [tex]\( \frac{1}{5} \)[/tex]
- Entertainment: [tex]\( \frac{2}{15} \)[/tex]
We need to sum these fractions. To do that, it's helpful to find a common denominator. The denominators we need to work with are 10, 4, 5, and 15. The least common multiple (LCM) of these numbers will be our common denominator.
The LCM of 10, 4, 5, and 15 is 60. Now, we convert each fraction to have this common denominator:
1. [tex]\( \frac{1}{10} \)[/tex] can be converted:
[tex]\[ \frac{1}{10} = \frac{1 \times 6}{10 \times 6} = \frac{6}{60} \][/tex]
2. [tex]\( \frac{1}{4} \)[/tex] can be converted:
[tex]\[ \frac{1}{4} = \frac{1 \times 15}{4 \times 15} = \frac{15}{60} \][/tex]
3. [tex]\( \frac{1}{5} \)[/tex] can be converted:
[tex]\[ \frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60} \][/tex]
4. [tex]\( \frac{2}{15} \)[/tex] can be converted:
[tex]\[ \frac{2}{15} = \frac{2 \times 4}{15 \times 4} = \frac{8}{60} \][/tex]
Next, we sum these converted fractions:
[tex]\[ \frac{6}{60} + \frac{15}{60} + \frac{12}{60} + \frac{8}{60} = \frac{6 + 15 + 12 + 8}{60} = \frac{41}{60} \][/tex]
So, the fractional part of their income that the family spends on these items altogether is [tex]\( \frac{41}{60} \)[/tex].
The correct answer is [tex]\( \boxed{\frac{41}{60}} \)[/tex].
