To tackle this problem, let's break it down step-by-step.
1. Identify the fractions involved:
- 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]
2. Find a common denominator for the fractions:
We need to align these fractions to a common base to add them. The denominators involved are 10, 4, 5, and 15. The least common denominator (LCD) for 10, 4, 5, and 15 is 60.
3. Convert each fraction to have the common denominator of 60:
- [tex]\( \frac{1}{10} = \frac{1 \times 6}{10 \times 6} = \frac{6}{60} \)[/tex]
- [tex]\( \frac{1}{4} = \frac{1 \times 15}{4 \times 15} = \frac{15}{60} \)[/tex]
- [tex]\( \frac{1}{5} = \frac{1 \times 12}{5 \times 12} = \frac{12}{60} \)[/tex]
- [tex]\( \frac{2}{15} = \frac{2 \times 4}{15 \times 4} = \frac{8}{60} \)[/tex]
4. Add the fractions together:
Now, we sum all these converted fractions:
[tex]\[
\frac{6}{60} + \frac{15}{60} + \frac{12}{60} + \frac{8}{60}
\][/tex]
Treating the numerators separately, we get:
[tex]\[
6 + 15 + 12 + 8 = 41
\][/tex]
So the total is:
[tex]\[
\frac{41}{60}
\][/tex]
5. Simplify the fraction (if necessary):
In this case, [tex]\( \frac{41}{60} \)[/tex] is already in its simplest form since 41 is a prime number and does not share any common factors with 60.
Hence, the fractional part of their income spent on these items altogether is [tex]\( \frac{41}{60} \)[/tex].
6. Verify the answer against the given options:
The correct answer is:
[tex]\[
\boxed{ \frac{41}{60} }
\][/tex]
