To add the mixed numbers (-4 \frac{17}{42}) and (3 \frac{3}{14}), follow these steps:
- Convert the mixed numbers into improper fractions.
- Find a common denominator.
- Add the fractions.
- Convert back to a mixed number if necessary.
Step 1: Convert to improper fractions
For (-4 \frac{17}{42}):
(-4 = -\frac{168}{42}) (since (-4 \times 42 = -168))
So, (-4 \frac{17}{42} = -\frac{168}{42} + \frac{17}{42} = -\frac{168 - 17}{42} = -\frac{151}{42}).
For (3 \frac{3}{14}):
(3 = \frac{42}{14}) (since (3 \times 14 = 42))
So, (3 \frac{3}{14} = \frac{42}{14} + \frac{3}{14} = \frac{42 + 3}{14} = \frac{45}{14}).
Step 2: Find a common denominator
The denominators are 42 and 14. The least common multiple (LCM) of 42 and 14 is 42.
Step 3: Convert ( \frac{45}{14} ) to have a denominator of 42
(\frac{45}{14} = \frac{45 \times 3}{14 \times 3} = \frac{135}{42}).
Step 4: Add the fractions
Now we can add the two fractions:
[
-\frac{151}{42} + \frac{135}{42} = \frac{-151 + 135}{42} = \frac{-16}{42}.
]
Step 5: Simplify the fraction
(-\frac{16}{42}) can be simplified:
(-\frac{16 \div 2}{42 \div 2} = -\frac{8}{21}).
Final Result
Thus, the sum of (-4 \frac{17}{42}) and (3 \frac{3}{14}) is:
[
-\frac{8}{21}.
]
