To solve the subtraction problem [tex]\( 6 \frac{25}{27} - \frac{4}{9} - 1 \frac{1}{3} \)[/tex], we need to follow a series of steps to handle mixed numbers and fractions correctly.
### Step 1: Convert Mixed Numbers to Improper Fractions
#### First Term:
The mixed number [tex]\( 6 \frac{25}{27} \)[/tex] can be converted to an improper fraction as follows:
[tex]\[ 6 \frac{25}{27} = 6 + \frac{25}{27} \][/tex]
To combine these, we can write 6 as a fraction with the denominator 27:
[tex]\[ 6 = \frac{6 \cdot 27}{27} = \frac{162}{27} \][/tex]
Thus:
[tex]\[ 6 \frac{25}{27} = \frac{162}{27} + \frac{25}{27} = \frac{187}{27} \][/tex]
#### Second Term:
The fraction [tex]\( \frac{4}{9} \)[/tex] is already in fractional form, so no conversion is needed.
#### Third Term:
The mixed number [tex]\( 1 \frac{1}{3} \)[/tex] can be converted to an improper fraction as follows:
[tex]\[ 1 \frac{1}{3} = 1 + \frac{1}{3} \][/tex]
To combine these, we can write 1 as a fraction with the denominator 3:
[tex]\[ 1 = \frac{3}{3} \][/tex]
Thus:
[tex]\[ 1 \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \][/tex]
### Step 2: Convert All Fractions to a Common Denominator
To perform the subtraction, we need to convert these fractions to a common denominator. The denominators we have are 27, 9, and 3. The least common multiple of these is 27.
- For the first term:
[tex]\[ \frac{187}{27} \][/tex]
- For the second term, convert [tex]\( \frac{4}{9} \)[/tex] to a denominator of 27:
[tex]\[ \frac{4}{9} = \frac{4 \cdot 3}{9 \cdot 3} = \frac{12}{27} \][/tex]
- For the third term, convert [tex]\( \frac{4}{3} \)[/tex] to a denominator of 27:
[tex]\[ \frac{4}{3} = \frac{4 \cdot 9}{3 \cdot 9} = \frac{36}{27} \][/tex]
### Step 3: Perform the Subtraction
Now we subtract these fractions:
[tex]\[ \frac{187}{27} - \frac{12}{27} - \frac{36}{27} \][/tex]
Combine the numerators over the common denominator 27:
[tex]\[ \frac{187 - 12 - 36}{27} = \frac{139}{27} \][/tex]
### Step 4: Convert Improper Fraction to Mixed Number (if desired)
To convert [tex]\( \frac{139}{27} \)[/tex] back to a mixed number:
[tex]\[ 139 \div 27 = 5 \text{ remainder } 4 \][/tex]
So,
[tex]\[ \frac{139}{27} = 5 \frac{4}{27} \][/tex]
Thus, the final result of the subtraction is:
[tex]\[ 6 \frac{25}{27} - \frac{4}{9} - 1 \frac{1}{3} = 5 \frac{4}{27} \][/tex]
For reference, the approximate decimal result is 5.148.
So, the complete detailed answer is:
[tex]\[ 6 \frac{25}{27} - \frac{4}{9} - 1 \frac{1}{3} = 5 \frac{4}{27} \approx 5.148 \][/tex]
