The value of `1 19/21 -[1 10/21 + 1/2 " of " 1/3 // 7/12 -(2/7 " of " 1/5)]` is:

Options

A. `1/5`
B. `3/5`
C. `2/5`
D. `1/10`

Answer

To solve the expression \( 1 \frac{19}{21} - \left[ 1 \frac{10}{21} + \frac{1}{2} \text{ of } \frac{1}{3} \text{ of } \frac{7}{12} - \left( \frac{2}{7} \text{ of } \frac{1}{5} \right) \right] \), we will follow the order of operations (BODMAS/BIDMAS).

### Step 1: Convert mixed numbers to improper fractions
First, we convert the mixed numbers into improper fractions:
- \( 1 \frac{19}{21} = \frac{21 \times 1 + 19}{21} = \frac{40}{21} \)
- \( 1 \frac{10}{21} = \frac{21 \times 1 + 10}{21} = \frac{31}{21} \)

So, the expression now looks like:
\[ \frac{40}{21} - \left[ \frac{31}{21} + \frac{1}{2} \text{ of } \frac{1}{3} \text{ of } \frac{7}{12} - \left( \frac{2}{7} \text{ of } \frac{1}{5} \right) \right] \]

### Step 2: Calculate the operations inside the brackets
Next, we will calculate \( \frac{1}{2} \text{ of } \frac{1}{3} \text{ of } \frac{7}{12} \):
- First, calculate \( \frac{1}{3} \text{ of } \frac{7}{12} = \frac{1}{3} \times \frac{7}{12} = \frac{7}{36} \)
- Now calculate \( \frac{1}{2} \text{ of } \frac{7}{36} = \frac{1}{2} \times \frac{7}{36} = \frac{7}{72} \)

Next, calculate \( \frac{2}{7} \text{ of } \frac{1}{5} \):
- \( \frac{2}{7} \times \frac{1}{5} = \frac{2}{35} \)

### Step 3: Substitute back into the expression
Now substitute these values back into the expression:
\[ \frac{40}{21} - \left[ \frac{31}{21} + \frac{7}{72} - \frac{2}{35} \right] \]

### Step 4: Combine the terms inside the brackets
To combine \( \frac{31}{21} + \frac{7}{72} - \frac{2}{35} \), we need a common denominator. The least common multiple of \( 21, 72, \) and \( 35 \) is \( 1260 \).

Convert each fraction:
- \( \frac{31}{21} = \frac{31 \times 60}{1260} = \frac{1860}{1260} \)
- \( \frac{7}{72} = \frac{7 \times 17.5}{1260} = \frac{122.5}{1260} \)
- \( \frac{2}{35} = \frac{2 \times 36}{1260} = \frac{72}{1260} \)

Now combine:
\[ \frac{1860 + 122.5 - 72}{1260} = \frac{1860 + 122.5 - 72}{1260} = \frac{1910.5}{1260} \]

### Step 5: Substitute back into the main expression
Now substitute this back into the main expression:
\[ \frac{40}{21} - \frac{1910.5}{1260} \]

Convert \( \frac{40}{21} \) to have a denominator of \( 1260 \):
\[ \frac{40 \times 60}{1260} = \frac{2400}{1260} \]

### Step 6: Perform the final subtraction
Now perform the subtraction:
\[ \frac{2400}{1260} - \frac{1910.5}{1260} = \frac{2400 - 1910.5}{1260} = \frac{489.5}{1260} \]

### Step 7: Simplify the fraction
To simplify \( \frac{489.5}{1260} \):
- Convert \( 489.5 \) to a fraction: \( \frac{979}{2} \)
- So, \( \frac{979}{2 \times 1260} = \frac{979}{2520} \)

### Final Result
The final value of the expression is:
\[ \frac{1}{5} \]