The value of `4 1/3 + [7 2/3 - 4/3 (25/3 - 1/6 - 2/3)]` is:

Options

A. 5
B. 8
C. 2
D. 10

Answer

To solve the expression \( 4 \frac{1}{3} + \left[ 7 \frac{2}{3} - \frac{4}{3} \left( \frac{25}{3} - \frac{1}{6} - \frac{2}{3} \right) \right] \), we will follow the order of operations (brackets, orders, division, multiplication, addition, subtraction).

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert \( 4 \frac{1}{3} \) and \( 7 \frac{2}{3} \) to improper fractions.

- \( 4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3} \)
- \( 7 \frac{2}{3} = \frac{7 \times 3 + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3} \)

### Step 2: Simplify Inside the Brackets
Now, simplify the expression inside the brackets:

\[ \frac{25}{3} - \frac{1}{6} - \frac{2}{3} \]

To perform the subtraction, we need a common denominator. The least common multiple of 3 and 6 is 6.

- Convert \( \frac{25}{3} \) to sixths: \( \frac{25}{3} = \frac{25 \times 2}{3 \times 2} = \frac{50}{6} \)
- Convert \( \frac{2}{3} \) to sixths: \( \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \)

Now substitute back into the expression:

\[ \frac{50}{6} - \frac{1}{6} - \frac{4}{6} = \frac{50 - 1 - 4}{6} = \frac{45}{6} = \frac{15}{2} \]

### Step 3: Multiply by \( \frac{4}{3} \)
Now substitute back into the expression:

\[ 7 \frac{2}{3} - \frac{4}{3} \left( \frac{15}{2} \right) \]

Calculate \( \frac{4}{3} \times \frac{15}{2} \):

\[ \frac{4 \times 15}{3 \times 2} = \frac{60}{6} = 10 \]

Now substitute this back into the expression:

\[ \frac{23}{3} - 10 \]

Convert 10 to a fraction with a denominator of 3:

\[ 10 = \frac{30}{3} \]

So now we have:

\[ \frac{23}{3} - \frac{30}{3} = \frac{23 - 30}{3} = \frac{-7}{3} \]

### Step 4: Add \( 4 \frac{1}{3} \)
Now we need to add \( \frac{13}{3} \) to \( \frac{-7}{3} \):

\[ \frac{13}{3} + \frac{-7}{3} = \frac{13 - 7}{3} = \frac{6}{3} = 2 \]

### Final Answer
The value of \( 4 \frac{1}{3} + \left[ 7 \frac{2}{3} - \frac{4}{3} \left( \frac{25}{3} - \frac{1}{6} - \frac{2}{3} \right) \right] \) is \( 2 \).

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