A fruit juice seller has `12 (5)/(7)` litres of juice in a container. He sells `9 (5)/(6)` litres of juice. He then refills the container with `8 (1)/(4)` litres of juice. How much juice is there in the container now?

Options

A. `11 (11)/(84)`
B. `9 (11)/(74)`
C. `13 (13)/(84)`
D. `11 (23)/(84)`

Answer

To solve the problem step by step, we will first convert the mixed fractions into improper fractions, perform the necessary arithmetic operations, and finally find the total amount of juice in the container.

### Step 1: Convert Mixed Numbers to Improper Fractions

1. **Convert `12 (5/7)` to an improper fraction:**
- Multiply the whole number (12) by the denominator (7):
\( 12 \times 7 = 84 \)
- Add the numerator (5) to this product:
\( 84 + 5 = 89 \)
- Thus, \( 12 (5/7) = \frac{89}{7} \)

2. **Convert `9 (5/6)` to an improper fraction:**
- Multiply the whole number (9) by the denominator (6):
\( 9 \times 6 = 54 \)
- Add the numerator (5) to this product:
\( 54 + 5 = 59 \)
- Thus, \( 9 (5/6) = \frac{59}{6} \)

3. **Convert `8 (1/4)` to an improper fraction:**
- Multiply the whole number (8) by the denominator (4):
\( 8 \times 4 = 32 \)
- Add the numerator (1) to this product:
\( 32 + 1 = 33 \)
- Thus, \( 8 (1/4) = \frac{33}{4} \)

### Step 2: Calculate the Remaining Juice After Selling

Now we need to subtract the amount sold from the initial amount:

\[ \text{Remaining Juice} = \frac{89}{7} - \frac{59}{6} \]

To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 6 is 42.

4. **Convert the fractions to have a common denominator:**
- For \( \frac{89}{7} \):
\[ \frac{89}{7} = \frac{89 \times 6}{7 \times 6} = \frac{534}{42} \]
- For \( \frac{59}{6} \):
\[ \frac{59}{6} = \frac{59 \times 7}{6 \times 7} = \frac{413}{42} \]

5. **Subtract the two fractions:**
\[ \frac{534}{42} - \frac{413}{42} = \frac{534 - 413}{42} = \frac{121}{42} \]

### Step 3: Add the Refilled Juice

Now we need to add the juice that was refilled:

\[ \text{Total Juice} = \frac{121}{42} + \frac{33}{4} \]

Again, we need a common denominator. The LCM of 42 and 4 is 84.

6. **Convert the fractions to have a common denominator:**
- For \( \frac{121}{42} \):
\[ \frac{121}{42} = \frac{121 \times 2}{42 \times 2} = \frac{242}{84} \]
- For \( \frac{33}{4} \):
\[ \frac{33}{4} = \frac{33 \times 21}{4 \times 21} = \frac{693}{84} \]

7. **Add the two fractions:**
\[ \frac{242}{84} + \frac{693}{84} = \frac{242 + 693}{84} = \frac{935}{84} \]

### Final Answer

The total amount of juice in the container now is \( \frac{935}{84} \) litres.

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