Tanya gives away to each of four girls `1/12, 5/18, 7/30, 7/48` of the apples in a basket and has only just enough apples to be able to do so without dividing an apple. The minimum number of apple had she ?

Options

A. 250
B. 720
C. 750
D. None of these

Answer

To find the minimum number of apples Tanya had, we need to determine the least common multiple (LCM) of the denominators of the fractions given. Here’s a step-by-step solution:

### Step 1: Identify the Denominators
The fractions given are:
- \( \frac{1}{12} \)
- \( \frac{5}{18} \)
- \( \frac{7}{30} \)
- \( \frac{7}{48} \)

The denominators are 12, 18, 30, and 48.

### Step 2: Find the LCM of the Denominators
To find the LCM, we can use the prime factorization method.

- **Prime Factorization**:
- \( 12 = 2^2 \times 3^1 \)
- \( 18 = 2^1 \times 3^2 \)
- \( 30 = 2^1 \times 3^1 \times 5^1 \)
- \( 48 = 2^4 \times 3^1 \)

- **LCM Calculation**:
- Take the highest power of each prime:
- For \( 2 \): highest power is \( 2^4 \) (from 48)
- For \( 3 \): highest power is \( 3^2 \) (from 18)
- For \( 5 \): highest power is \( 5^1 \) (from 30)

Thus, the LCM is:
\[ LCM = 2^4 \times 3^2 \times 5^1 \]

### Step 3: Calculate the LCM
Now we calculate:
\[ 2^4 = 16 \]
\[ 3^2 = 9 \]
\[ 5^1 = 5 \]

Now, multiply these together:
\[ LCM = 16 \times 9 \times 5 \]

Calculating step-by-step:
1. \( 16 \times 9 = 144 \)
2. \( 144 \times 5 = 720 \)

Thus, the LCM of 12, 18, 30, and 48 is 720.

### Step 4: Conclusion
The minimum number of apples Tanya had is 720.

### Final Answer:
Tanya had a minimum of **720 apples**.

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