A cake is shared among 5 friends. Four friends take `(1)/(8),(1)/(6),(5)/(12),(1)/(12)` part of the cake, respectively. How much part of the cake does the fifth friend get ?

Options

A. `(5)/(24)`
B. `(3)/(8)`
C. `(1)/(6)`
D. `(1)/(4)`

Answer

To solve the problem of how much part of the cake the fifth friend gets, we can follow these steps:

### Step 1: Identify the parts of the cake taken by the first four friends.
The parts taken by the four friends are:
- First friend: \( \frac{1}{8} \)
- Second friend: \( \frac{1}{6} \)
- Third friend: \( \frac{5}{12} \)
- Fourth friend: \( \frac{1}{12} \)

### Step 2: Find a common denominator for the fractions.
The denominators are 8, 6, 12, and 12. The least common multiple (LCM) of these numbers is 24.

### Step 3: Convert each fraction to have the common denominator of 24.
- For \( \frac{1}{8} \):
\[ \frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24} \]
- For \( \frac{1}{6} \):
\[ \frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24} \]
- For \( \frac{5}{12} \):
\[ \frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24} \]
- For \( \frac{1}{12} \):
\[ \frac{1}{12} = \frac{1 \times 2}{12 \times 2} = \frac{2}{24} \]

### Step 4: Add the fractions together.
Now we add the converted fractions:
\[ \frac{3}{24} + \frac{4}{24} + \frac{10}{24} + \frac{2}{24} = \frac{3 + 4 + 10 + 2}{24} = \frac{19}{24} \]

### Step 5: Subtract the total from 1 to find the part for the fifth friend.
The total part of the cake taken by the first four friends is \( \frac{19}{24} \). Therefore, the part of the cake that the fifth friend gets is:
\[ 1 - \frac{19}{24} = \frac{24}{24} - \frac{19}{24} = \frac{5}{24} \]

### Final Answer:
The fifth friend gets \( \frac{5}{24} \) of the cake.

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