A person distributes his pens among four friends A, B, C, D in the ratio `1/3 : 1/4 : 1/5 : 1/6`. What is the minimum number of pens that the person should have?

Options

A. 57
B. 65
C. 75
D. 45

Answer

To solve the problem of distributing pens among four friends A, B, C, and D in the ratio \( \frac{1}{3} : \frac{1}{4} : \frac{1}{5} : \frac{1}{6} \), we will follow these steps:

### Step 1: Convert the Ratios to a Common Format
The given ratios are \( \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6} \). To compare these ratios easily, we can find a common denominator.

### Step 2: Find the Least Common Multiple (LCM)
The denominators are 3, 4, 5, and 6. We need to find the LCM of these numbers.

- The prime factorization of each number is:
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 6 = 2^1 \times 3^1 \)

- The LCM is found by taking the highest power of each prime:
- \( 2^2 \) from 4
- \( 3^1 \) from 3 or 6
- \( 5^1 \) from 5

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 3: Convert Ratios to Whole Numbers
Now we can express the ratios in terms of a common multiple:
- For A: \( \frac{1}{3} \) of 60 = 20
- For B: \( \frac{1}{4} \) of 60 = 15
- For C: \( \frac{1}{5} \) of 60 = 12
- For D: \( \frac{1}{6} \) of 60 = 10

So, the distribution of pens is:
- A = 20 pens
- B = 15 pens
- C = 12 pens
- D = 10 pens

### Step 4: Calculate Total Pens
Now, we add the number of pens each friend receives:
\[ \text{Total pens} = 20 + 15 + 12 + 10 = 57 \]

### Step 5: Determine the Minimum Number of Pens
The total number of pens distributed is 57. Since we have expressed the distribution in terms of a common multiple (60), the minimum number of pens the person should have is:
\[ \text{Minimum number of pens} = 57 \]

### Final Answer
Thus, the minimum number of pens that the person should have is **57**.

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