A person distributes his pens among four friends $A, B, C$ and $D$ in the ratio $\frac{1}{3} : \frac{1}{4} : \frac{1}{5} : \frac{1}{6}$. What is the minimum number of pens that the person should have?

### Concept & Integer Ratios Ratios expressed as fractions must be converted to whole numbers when dealing with indivisible discrete items (like pens). The minimum total quantity is the sum of these simplified whole-number ratio units. ### Step-by-Step Solution Given: Ratio of distribution = $\frac{1}{3} : \frac{1}{4} : \frac{1}{5} : \frac{1}{6}$ 1. To convert these fractions into whole numbers, find the Least Common Multiple (LCM) of the denominators ($3, 4, 5, 6$). LCM(3, 4, 5, 6) = 60 2. Multiply each fraction in the ratio by the LCM: $A = \frac{1}{3} \times 60 = 20$ $B = \frac{1}{4} \times 60 = 15$ $C = \frac{1}{5} \times 60 = 12$ $D = \frac{1}{6} \times 60 = 10$ 3. The simplified integral ratio is $20 : 15 : 12 : 10$. 4. Because a pen cannot be broken into fractions, the actual number of pens given to each friend must be a multiple of these ratio units. 5. To find the *minimum* total number of pens, we assume the multiplier is 1. Minimum pens = $20 + 15 + 12 + 10 = 57$. ### Exam Strategy & Shortcut Whenever a ratio of discrete items is given in unit fractions, rapidly find the LCM of denominators, establish the integer ratio, and sum the parts. That sum represents the absolute minimum possible quantity. ### Common Pitfall Summing the fractions directly ($\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}$) instead of converting them into an integer ratio scale first. ### Final Answer Therefore, the correct answer is **57**.