This question involves the relationship between time taken to complete a work and the efficiency of the individuals doing the work. For a fixed amount of work, efficiency is inversely proportional to the time taken.

Understanding Time and Efficiency in Work Problems

When we talk about time and work, efficiency is defined as the amount of work done per unit of time. If a person takes less time to complete a certain amount of work, it means they are more efficient. Conversely, if they take more time, they are less efficient.

The relationship can be expressed as:

$$Efficiency \propto \frac{1}{Time}$$

This means that if the time taken doubles, the efficiency halves, and if the time taken halves, the efficiency doubles, assuming the total work remains constant.

Calculating the Efficiency Ratio from Time Ratio

We are given the ratio of time taken by A, B, and C to complete a work:

Time ratio of A : B : C = $T_A : T_B : T_C = 4 : 3 : 6$.

Since efficiency is inversely proportional to time, the ratio of their efficiencies will be the inverse ratio of their times.

Efficiency ratio of A : B : C = $E_A : E_B : E_C = \frac{1}{T_A} : \frac{1}{T_B} : \frac{1}{T_C}$

Substituting the given time ratio:

$$E_A : E_B : E_C = \frac{1}{4} : \frac{1}{3} : \frac{1}{6}$$

Converting Fractional Ratio to Integer Ratio

To express the ratio $\frac{1}{4} : \frac{1}{3} : \frac{1}{6}$ as a ratio of integers, we need to multiply each fraction by the least common multiple (LCM) of the denominators (4, 3, and 6).

• Denominators are 4, 3, and 6.
• Prime factorization: $4 = 2^2$, $3 = 3^1$, $6 = 2 \times 3$.
• LCM(4, 3, 6) = $2^2 \times 3 = 4 \times 3 = 12$.

Now, multiply each term in the ratio by 12:

• For A: $12 \times \frac{1}{4} = 3$
• For B: $12 \times \frac{1}{3} = 4$
• For C: $12 \times \frac{1}{6} = 2$

So, the efficiency ratio of A : B : C is $3 : 4 : 2$.

Summary of Steps

Here's a quick summary of the process:

1. Identify the given ratio of time taken: $T_A : T_B : T_C = 4 : 3 : 6$.
2. Understand the inverse relationship: Efficiency ratio is the reciprocal of the time ratio, i.e., $E_A : E_B : E_C = \frac{1}{T_A} : \frac{1}{T_B} : \frac{1}{T_C}$.
3. Write the fractional ratio: $E_A : E_B : E_C = \frac{1}{4} : \frac{1}{3} : \frac{1}{6}$.
4. Find the LCM of the denominators (4, 3, 6), which is 12.
5. Multiply each term of the fractional ratio by the LCM to get the integer ratio: $12 \times \frac{1}{4} : 12 \times \frac{1}{3} : 12 \times \frac{1}{6} = 3 : 4 : 2$.

The ratio of their efficiency is $3 : 4 : 2$. This corresponds to option 1.

Revision Table: Key Concepts in Time and Work

Additional Information: Applications of Efficiency and Time Ratios

Understanding the relationship between time and efficiency is crucial for solving various time and work problems. These concepts are frequently tested in competitive exams. Here are some related points:

• If the efficiency ratio of two people is $E_1 : E_2$, the ratio of time taken by them to complete the same work is $T_1 : T_2 = \frac{1}{E_1} : \frac{1}{E_2}$. This is the inverse relationship in action.
• If multiple people work together, their efficiencies are added up to find the combined efficiency.
• The total work can often be calculated by multiplying the efficiency of a person by the time they take ($Work = Efficiency \times Time$). Using the efficiency ratio and total time, one can find the time taken by individuals or groups to complete a task.
• Problems might involve comparing efficiencies, calculating time taken by different groups, or determining the share of wages based on the work done (which is often proportional to efficiency).