Work and Time Problem Analysis

This problem involves the concept of work and time, where the efficiency of individuals affects the time taken to complete a task. The key relationship is between work rate, time, and the total work done. We can assume the total task is one unit of work.

Let's define the terms:

• Work Rate: The amount of work done per unit of time (e.g., task per hour).
• Time: The duration taken to complete the work.
• Total Work: The entire task to be completed (which we normalize to 1 unit).

The formula connecting these is: $\text{Work Rate} \times \text{Time} = \text{Total Work}$. Since Total Work is 1, we have $\text{Work Rate} = 1 / \text{Time}$. Conversely, $\text{Time} = 1 / \text{Work Rate}$.

Relating Anu's and Binni's Work Efficiency

We are told that Anu is four times as good as Binni in completing a task. This means Anu's work rate is four times Binni's work rate.

Let $R_A$ be Anu's work rate and $R_B$ be Binni's work rate.

According to the question: $\text{Anu's Work Rate} = 4 \times \text{Binni's Work Rate}$, which is written as $R_A = 4 R_B$.

Let $T_A$ be the time Anu takes alone to complete the task and $T_B$ be the time Binni takes alone. Using the relationship $\text{Work Rate} = 1 / \text{Time}$, we have $R_A = 1/T_A$ and $R_B = 1/T_B$.

Substituting these into the rate relationship: $1/T_A = 4 (1/T_B)$. This simplifies to $T_B = 4 T_A$. This means Binni takes four times as long as Anu to complete the task alone, which makes sense because Anu is four times more efficient.

Calculating Combined Work Rate

When Anu and Binni work together, their individual work rates add up to form a combined work rate. Let $R_{Together}$ be their combined work rate.

$R_{Together} = R_A + R_B$

We are given that together they finish the same task in 7 hours. Using the formula $\text{Work Rate} = 1 / \text{Time}$, their combined work rate is $1/7$ (task per hour).

So, $R_{Together} = 1/7$.

Setting up the Equation to Find Anu's Time

We have the combined work rate equation: $R_A + R_B = 1/7$.

We also know $R_A = 1/T_A$ and $R_B = 1/T_B$. Substituting these into the equation:

$\frac{1}{T_A} + \frac{1}{T_B} = \frac{1}{7}$

From the efficiency relationship, we established that $T_B = 4 T_A$. Substitute this into the equation:

$\frac{1}{T_A} + \frac{1}{4T_A} = \frac{1}{7}$

Solving for Anu's Time Alone ($T_A$)

Now, we solve the equation for $T_A$:

Combine the terms on the left side by finding a common denominator, which is $4T_A$:

$\frac{4}{4T_A} + \frac{1}{4T_A} = \frac{1}{7}$

$\frac{4 + 1}{4T_A} = \frac{1}{7}$

$\frac{5}{4T_A} = \frac{1}{7}$

Now, cross-multiply:

$5 \times 7 = 1 \times 4T_A$

$35 = 4T_A$

To find $T_A$, divide both sides by 4:

$T_A = \frac{35}{4}$

So, Anu alone will complete the task in $\frac{35}{4}$ hours.

We found $T_A = \frac{35}{4}$. Let's verify this. If $T_A = 35/4$, then $R_A = 4/35$. $T_B = 4T_A = 4 \times (35/4) = 35$. So $R_B = 1/35$. Combined rate $R_A + R_B = 4/35 + 1/35 = 5/35 = 1/7$. Time taken together is $1 / (1/7) = 7$ hours. This matches the information given in the question.

Revision Table: Key Concepts in Work and Time

Additional Information: Work and Time Formulas

Here are some general formulas useful for solving work and time problems:

• If a person can do a piece of work in $T$ days, then in 1 day they do $1/T$ of the work.
• If a person does $1/T$ of the work in 1 day, they can complete the work in $T$ days.
• If A can do a work in $T_A$ days and B can do it in $T_B$ days, then working together, they can complete the work in $T_{together}$ days, where $\frac{1}{T_{together}} = \frac{1}{T_A} + \frac{1}{T_B}$. This can be simplified to $T_{together} = \frac{T_A \times T_B}{T_A + T_B}$.
• If there are more than two people, say A, B, and C, taking $T_A$, $T_B$, and $T_C$ days respectively, then $\frac{1}{T_{together}} = \frac{1}{T_A} + \frac{1}{T_B} + \frac{1}{T_C}$.