Understanding the Work and Time Problem

This problem involves concepts of work and time, specifically how different people with varying efficiencies complete a task together and alone. We are given information about the relative efficiency of two workers (A and B), the time one worker (C) takes alone, and the time all three take together. We need to find the time taken by worker A alone.

Key Concepts: Work Rate

The key concept here is the work rate. The work rate is the amount of work done per unit of time (in this case, per day). If a person takes $T$ days to complete a job, their work rate per day is $\frac{1}{T}$. The total work is considered as 1 unit (painting the whole house).

• Work Rate ($R$) $= \frac{1}{\text{Time taken}}$
• Total Work $= \text{Total Rate} \times \text{Total Time}$
• If multiple people work together, their rates add up: $R_{total} = R_1 + R_2 + R_3 + ...$

Translating the Given Information into Equations

Let:

• $T_A$ be the number of days A takes to paint the house alone.
• $T_B$ be the number of days B takes to paint the house alone.
• $T_C$ be the number of days C takes to paint the house alone.

Their respective daily work rates are $\frac{1}{T_A}$, $\frac{1}{T_B}$, and $\frac{1}{T_C}$.

From the question:

1. "A is thrice as good a workman as B": This means A's work rate is 3 times B's work rate.

   $\text{Rate of A} = 3 \times \text{Rate of B}$

   $\frac{1}{T_A} = 3 \times \frac{1}{T_B}$

   This implies $T_B = 3 \times T_A$. B takes three times as long as A because A is three times as efficient.

2. "C alone takes 48 days to paint a house":

   $T_C = 48$ days

   Rate of C ($R_C$) $= \frac{1}{48}$ per day.

3. "All three A, B and C working together take 16 days to paint the house":

   When A, B, and C work together, their rates add up. The combined rate is $\frac{1}{16}$ per day (since they complete the work in 16 days).

   $\text{Rate of A} + \text{Rate of B} + \text{Rate of C} = \frac{1}{16}$

   $\frac{1}{T_A} + \frac{1}{T_B} + \frac{1}{T_C} = \frac{1}{16}$

Solving for the Time A Takes Alone

Now we substitute the known values and relationships into the combined rate equation:

We know $T_B = 3T_A$ and $T_C = 48$. Substitute these into the equation:

$\frac{1}{T_A} + \frac{1}{3T_A} + \frac{1}{48} = \frac{1}{16}$

Combine the terms involving $T_A$ on the left side. The common denominator for $T_A$ and $3T_A$ is $3T_A$:

$\frac{3}{3T_A} + \frac{1}{3T_A} + \frac{1}{48} = \frac{1}{16}$

$\frac{3 + 1}{3T_A} + \frac{1}{48} = \frac{1}{16}$

$\frac{4}{3T_A} + \frac{1}{48} = \frac{1}{16}$

Now, isolate the term with $T_A$ by subtracting $\frac{1}{48}$ from both sides:

$\frac{4}{3T_A} = \frac{1}{16} - \frac{1}{48}$

To subtract the fractions on the right side, find a common denominator. The common denominator for 16 and 48 is 48 ($16 \times 3 = 48$).

$\frac{1}{16} = \frac{1 \times 3}{16 \times 3} = \frac{3}{48}$

So the equation becomes:

$\frac{4}{3T_A} = \frac{3}{48} - \frac{1}{48}$

$\frac{4}{3T_A} = \frac{3 - 1}{48}$

$\frac{4}{3T_A} = \frac{2}{48}$

Simplify the fraction on the right side ($\frac{2}{48} = \frac{1}{24}$):

$\frac{4}{3T_A} = \frac{1}{24}$

Cross-multiply to solve for $T_A$:

$4 \times 24 = 1 \times 3T_A$

$96 = 3T_A$

Divide by 3 to find $T_A$:

$T_A = \frac{96}{3}$

$T_A = 32$ days

So, it will take 32 days for A alone to paint the house.

Verification

Let's check if this answer is consistent with the problem statement:

• If $T_A = 32$ days, Rate of A ($R_A$) = $\frac{1}{32}$ per day.
• A is thrice as good as B, so $R_A = 3R_B$. $\frac{1}{32} = 3R_B \implies R_B = \frac{1}{96}$. Time for B ($T_B$) = 96 days.
• Rate of C ($R_C$) = $\frac{1}{48}$ per day (given).
• Combined rate: $R_A + R_B + R_C = \frac{1}{32} + \frac{1}{96} + \frac{1}{48}$.
• Find a common denominator (96): $\frac{3}{96} + \frac{1}{96} + \frac{2}{96} = \frac{3+1+2}{96} = \frac{6}{96} = \frac{1}{16}$.
• Combined time = $\frac{1}{\text{Combined Rate}} = \frac{1}{1/16} = 16$ days.

The calculated combined time (16 days) matches the given information, so our answer for the time A takes alone is correct.

Revision Table: Work and Time Concepts

Additional Information: Efficiency and Time

Efficiency and time taken to complete a task are inversely proportional. If a person is twice as efficient as another, they will take half the time to complete the same amount of work. In this problem, A is thrice as efficient as B, which is why A takes $\frac{1}{3}$ the time B takes ($T_A = \frac{1}{3} T_B$, or $T_B = 3 T_A$). Understanding this inverse relationship is crucial for solving work and time problems.

Problems involving work and time often require setting up equations based on the rates of individual workers and their combined rate. Converting the word problem statements into mathematical expressions is the key first step.