Consider the given question and decide which of the following statements is sufficient to answer the question. How long will X take to build the wall? Statements: 1. X and Y together can build the wall in 8 days. 2. X takes 5 days more than Y to build the wall.
Both 1 and 2 are sufficient to answer the question.
Solving Time and Work Data Sufficiency Question
This is a Data Sufficiency question where we need to determine if the given statements, individually or together, provide enough information to answer the main question: How long will X take to build the wall?
Understanding the Question
The question asks for the time taken by person X alone to build a wall. Let's denote the time taken by X as $T_X$ and the time taken by Y as $T_Y$. We need to find the value of $T_X$.
Analyzing Statement 1: X and Y Together
Statement 1 says: "X and Y together can build the wall in 8 days."
In terms of work rate, if the total work is 1 unit (building the wall), the work done by X in one day is $1/T_X$, and the work done by Y in one day is $1/T_Y$. When they work together, their rates add up.
The combined rate of X and Y is $1/8$ work per day. So, we can write the equation:
\(\frac{1}{T_X} + \frac{1}{T_Y} = \frac{1}{8}\)
This equation has two unknown variables, $T_X$ and $T_Y$. We cannot find a unique value for $T_X$ using this equation alone. For example, if $T_X = 12$, then $1/12 + 1/T_Y = 1/8$, which gives $1/T_Y = 1/8 - 1/12 = (3-2)/24 = 1/24$, so $T_Y = 24$. If $T_X = 10$, then $1/10 + 1/T_Y = 1/8$, which gives $1/T_Y = 1/8 - 1/10 = (5-4)/40 = 1/40$, so $T_Y = 40$. Since $T_X$ can be different values, Statement 1 alone is not sufficient.
Analyzing Statement 2: X's Time vs Y's Time
Statement 2 says: "X takes 5 days more than Y to build the wall."
This statement gives a relationship between $T_X$ and $T_Y$:
\(T_X = T_Y + 5\)
This is a single equation with two unknown variables, $T_X$ and $T_Y$. We cannot find a unique value for $T_X$ using this equation alone. For any value of $T_Y$ (greater than 0), we can find a corresponding value for $T_X$. For example, if $T_Y = 10$, $T_X = 15$. If $T_Y = 20$, $T_X = 25$. Since $T_X$ can be different values, Statement 2 alone is not sufficient.
Combining Statements 1 and 2
Now let's consider both statements together. We have the following system of two equations with two unknowns:
- \(\frac{1}{T_X} + \frac{1}{T_Y} = \frac{1}{8}\)
- \(T_X = T_Y + 5\)
We can substitute the expression for $T_X$ from equation (2) into equation (1):
\(\frac{1}{T_Y + 5} + \frac{1}{T_Y} = \frac{1}{8}\)
We can solve this equation for $T_Y$. First, find a common denominator on the left side:
\(\frac{T_Y + (T_Y + 5)}{T_Y(T_Y + 5)} = \frac{1}{8}\)
\(\frac{2T_Y + 5}{T_Y^2 + 5T_Y} = \frac{1}{8}\)
Now, cross-multiply:
\(8(2T_Y + 5) = 1(T_Y^2 + 5T_Y)\)
\(16T_Y + 40 = T_Y^2 + 5T_Y\)
Rearrange the terms to form a quadratic equation:
\(T_Y^2 + 5T_Y - 16T_Y - 40 = 0\)
\(T_Y^2 - 11T_Y - 40 = 0\)
This is a standard quadratic equation in the form \(aT_Y^2 + bT_Y + c = 0\), where \(a=1\), \(b=-11\), and \(c=-40\). We can use the quadratic formula to find the value(s) of $T_Y$:
\(T_Y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
\(T_Y = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(1)(-40)}}{2(1)}\)
\(T_Y = \frac{11 \pm \sqrt{121 + 160}}{2}\)
\(T_Y = \frac{11 \pm \sqrt{281}}{2}\)
Since time must be a positive value, we consider the positive root:
\(T_Y = \frac{11 + \sqrt{281}}{2}\)
Once we have a unique value for $T_Y$, we can find $T_X$ using the relationship from Statement 2:
\(T_X = T_Y + 5\)
\(T_X = \left(\frac{11 + \sqrt{281}}{2}\right) + 5\)
\(T_X = \frac{11 + \sqrt{281} + 10}{2}\)
\(T_X = \frac{21 + \sqrt{281}}{2}\)
Since we were able to find a unique, positive value for $T_X$ by combining both statements, the statements together are sufficient to answer the question.
Conclusion on Sufficiency
Statement 1 alone is not sufficient.
Statement 2 alone is not sufficient.
Statements 1 and 2 together are sufficient.
Therefore, both statements are needed to answer the question.
| Statement(s) Used | Sufficient? | Reason |
|---|---|---|
| Statement 1 Alone | No | One equation with two unknowns ($T_X, T_Y$). Cannot find unique $T_X$. |
| Statement 2 Alone | No | One equation with two unknowns ($T_X, T_Y$). Cannot find unique $T_X$. |
| Statements 1 & 2 Together | Yes | Two independent equations with two unknowns ($T_X, T_Y$). Can solve for unique $T_X$. |
Revision Table: Key Concepts for Data Sufficiency
| Concept | Description | Application Here |
|---|---|---|
| Data Sufficiency | Determine if given data is sufficient to answer a question, not necessarily find the answer itself. | We checked if we could find a unique value for X's time. |
| Sufficiency Criteria | A statement/combination is sufficient if it leads to a unique answer to the question. | Statements 1 and 2 together led to a unique value for \(T_X\). |
| Work Rate | The amount of work done per unit of time (e.g., 1/Time taken for whole work). | Used \(1/T_X\), \(1/T_Y\), \(1/8\) as daily work rates. |
| Combined Work Rate | Sum of individual work rates: \(R_{total} = R_1 + R_2 + ...\). | Used \(\frac{1}{T_X} + \frac{1}{T_Y} = \frac{1}{8}\). |
Additional Information on Work and Time Problems
Work and Time problems often involve calculating the time taken by individuals or groups to complete a task, based on their work rates. The key idea is that if a person takes \(T\) days to complete a job, their rate of work is \(1/T\) job per day.
- If two people, A and B, can complete a job in \(T_A\) and \(T_B\) days respectively, their combined rate is \(\frac{1}{T_A} + \frac{1}{T_B}\) per day. The time taken by A and B together is \(T_{A+B}\), where \(\frac{1}{T_{A+B}} = \frac{1}{T_A} + \frac{1}{T_B}\).
- If a person's work rate is constant, the total work done is equal to the rate multiplied by the time taken (Work = Rate \(\times\) Time).
- When comparing the time taken by two people, like in Statement 2 ($T_X = T_Y + 5$), this translates directly into a relationship between their times.
In Data Sufficiency questions involving Work and Time, you set up equations based on the statements and check if you can solve for the variable asked in the main question. You don't always need to calculate the final numerical value, just confirm that it is uniquely determinable.
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Important Questions from Statements and Conclusions
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