One question and three statements are numbered (I), (II) and (III). You have to decide whether the facts given in the statements are sufficient to answer the question given below. Question: What is the length of the rectangle? Statement: I. The perimeter of the rectangle is 80 meters. II. The sum of the adjacent sides is 40 meters. III. The width of the rectangle is 10 meters. Choose the correct option. A. Only I is sufficient. B. Only I and II are sufficient. C. III and one from I or II are sufficient. D. II and one from I or III are sufficient.
C
The problem asks whether the given statements are sufficient to determine the length of a rectangle. Let's denote the length of the rectangle as \(L\) and the width as \(W\).
Understanding Rectangle Properties
A rectangle has four sides, with opposite sides being equal in length. The adjacent sides meet at a right angle. The key properties mentioned in the statements are:
- Perimeter: The total distance around the outside of the rectangle. Formula: Perimeter \(P = 2 \times (L + W)\).
- Sum of adjacent sides: The sum of the length and the width. Formula: Sum \(= L + W\).
- Width: The shorter of the two adjacent sides (or either side if it's a square).
Analyzing Each Statement for Sufficiency
We need to check if each statement alone, or in combination with others, provides enough information to find a unique value for \(L\).
Statement I: The perimeter of the rectangle is 80 meters.
Given: \(P = 80\) meters.
Using the perimeter formula: \(80 = 2 \times (L + W)\)
Dividing by 2: \(40 = L + W\)
This statement gives us the sum of the length and the width. However, it does not give us the individual values of \(L\) or \(W\). For example, if \(L=30\), \(W\) must be 10, but if \(L=35\), \(W\) must be 5. There are multiple possible pairs of \(L\) and \(W\) that sum to 40. Therefore, Statement I alone is not sufficient to find the length \(L\).
Statement II: The sum of the adjacent sides is 40 meters.
Given: \(L + W = 40\) meters.
This statement is identical in meaning to what we derived from Statement I. It gives us the sum of the length and the width but not the individual values of \(L\) or \(W\). Similar to Statement I, this does not provide a unique value for \(L\). Therefore, Statement II alone is not sufficient to find the length \(L\).
Statement III: The width of the rectangle is 10 meters.
Given: \(W = 10\) meters.
This statement gives us the value of the width. However, it tells us nothing about the length. The length could be any value (e.g., 5 meters, 20 meters, 100 meters), and the width would still be 10 meters. Therefore, Statement III alone is not sufficient to find the length \(L\).
Analyzing Combinations of Statements
Since no single statement is sufficient, let's consider combining them.
Statements I and II together:
From I: \(L + W = 40\)
From II: \(L + W = 40\)
These two statements provide the same information. Combining them does not give us enough information to find unique values for \(L\) and \(W\). Therefore, Statements I and II together are not sufficient.
Statements I and III together:
From I: \(L + W = 40\)
From III: \(W = 10\)
We have a system of two equations with two variables:
1. \(L + W = 40\)
2. \(W = 10\)
Substitute the value of \(W\) from equation (2) into equation (1):
\(L + 10 = 40\)
Subtract 10 from both sides:
\(L = 40 - 10\)
\(L = 30\)
With Statements I and III, we can find a unique value for the length \(L\). Therefore, Statements I and III together are sufficient.
Statements II and III together:
From II: \(L + W = 40\)
From III: \(W = 10\)
This is the same system of equations as when combining I and III. Substitute \(W = 10\) into \(L + W = 40\):
\(L + 10 = 40\)
\(L = 30\)
With Statements II and III, we can find a unique value for the length \(L\). Therefore, Statements II and III together are sufficient.
Evaluating the Options
We found that Statement III combined with either Statement I or Statement II is sufficient to determine the length of the rectangle.
- Option A: Only I is sufficient. (False)
- Option B: Only I and II are sufficient. (False)
- Option C: III and one from I or II are sufficient. This means (III and I) OR (III and II) are sufficient. We showed that both (III and I) and (III and II) are sufficient. This option correctly describes our findings. (True)
- Option D: II and one from I or III are sufficient. This means (II and I) OR (II and III) are sufficient. We showed that (II and I) is not sufficient, but (II and III) is sufficient. Since the statement uses "and one from", it implies that II combined with either I or III must be sufficient. However, the phrasing "II and one from I or III are sufficient" might be interpreted as "is the combination of (II and I) sufficient, or is the combination of (II and III) sufficient?". In typical data sufficiency problems, the options specify the *minimum* set of statements. Option C states that using III *along with either* I or II works, which is what we found. Option D states that using II *along with either* I or III works. While II and III work, II and I do not. Option C is the more accurate and typical phrasing indicating that Statement III is essential, and either Statement I or II can be used with it.
Based on our analysis, the length can be determined if we have Statement III and either Statement I or Statement II.
| Combination | Sufficiency | Reason |
|---|---|---|
| I alone | Not Sufficient | \(L+W=40\), many possible \(L, W\) pairs. |
| II alone | Not Sufficient | \(L+W=40\), same as I. |
| III alone | Not Sufficient | \(W=10\), \(L\) can be anything. |
| I and II | Not Sufficient | Both give \(L+W=40\), same info. |
| I and III | Sufficient | \(L+W=40\) and \(W=10\) gives \(L=30\). |
| II and III | Sufficient | \(L+W=40\) and \(W=10\) gives \(L=30\). |
The correct condition for sufficiency is having Statement III along with either Statement I or Statement II.
Conclusion on Data Sufficiency
Statement III ($W=10$) provides a specific value for the width. Both Statement I ($L+W=40$) and Statement II ($L+W=40$) provide the sum of length and width. Having the sum ($L+W$) and one of the variables ($W$) allows us to solve for the other variable ($L$). Therefore, combining Statement III with either Statement I or Statement II gives us enough information.
Thus, Statement III and one from Statement I or II are sufficient to find the length of the rectangle.
Revision Table: Rectangle Data Sufficiency
| Statement(s) Used | Information Gained | Can Length be Found? |
|---|---|---|
| I | \(L+W = 40\) | No |
| II | \(L+W = 40\) | No |
| III | \(W=10\) | No |
| I & II | \(L+W = 40\) | No |
| I & III | \(L+W = 40\) AND \(W=10\) | Yes, \(L=30\) |
| II & III | \(L+W = 40\) AND \(W=10\) | Yes, \(L=30\) |
Additional Information: Data Sufficiency Concepts
Data sufficiency questions test your ability to determine whether the information given in the statements is enough to answer the question, without necessarily finding the answer itself. The general steps are:
- Read the question carefully and understand what needs to be determined.
- Analyze each statement individually to see if it alone is sufficient.
- If no single statement is sufficient, combine pairs of statements and check for sufficiency.
- If pairs are not sufficient, combine all statements (though often the answer is found before this step).
- Select the option that correctly describes the minimum set of statements required for sufficiency.
In this rectangle length problem, sufficiency is achieved when we have an equation relating L and W (like from I or II) and a specific value for one of the variables (like from III).
Important Questions from Data Sufficiency
Which of the following statement is correct?
I. The value of 1002 - 992 + 982 - 972 + 962 - 952 + 942 - 932 + ...... + 222 - 212 is 4840.
II. The value of
\(\left(k^2+ \frac{1}{k^2} \right) \left(k- \frac{1}{k} \right) \left(k^4+ \frac{1}{k^4} \right) \) \(\left(k+ \frac{1}{k} \right)\left(k^4- \frac{1}{k^4} \right)\) is \(k^{16}- \frac{1}{k^{16}}\).
Given below is a question followed by two statements I and II. Read both statements carefully to decide which one of them is sufficient to answer the question.
Six people - A, B, C, D, E and F are sitting in a straight line facing north. Who sits to the immediate left of D?
(I) A sits second from one of the extreme ends of the line. Only two people sit between A and B. E sits to the immediate right of C.
(II) C sits third from the left end of the line. Only one person sits between C and F. E sits third to the right of F.
Please read the below equation and select an appropriate option from the following.
x + y = 10 ; y = x - 2
Quantity A is x.
Quantity B is y.
A lady wants to buy the following items in the given price range-
1. Tomato between Rs. 40 and Rs. 45 per kg.
2. Grapes in the price range of Rs. 80 and Rs. 90 per kg.
3. Milk packets at Rs. 23 per liter.
In which shop from the following will she definitely get all her items/
A. Shop S sells tomato at Rs. 22.5 per half kg, Grapes at Rs. 82 per kg and milk at Rs. 24 per liter.
B. Shop H sells grapes at Rs. 21 per quarter kg, milk at 12.5 per half litre and tomato at Rs. 22 per half kg.
C. Shop O sells milk at Rs. 11.5 per half litre, tomato at 21 per half kg and grapes at Rs. 43 per half kg.
D. Shop P sells tomato at Rs. 23.5 per half kg, grapes at Rs. 85 per kg and milk at Rs. 23 per litre.You are given a question and two statements. Identify which of the statements is/are sufficient to answer the question.
Question:
When is Yuvi's wedding anniversary?
Statements:
1: On the 7 th day of a month.
2. The month has 29 days only in a leap year.
