**Example 1:** If *p* is a prime number between 10 and 20, what is the value of *p*?

*p* > 12
*p* < 17

**Sol:** In this question, you are being asked to determine the value of p. This means the exact value that p represents. One and only one number is an acceptable answer to a question so phrased. What would you need to know to answer this? From the definition of a prime number, you can easily figure out that there are four prime numbers between 10 and 20: 11, 13, 17 and 19. Now, analyze the two statements and check whether either or both can give you the information you need. Statement A tells you that p is greater than 12, which does not answer the question uniquely. It gives three possible answers: 13, 17 and 19. Hence, statement A alone is not sufficient. Now, statement B by itself is also not sufficient as- if p is less than 17, there are two possible answers: 11 and 13. Finally, if you combine both the statements, you know that p is greater than 12 and that p is less than 17; only one possible number that fits in these conditions is 13. Hence, the question can be solved with the help of both statements together.

**Example 2:** Three packages have a combined weight of 96 kg. What is the weight of the heaviest?

A. One package weighs 24 kg.

B. One package weighs 48 kg.

**Sol:** Statement A is not sufficient to determine the weight of the heaviest package. It implies only that the combined weight of the other two packages is 72 kg. Statement B alone is sufficient for it implies that the combined weight of two of the packages is only 48 kg. Since the weight of the 48 kg packages is equal to the combined weight of the other two packages, the heaviest package must weigh 48 kg. So, we can see that statement B alone is sufficient to answer the question but statement A alone is not.

**Example 3:**** **Rahul ranks 30^{th} from the last in a class. What is his rank from the top?

- There are 48 students in the class.
- Rohit who ranks 13
^{th} in the same class, ranks 36^{th} from the last.

**Sol:** To solve this question, we need total number of students in the class.

From A, we conclude that in a class of 48 students, Rahul ranks 30^{th} from the last and hence 19^{th} from the top.From B, we conclude that there are 12 students above and 35 students below Rohit in rank. Thus, there are (12 + 1 + 35) = 48 students in the class.So, Rahul who ranks 30^{th} from the last, is 19^{th} from the top. So each statement alone is sufficient.

**Example 4:** Is *x* greater than 0?

*x*^{3} is less than 0.
- 3
*x* = -3.

**Sol**: Statement A establishes that *x*^{3} < 0, so *x* itself must be a negative number. Statement A alone, therefore, is sufficient to establish that the answer to the question is “no, *x* is not greater than 0”. Similarly, statement B alone also establishes that *x* is negative. Watch out! Some test-takers would, give reasoning (incorrectly) that since the answer to the question is “no”, the information is not sufficient. In fact, the information is sufficient to give a definite negative answer to the question.

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**Example 5:** At a clothing store, Amit spent Rs. 130. How many of the articles of clothing that Amit purchased were priced at Rs. 15?

- Amit purchased only articles costing Rs. 15 and Rs. 20.
- Amit purchased more than two Rs. 20 articles.

**Sol**: Statment A Alone is not sufficient as there are many possibilities.

Rs 20 articles |
Rs 15 articles |

5 |
2 |

2 |
6 |

Statement B alone is not sufficient because there are many possibilities. But if we combine both statements, an answer can be obtained. Since the articles of clothing are whole articles (no fraction allowed), Amit purchased exactly two Rs. 15 articles. So the correct answer is obtained by combining the two statements.

**Example 6:** If *x* and *y* are integers, what is the value of *x*^{2}?

*xy* = 7.
*y *< 4.

**Sol**: It's probably clear that neither statement alone gives enough information to answer the question. However, you might be tempted to answer that both statements combined are sufficient, reasoning as follows: " Statement A tells us that the product of *x* and *y* equals 7, and since 7 is a prime number, the only two numbers we can multiply together to get 7 are 1 and 7. Then statement B tells us that *y* is less than 4.

Combining these two facts, we conclude that *y* must equal 1, making *x* equal to 7."

Not so! The possibility overlooked here is that *x* and *y* might both be negative integers, specifically, –1 and –7. When multiplied together, these two integers have a product of +7, just as statement A states. And both have a value less than 4, in accordance with statement B. Therefore, even when we combine the two numbered statements, we cannot determine with certainty the value of *x*, which could be equal to 7, –1 or –7. Thus, the answer cannot be obtained even by combining both the statements.

**Example 7: **Is *xy* > 0?

*xy*^{2 }> 0
*x*^{2}*y*^{3 }< 0

**Sol**: Any quantity raised to the second power (or other positive even power) generates a positive result. But a quantity raised to the third power (or other positive odd power) may be positive or negative depending on the sign of the original quantity. Thus, statement A alone is not sufficient to establish the sign of the *xy*, but it does establish that *x* is positive. Statement B alone is not sufficient to establish the sign of *xy*, but it does establish that *y*^{3} and therefore *y* is negative. So, both statements taken together establish that *xy* < 0. Hence, the answer is obtained by combining both the statements.

**Example 8:** An item is discounted by 15 % from its usual selling price. What is the usual selling price?

- The value of the discount is Rs. 450.
- The discounted price is Rs. 2550.

**Sol**: Statement A alone is sufficient, since 15% of the usual selling price is Rs. 450, value of the discount:

0.15 x Usual Selling Price = Rs. 450. Usual Selling Price = Rs. 450 / 0.15 = Rs. 3000

Statement B alone is also sufficient, for the discounted price is equal to the usual selling price minus the discount; and the discount can be expressed as a percentage of the usual selling price.

Discount Price = Usual Selling Price – (0.15 Usual Selling Price) ⇒ Discount = 0.85 Usual Selling Price

⇒ 0.85 Usual Selling Price = Rs. 2550 ⇒ Usual Selling Price = Rs. 3000. So each statement alone is sufficient.