# Sample Problem Solving Questions

**Question 1.**

Of the 52 students in a class, 22 studied Physics for at least two years, 31 were male, and 14 studied Physics for less than two years and were female. How many students in the class studied Physics for at least 2 years and were male?

(A) 9

(B) 11

(C) 15

(D) 17

(E) 21

**Explanation:**

The problem categorizes the students by two distinct categories: whether the students were male or female and whether the students studied Physics for at least two years. Draw a table to summarize the given information.

Studied Physics for at least 2 years | Studied Physics for less than 2 years | Total | |
---|---|---|---|

Male | 31 | ||

Female | 14 | ||

Total | 22 | 52 |

According to the given information of the 52 students 22 had studied physics for at least two years, so the remaining 30 students studied Physics for less than 2 years. Of those who studied Physics for less than 2 years, 14 were female, so 30 − 14 = 16 were male students. Out of total 31 male students, 31 − 16 = 15 studied Physics for at least two years. The results are shown in the following table.

Studied Physics for at least 2 years | Studied Physics for less than 2 years | Total | |
---|---|---|---|

Male | 15 | 16 | 31 |

Female | 14 | ||

Total | 22 | 30 | 52 |

The correct answer is C.

**Question 2.**

Which of the following must be true if the average (arithmetic mean) of seven consecutive integers is an even number?

I. The sum of the integers is an even number.

II. The smallest integer is an even number.

III. The product of the largest and smallest number is an odd number.

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I and III only

**Explanation:**

This is one of those problems that can be easily solved by picking numbers.

Let us choose an easy set of numbers to use for this problem. We can pick any set of consecutive integer, and the only restriction is that the average of these integers should be even. For an odd number of consecutive integers, the average is simply the middle number of the set. The smallest set we can pick is 1,2,3,4,5,6,7. The average of this set (middle number in the set of consecutive integers) is 4, which is even, so we are good to go.

Statement I. The sum of the integers is 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28, which is even. Statement I must be true. We can now eliminate choices B and C as they do not contain statement I.

Statement II. The smallest of the integers, 1, is odd, which makes Statement II incorrect. Eliminate choice D.

Statement III. The product of the smallest and largest integer is 1 × 7 = 7, which is odd. This makes Statement III correct. Therefore, statements I and III must be true and the correct answer is E.

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