GMAT Descriptive Statistics : Averages

Question

If the average of 5 positive integers is 40 and the difference between the largest and the smallest of these 5 numbers is 10, what is the maximum value possible for the largest of these 5 integers?
A. 50
B. 52
C. 49
D. 48
E. 44

Correct Answer : Choice D. 48

Explanation

The average of 5 positive integers is 40. i.e., the sum of these integers = 5*40 = 200
 
Let the least of these 5 numbers be x.
Then the largest of these 5 numbers will be x + 10.
 
If we have to maximize the largest of these numbers, we have to minimize all the other numbers.
 
That is 4 of these numbers are all at the least value possible = x.
So, x + x + x + x + x + 10 = 200
Or x = 38.
So, the largest of these 5 integers is 48.

You could get additional questions on GMAT descriptive statistics on our question bank.

Or you could download and buy an eBook on Descriptive Statistics

Geometry Triangles Data Sufficiency

Here is a DS question on Geometry on basic properties of triangles.

Is triangle ABC with sides a, b and c acute angled?

1. Triangle with sides a^2, b^2, c^2 has an area of 140 sq cms.

2. Median AD to side BC is equal to altitude AE to side BC.

The correct answer is Choice A. Statement 1 alone is sufficient, while statement 2 is not.

Explanation

From the information given we have to determine if the given triangle is an acute angled triangle.


The information given to us is the measure of the sides of the triangle.


If a, b and c measure the sides of a triangle, and let us say 'a' is the longest side of the triangle, then
1. the triangle is acute angled if a^2 < b^2 + c^2
2. right angled if a^2 = b^2 + c^2 and
3. obtuse angled if a^2 > b^2 + c^2

Now let us evaluate the statements given to us

Statement 1: Triangle with sides a^2, b^2, c^2 has an area of 140 sq cms.
The statement provides us with one valuable information. We can form a triangle with sides a^2, b^2, c^2. 

For any triangle we know that sum of two sides is greater than the third side.
So,  a^2 < b^2 + c^2

That is the condition to be satisfied for a triangle with sides a, b and c to be an acute angled triangle.
Statement 1 answers in the positive and the information is sufficient.

Statement 2: Median AD to side BC is equal to altitude AE to side BC 
We can infer that the triangle is either equilateral or isosceles. An equilateral triangle is definitely an acute angled triangle. However, an isosceles triangle need not be an acute angled triangle.

So, statement 2 is not sufficient.

Statement 1 is sufficient, while statement 2 is not sufficient. Choice A is the correct answer.  

Mean, Median - GMAT Descriptive Statistics

Hi

Questions testing you understanding of the link betwen Mean and Median are frequently tested in the GMAT.

Here is a basic but interesting question on the combo.

Question

Positive integers from 1 to 45, inclusive are placed in 5 groups of 9 each. What is the highest possible average of the medians of these 5 groups?
A. 25
B. 31
C. 15
D. 26
E. 23

Correct Answer
Choice B. 31 is the highest possible average of the medians.

Explanatory Answer
We need to maximize the median in each group in order to maximize the average of all the medians.

The highest possible median is 41 as there should be 4 numbers higher than the median in the group of 9.

So, if we have a group that has a, b, c, d, 41, 42, 43, 44, 45, the median will be 41.
In this set, it is essential not to use any more high values on a, b, c, or d as these do not affect the median.
So, the median of a group that comprises 1, 2, 3, 4, 41, 42, 43, 44, 45 will be 41.

The next group can be 5, 6, 7, 8, 36, 37, 38, 39, 40. The median will be 36.

To maximize medians in all the 5 groups, the medians of the 5 groups will have to be 21, 26, 31, 36 and 41.

The highest possible average of the medians will be the average of these 5 numbers = 31. 

GMAT Geometry DS

Directions

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Question
Is triangle ABC obtuse angled?
I)  a^2 + b^2 > c^2
II) The center of the circle circumscribing the triangle does not lie inside the triangle

The correct answer is Choice E. Data is insufficient.

Explanatory Answer
In an obtuse angled triangle, if 'c' is the longest side, then c^2 > a^2 + b^2

Statement 1
a^2 + b^2 > c^2.
We have no information about whether c is the longest side in the triangle. Hence, we cannot answer the question. Statement 1 is INSUFFICIENT.

Statement 2
The center of the circle circumscribing the triangle does not lie inside the triangle.

> For an acute angled triangle, the center of the circle circumscribing the triangle lies inside the triangle.
> For a right triangle, the center of the circle circumscribing the triangle lies at the mid point of the hypotenuse.
> For an obtuse angled triangle, the center of the circle circumscribing the triangle lies outside the triangle.

From statement 2, we can deduce that the triangle is not an acute angled triangle. It may be a right angled triangle or an obtuse angled triangle. Hence, statement 2 is also INSUFFICIENT.

Combining the two statements, we cannot deduce anything more than what we could deduce using the information from the two statements independently.

Hence, Choice E is the correct answer.

You could get additional GMAT Data Sufficiency Practice questions here.

GMAT Coordinate Geometry

Here is a question in coordinate geometry. The primary idea tested by this question is one's understanding about equations of straight lines.

Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (8, 0) and the point (0, 12) is the lowest among all elements in set S. How many such points P exist in set S?
A. 1
B. 5
C. 11
D. 8
E. 3

The correct answer is Choice E. 3 points.

Explanatory Answer

The sum of the distances from point P to the other two points will be at its lowest only when point P lies on the line segment joining the points (8, 0) and (0, 12).

The equation of the line segment joining the points (8, 0) and (0, 12) is

Or the equation is 12x + 8y = 96 or 3x + 2y = 24.

We know the elements of set S contain points whose abscissa and ordinate are both natural numbers.

The equation of the line is 3x + 2y = 24 and hence, x will take even values while y will take values that are multiples of 3.

The values are x = 2, y = 9; x = 4, y = 6; x = 6, y = 3.

Hence, there are 3 such points that exist in set S.

Number Properties - Product of factors

What is the product of all positive factors of 18?
A. 324
B. 5832
C. 39
D. 1521
E. 3042

The correct answer is 5832. Choice B.

Let us first focus on how to find the answer methodically and then understand why the method works.

The How?

Step 1: Find the number of factors for 18.
Step 1a. Express 18 as a product of its prime factors. 18 = 2 * 3^2
Step 1b. Increment the powers of each of the prime factors. The power of 2 is 1. The power of 3 is 2. After incrementing, we get (1 + 1) and (2 + 1)
Step 1c. Number of factors = Product of the incremented numbers = (1 + 1)(2 + 1) = 2 * 3 = 6

Step 2: The product of all factors of 18 = 18^(number of factors/2)
The product = 18^(6/2) = 18^3 = 5832.

The Why?
The factors of 18 are 1, 2, 3, 6, 9 and 18.
Product of the factors = 1*2*3*6*9*18.
We can rewrite this as (1*18)*(2*9)*(3*6) = 18 * 18 * 18.

So, if a positive integer 'n' has 'x' factors, then the product of all its factors = n^(x/2).

What if the number has odd number of factors?
Try the method for a perfect square such as 36 or 16.

GMAT DS : Modulus

Is xy < 0?

(1) 5|x| + |y| = 0

(2) |x| + 5|y| = 0