GMAT Word Problem : Quadratic Equations

This question is a word problem. A problem solving question that tests your ability to frame an equation and solve it to get the answer.

Question

3 women and a few men participated in a chess tournament. Each player played two matches with each of the other players. If the number of matches that men played among themselves is 78 more than those they played with the women, how many more men than women participated in the tournament?
A. 11
B. 13
C. 14
D. 10
E. 8

Correct Answer : 10. Choice D

Explanatory Answer

Let the number of men who participated in the tournament be 'n'

Each player played two matches against all the other players.

So, the n men would have played 2 * 3 * n = 6n matches against the women.

Each of the n men would have played two matches among themselves.
Each man plays with each of the other (n - 1) players.

So, the number of matches among the men = n(n -1)

We know that the men have played 78 more matches among themselves than against the women.
i.e., n(n -1) = 78 + 6n
or n² - n - 6n - 78 = 0
or n² - 7n - 78 = 0
Factorizing, we get (n - 13)(n + 6) = 0
or n = 13 or n = -6.

n cannot be negative. Hence, n = 13.
Number of men = 13. Number of women = 3.

So, there are 10 more men than the number of women in the tournament.

DS Number Properties - Divisibility - Prime Divisors

This is an interesting Data Sufficiency question that tests your understanding of divisibility, indices and prime factors.

Question
If n is an integer, is n³ divisible by 54?
 1. n² is divisible by 6.
 2. n³ is divisible by 36.

Correct Answer : Choice D. Each statement is independently sufficient to answer the question.

Explanatory Answer

For "is" questions in DS, we need to answer with a clear Yes or a clear No. If the data in the statements does not lead to arriving at a definite Yes or No, the data is insufficient.

We know from the question stem that n is an integer.

I. Let us look at statement 1 alone : n² is divisible by 6.
If n² is divisible by 6, then n² is divisible by both 2 and 3 - the prime factors of 6.
But, we know that n is an integer.
Therefore, n² will be of the form p₁^a * p₂^b, where p₁ and p₂ are prime factors of n and a and b are even.
Hence, we can deduce that when  n² is expressed in terms of its prime factors, the power of 2 and 3 in it will be even.
So,  n² will be divisible by both 2² and 3². Hence, n is divisible by 2 and 3.

If n is divisible by 2 and 3, then n³ will be divisible by 2³ and 3³ or by 216.
If n³ is divisible by 216, it will certainly be divisible by any factor of 216 - and therefore by 54.

Statement 1 alone is sufficient.
Answer is either choice A or choice D.

II. Let us look at statement 2 alone :  n³ is divisible by 36.
For integer n, when n³ is expressed in terms of its prime factors, the powers of the prime factors will be multiples of 3.
So, if n³ is divisible by 36 or 2² * 3², we can deduce that n³ is actually divisible by 2³ and 3³ as the power of 2 and 3 should be a multiple of 3.
If n³ is divisible by 2³ and 3³, it is divisible by 2³ and 3³ or by 216.

Statement 2 alone is sufficient.

Because each of these statements is independently sufficient, Choice D is the answer.

GMAT DS - Inequalities, Descriptive Statistics

This data sufficiency question tests your understanding of Inequalities and descriptive statistics.

Question

Is 'b' the median of 3 numbers a, b, and c?
1. b/a = c/b
2. ab < 0

The correct answer is C. Both the statements together are sufficient to answer the question.

Explanatory answer
When the numbers a, b, and c are arranged in an ascending order, the middle number is the median. We need to determine if 'b' is the median of these 3 numbers.

Statement 1:
b/a = c/b
i.e., b^2 = ac
So, we can conclude that a, b and c are in a geometric progression with 'b' as their geometric mean.

For 3 positive numbers a, b and c that are in a geometric progression, b will be the geometric mean and the median.

However, we do not know if all 3 numbers a, b and c are positive
Hence, we cannot determine if 'b' is the median of these 3 numbers.
Hence, Statement 1 alone is NOT sufficient.

Statement 2:
ab < 0

The product of two numbers is negative if one of the numbers is negative and the other is positive. So, from this statement we can conclude that one of a or b is negative.
However, that is not sufficient to determine whether b is the median of the 3 numbers.

For instance, a = -4, b = 5 and c = 10, then b will be the median.
Conversely, a = -4 , b = 5 and c = -15, then a will be the median.

Hence, statement 2 alone is NOT sufficient.

Combining the two statements

We know from statement 1 that b is the geometric mean of a, b and c.
We know from statement 2 that one of a or b is negative.
Therefore, we can conclude that not all three numbers - a, b and c are positive.

If one or more of the 3 numbers happen to be negative numbers, then b will not be the median of these numbers.

We can therefore, answer conclusively using the two statements that 'b' is not the median.

Hence, combining the information given in the two statements is SUFFICIENT to answer the question.

Choice C is the answer.

GMAT Inequalities - Data Sufficiency

Inequalities and Data Sufficiency are favorite combinations in the GMAT quant section.

Here is a relatively easy question

Is 'a' positive?
1. a - b > 0
2. 2a - b > 0

Correct Answer is Choice E. The data is insufficient.

Explanatory Answer

Let us evaluate statement 1 alone.  a - b > 0.
From this statement we can conclude that a > b. But we cannot gain any insight about whether a is positive.

Here are two possible scenarios where the statement is true without helping us arrive at any conclusion.

Both a and b could be negative and a could be greater than b. For instance, a = -5 and b = -10. a - b > 0. 'a' is negative.

Alternatively a could be positive. For instance, a = 10 and b could be 3. a - b > 0 and 'a' is positive.

So, statement 1 alone is NOT SUFFICIENT.

Now, let us evaluate statement 2 alone. 2a - b > 0.
From this statement we can conclude that 2a > b. However, we cannot gain any insight about whether a is positive.

Let us check out the following two scenarios.
1. Let a = -3, b = -100, 2a = -6. 2a > b and a is negative.
2. Let a = 10, b = 12 and therefore, 2a = 20. 2a > b and a is positive.

So, statement 2 alone is NOT SUFFICIENT.

Combining the two statements, we know a - b > 0 and 2a - b > 0.
Let us look at the following two scenarios.
1. a = -3, b = -100 and 2a = -6. a > b and 2a > b. a is negative.
2. a = 20, b = 15 and 2a = 40. a > b and 2a > b. However, a is positive.

Even after combining the data in the two statements, we cannot conclude whether a is positive.

Hence, Choice E is the correct answer.

You can find additional GMAT Inequalities Practice questions here.

Coordinate Geometry DS - Lines and Circles

A data sufficiency question in coordinate geometry

Does the line x + y = 6 intersect or touch circle C with radius 5 units?

1. Center of the circle lies in the third quadrant.
2. Point (-4, -4) does not lie inside the circle.

Correct Answer is Choice E. The data is INSUFFICIENT.

Explanatory Answer

The line will intersect or touch the circle if the distance between the center of the circle and any point on the line is less than or equal to the radius of the circle.

Let us a pick a point on the line - say A(3, 3).

Statement 1: The center of the circle lies in the third quadrant.

The center could be at O₁(-0.5, -0.5) or could be at O₂(-10, -10).
Case 1: If the center is at O₁(-0.5, -0.5), then the distance between O₁A is less than 5 units, the radius of the circle. So, the line will intersect with the circle.

Case 2: On the other hand if the center is at O₂(-10, -10), then the distance between O₂A will be greater than 5 units, the radius of the circle. So, the line will neither touch nor intersect with the circle.

Hence, from statement 1 we cannot answer the question. Data INSUFFICIENT.

Statement 2: Point (-4, -4) does not lie inside the circle.

The distance between the center of the circle and (-4, -4) is more than 5 units.

Case 1: The center of the circle could be at (0, 0) and point (-4, -4) will lie outside the circle. However, the distance between the center and point A(3, 3) is less than 5 units. Hence, the line will intersect with the circle.

Case 2: Conversely, the center of the circle could be at (-10, -10). Point (-4, -4) will still lie outside the circle and the distance between point A(3,3 ) and the center (-10, -10) will be more than 5 units. Hence, the line will neither touch nor intersect with the circle.

Hence, from statement 2 we cannot answer the question. Data INSUFFICIENT.

Combining the two statements -  Center of the circle lies in the third quadrant and Point (-4, -4) does not lie inside the circle.

Consider the two following options for the center of the circle.

The center could be at O₁(-0.1, -0.1) or could be at O₂(-10, -10).
Case 1: The distance between  O₁(-0.1, -0.1) and (-4, -4) is more than 5 units and that between O₁(-0.1, -0.1) and (3, 3) is less than 5 units. Hence, all conditions stated in the two statements are satisfied and the line intersects with the circle.
Case 2: The distance between  O₂(-10, -10) and (-4, -4) is more than 5 units and that between O₂(-10, -10) and (3, 3) is also more than 5 units. Hence, all conditions stated in the two statements are satisfied - but the line does not intersect or touch the circle.

Therefore, using the statements independently or together we will not be able to answer the question.

Choice E is the answer.

DS Inequalities - Modulus

Here is a data sufficiency question that combines the concepts of modulus, indices and inequalities.

Is |a| > a?
1. a² < a
2. (a/2) > (2/a)

The magnitude of 'a' will be greater than a only if 'a' is a negative number. For positive numbers the magnitude of 'a' will be equal to 'a'.

So, what we have to determine using the two statements is whether a is negative.

Statement 1: a² < a

This inequality holds good only when 0 < a < 1.  i.e., we can conclude that a is positive. 
Hence, we can answer the question - whether a is negative with a definite NO.

Therefore, statement 1 is SUFFICIENT.

Statement 2: (a/2) > (2/a)

a is a variable and can therefore, take both positive and negative values.

We can derive two options from the information given in statement 2

Option 1:  a² > 4 only if a > 0
i.e.,  a² - 4 > 0 and a > 0
or (a + 2) (a - 2) > 0 and a > 0
or a > 2 or a < -2 and a > 0

Given that option 1 holds good only if a > 0, a > 2 and a cannot be less than -2. 
Option 1 therefore, points to the result that a is positive.

Option 2:  a² < 4 if a < 0
i.e.,  a² - 4 < 0 and a < 0
or (a + 2) (a - 2) < 0 and a < 0
or -2 < a < 2 and a < 0

Given that option 2 holds good only if a < 0, the range of values that a can take narrows down to -2 < a < 0.
Option 2 therefore, points to the result that a is negative.

Statement 2 leaves us with both the possibilities - 'a' could be positive or 'a' could be negative. 

Therefore, statement 2 is NOT SUFFICIENT.

Choice A is the correct answer.

Quadratic Equation PS - Sum of roots

This question is an interesting quadratic equation question that ties in concepts relating to factors in number properties.

While solving questions of this kind exercise caution with what the question asks. You may do everything right and still end up with an incorrect answer.

Question

x² + bx + 72 = 0 has two distinct integer roots; how many values are possible for 'b'?
A. 3
B. 12
C. 6
D. 24
E. 8

Correct Answer : 12 values. Correct Choice : B

Explanatory Answer

In any quadratic equation of the form ax² + bx + c = 0, (-b/a) represents the value of the sum of the roots and c/a represents the value of the product of the roots.

In the equation given in the question, the product of roots = 72/1 = 72.

We have been asked to find the number of values that 'b' can take.

If we determine all possible combinations for the roots of the quadratic equation, we can find out the number of values that 'b' can take.

The question states that the roots are integers.
If the roots are r₁ and r₂, then r₁ * r₂ = 72, where both r₁ and r₂ are integers.

Combinations of integers whose product is 72 are : (1, 72), (2, 36), (3, 24), (4, 18), (6, 12) and (8, 9) where both r₁ and r₂ are positive. 6 combinations.

For each of these combinations, both r₁ and r₂ could be negative and their product will still be 72.

i.e., r₁ and r₂ can take the following values too : (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12) and (-8, -9). 6 combinations.

Therefore, 12 combinations are possible where the product of r₁ and r₂ is 72.

Hence, 'b' will take 12 possible values.

Alternative Approach

If a positive integer 'n' has 'x' integral factors, then it can be expressed as a product of two number is x/2 ways.

So, as a first step let us find the number of factors for 72.

Step 1: Express 72 as a product of its prime factors. 2³ * 3²

Step 2: Number of factors = (3 + 1)*(2 + 1) = 12 (Increment the powers of each of the prime factors by 1 and multiply the result)

i.e., 72 has a 12 positive integral factors.

Hence, it can be expressed as a product of two positive integers in 6 ways. For each such combination, we can have a combination in which both the factors could be negative. Therefore, 6 more combinations - taking it to total of 12 combinations.