Math Revolution DS Expert - Ask Me Anything about GMAT DS

[GMAT math practice question]

(geometry) Is triangle ABC a right triangle?

1) The perimeter of triangle ABC is \(12\)

2) The length of the shortest side of triangle ABC is \(3\)

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

If a line is tangent to a circle at a point on the circle, the line is uniquely determined. Thus, condition 1) is sufficient.

Since \((5,5)\) is outside the circle, there are two tangents to the circle passing through this point.

Thus, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since questions about triangles require 3 variables, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Let \(x, y\) and \(z\) be the side-lengths of triangle ABC, and suppose \(x ≤ y ≤ z.\)

Then \(x = 3\) and \(x + y + z = 12.\)

If \(y = 4\) and \(z = 5\), triangle ABC has side lengths \(3, 4\) and \(5\). Thus it is a right triangle, and the answer is ‘yes’.

If \(y = 4.5\) and \(z = 4.5\), triangle ABC has side lengths \(3, 4.5\) and \(4.5.\) It is not a right triangle, and the answer is ‘no’.

Since the two conditions don’t yield a unique answer when applied together, they are not sufficient,

Therefore, E is the answer.
Answer: E

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

[GMAT math practice question]

(number properties) If \(n\) is a positive integer, is \(n^2+2n+44\) divisible by \(4\)?

1) \(n\) is an even integer.

2) \(n^2\) is divisible by \(144\)

[GMAT math practice question]

(number properties) If \(\frac{k}{mn}\), where \(k, m\) and \(n\) are positive integers, is a fraction in its lowest terms, is \(\frac{k}{mn}\) a terminating decimal?

1) \(\frac{1}{m}\) is a terminating decimal

2) \(\frac{1}{n}\) is a terminating decimal

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Asking if “\(n^2+2n+44\) is divisible by \(4\)” is equivalent to asking if \(n\) is an even integer, since \(44\) is a multiple of \(4\) and \(n^2+2n = n(n+2)\) is a multiple of \(4\) when \(n\) is an even number.
Thus, condition 1) is sufficient.

Condition 2)
Since \(n^2\) is divisible by \(144\), \(n\) is divisible by \(12\) and \(n\) is an even number.
Thus, condition 2) is also sufficient.

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).

[GMAT math practice question]

(number properties) What is the value of \(n\)?

1) \(n\) is the product of \(2\) different prime numbers less than \(15\)

2) \(n\) and \(210\) are relatively prime

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

In order for \(\frac{k}{mn}\) to a terminating decimal, \(mn\) must have no prime factors other than \(2\) and \(5\). This implies that neither \(m\) nor \(n\) have prime factors other than \(2\) and \(5\). Thus, we need both conditions 1) & 2) together.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(absolute value) What is the value of \(x\)?

\(1) |x+y|=|x-y|\)

\(2) |y| > y\)

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have \(1\) variable (\(n\)) and \(0\) equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Both \(n = 6\) and \(n = 10\) are products of two different prime numbers less than \(15\). Thus, condition 1) is not sufficient since it does not yield a unique solution.

Condition 2)
Both \(n = 11\) and \(n = 13\) are relatively prime to \(210\). Thus, condition 2) is not sufficient since it doesn’t yield a unique solution.

Conditions 1) & 2)
The prime numbers less than \(15\) are \(2, 3, 5, 7, 11\) and \(13\).

Since \(n\) and \(210 = 2*3*5*7\) are relatively prime, we must have \(n = 11*13 = 143.\)

Both conditions 1) & 2) together are sufficient since they yield a unique solution.

Therefore, C is the answer.
Answer: C

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

[GMAT math practice question]

(number properties) \(n\) is a positive odd integer. What is the value of \(n\)?

1) \(n, n + 2, n + 4\) are prime numbers.

2) \(n - 1\) is an even prime number.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.


Since we have \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)

\(|x+y|=|x-y|\)

\(=> |x+y|^2=|x-y|^2\)

\(=> (x+y)^2=(x-y)^2\)

\(=> x^2+2xy+y^2=x^2-2xy+y^2\)

\(=> 4xy = 0\)

\(=> xy = 0\)

\(=> x = 0\)or \(y = 0\)

Condition 1) is equivalent to \(x = 0\) or \(y = 0\). Condition 1) is not sufficient.

Condition 2) is equivalent to \(y < 0\). It follows from condition 2) that \(x = 0\) since \(y\) can’t be zero.

Both conditions 1) & 2) together are sufficient since they yield a unique solution.

Since this question is an absolute value question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

However, as we have seen above, neither condition 1) nor condition 2) is sufficient on its own.

Therefore, C is the answer.
Answer: C

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Condition 1):
The triple \(3, 5\) and \(7\) is the unique triple made up of three consecutive odd prime numbers number. So, \(n = 3\).

Thus, condition 1) is sufficient.

Condition 2):
Since \(n – 1 = 2\) is the unique even prime number, we must have \(n = 3\). Thus, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(absolute value) Is \(x > 0\)?

\(1) |x| + |y| > |x + y|\)

\(2) |y| > y\)

[GMAT math practice question]

(number properties) If \(m\) and \(n\) are positive integers, is \(\frac{m}{n}\) a terminating decimal?

1) \(m\) is divisible by \(9\)

2) \(n\) is divisible by \(30\)

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Since we have \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2):

Condition 1) is equivalent to the requirement that \(xy < 0\) and condition 2) is equivalent to the requirement that \(y < 0.\) Thus, both conditions together are sufficient.

Since this question is an inequality question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If \(x = 1, y = -1\), then \(x > 0\) and the answer is ‘yes’.
If \(x = -1, y = 1,\) then \(x < 0\) and the answer is ‘no’.
Condition 1) is not sufficient, since it doesn’t yield a unique solution.

Condition 2)
Since condition 2) doesn’t provide any information about \(x\), it is not sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

[GMAT math practice question]

(number properties) If \(m\) and \(n\) are non-negative integers, what is the value of \(mn\)?

\(1) 3^m=5^n\)

\(2) |m|=- √n\)