Overview of GMAT Math Question Types and Patterns on the GMAT

[GMAT math practice question]

(Geometry) The figure shows □ABCD with AB = 8, BC = 17 and CD = 9 and ∠BAC = ∠ADC = 90. What is the area of □ABCD?

[/header2]

A. 81
B. 92
C. 103
D. 114
E. 125

=>

AC^2 = 17^2 – 8^2 = (17 + 8)(17 - 8) = 25·9 = 5^2·3^2 = 15^2.
Thus, we have AC = 15.
AD^2 = AC^2 - CD^2 = 15^2 - 9^2 = (15 + 9)(15 - 9) = 24·6 = 2^2·6^2 = 12^2.
Thus, we have AD = 12.

□ABCD = △ABC + △ACD = (1/2)·8·15 + (1/2)·12·9 = 60 + 54 = 114.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(Function) What is the value of the function f(x)?

1) f(2020x + f(0)) = 2020x^2, x is a real number.
2) f(x) is a polynomial function.

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since a function has many variables to determine, E is most likely 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) together give us:

Assume t = 2020x + f(0).
Then we have x = (t - f(0))/2020.
f(t) = 2020[(t - f(0))/2020]^2 = [t - f(0)]^2/2020.
When we replace t by 0, we have f(0) = (f(0))2/2020 or (f(0))^2 – 2020f(0) = 0
Then we have f(0)(f(0) - 2020) = 0.
Thus f(0) = 0 or f(0) = 2020.

Both conditions 1) and 2) together 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) x and y are integers. What is the value of x + y?

1) xy = 1008.
2) The greatest common divisor of x and y is 6.

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (x and y) and 0 equations, C is most likely 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) together give us two solutions of x = 6·1, y = 6·28, x + y = 174 and x = 6·4, y = 6·7, x + y = 66.

Since the greatest common divisor of x and y is 6, we can assume that x = 6a, and y = 6b where a and b are relatively prime.
x·y = 6a·6b = 1008 = 6·6·28.
Then we have ab = 28.
(1, 28) and (4, 7) are possible pairs for (a, b).
If a = 1 and b = 28, we have x = 6·1 = 6, y = 6·28 = 168 and x + y = 174.
If a = 4 and b = 7, we have x = 6·4 = 24, y = 6·7 = 42 and x + y = 66.

The answer is not unique, and conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, E is the answer.
Answer: E

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]

(Proportion) The figure shows points A, B, C, and D on the number line. The coordinate of point A is -5, and that of point B is 4. What is the coordinate of point C?



1) AC:CD = 1:2
2) CD:DB = 2:3

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Assume c and d are coordinates of points C and D, respectively.

Since we have 2 variables (c and d) and 0 equations, C is most likely 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) together give us:
Since AC:CD = 1:2, we have AC = c – (-5) = c + 5, CD = d – c and (c + 5) : (d – c) = 1:2, which is equivalent to (d – c) = 2(c + 5), d – c = 2c + 10, or d = 3c + 10.
Since CD:DB = 2:3, we have CD = d – c, DB = 4 – d and (d – c) : (4 – d) = 2:3, which is equivalent to 2(4 - d) = 3(d - c), 8 - 2d = 3d – 3c or 5d = 3c + 8.
Then we have 5d = 5(3c + 10) = 15c + 50 = 3c + 8 or 12c = -42.
Thus, we have c = -7/2.

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(Number Properties) f(n) denotes the number of positive factors of a positive integer n. What is f(f(480))?

A. 8
B. 14
C. 16
D. 20
E. 24

=>

Remember the formula for the number of factors of a positive integer N = p^a·q^b·r^c is (a + 1)(b + 1)(c + 1).
Since 480 has the prime factorization 480 = 32·3·5 = 2^5·3^1·5^1, 480 has (5 + 1)(1 + 1)(1 + 1) = 6·2·2 = 24 factors and we have f(480) = 24.
Since 24 has the prime factorization 24 = 8·3 = 2^3·3^1, 24 has (3 + 1)(1 + 1) = 4·2 = 8 factors and we have f(f(480)) = f(24) = 8.

Therefore, the answer is A.
Answer: A

[GMAT math practice question]

(Inequalities) m is an integer where m = x + y. What is the minimum value of m?

1) 5 < x < 8.
2) -7 < y < -4.

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 3 variables (x, y, and m) and 1 equation, C is most likely 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) together give us:
When we add the inequalities 5 < x < 8 and -7 < y < -4, we have -2 < x + y < 4 or -2 < m < 4.
Since m is an integer, we have -1 ≤ m ≤ 3.
Thus, the minimum value of m is -1.

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

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.

Let’s look at the condition 1). Since it doesn’t have any information regarding the variable y, the answer is not unique, and the condition is not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2). Since it doesn’t have any information regarding the variable x, the answer is not unique, and the condition is not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) together are sufficient.
Therefore, C is the correct 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) n(A) denotes the number of positive divisors of a positive integer A. What is the smallest possible value of x satisfying n(280)÷n(30)·n(x) = 12?

A. 12
B. 14
C. 16
D. 18
E. 20

=>

If n = p^a·q^b·r^c has a prime factorization with different prime numbers p, q, and r, then the number of factors of n is (a + 1)(b + 1)(c + 1).

Since 280 = 2^3·5^1·7^1, we have n(280) = (3 + 1)(1 + 1)(1 + 1) = 4·2·2 = 16.
Since 30 = 2^1·3^1·5^1, we have n(30) = (1 + 1)(1 + 1)(1 + 1) = 2·2·2 = 8.

Then we have n(280)n(x)/n(30) = 16n(x)/8 = 2n(x) = 12 or n(x) = 6.

We have two cases of integers with 6 factors. They are p^2q^1 or p^5 where p and q are different prime numbers since (2 + 1)(1 + 1) = 6 and (5 + 1) = 6.

For x = p^2q^1, when we have p = 2 and q = 3, we have the smallest number 2^2·3^1 = 12.
For x = p^5, when we have p = 2, we have the smallest number 2^5 = 32.
The smallest value of x is 12.

Therefore, A is the correct answer.
Answer: A

Solution:

f(4) = f(2·2) = f(2) + f(2) = 1 + 1 = 2.
f(8) = f(2·4) = f(2) + f(4) = 1 + 2 = 3.

Therefore, C is the correct answer.

Answer: C.

(Algebra) We have \(\frac{x}{2}\) = \(\frac{y}{3}\). What is the value of \(\frac{2x}{(x+y)}\) + \(\frac{3y}{(x-y)}\) + \(\frac{x^2}{(x^2-y^2 )}\)?


Solution:


When we assume \(\frac{x}{2}\) = \(\frac{y}{3}\) = k, we have x = 2k and y = 3k.

\(\frac{2x}{(x+y)}\) + \(\frac{3y}{(x-y)}\) + \(\frac{x^2}{(x^2-y^2 )}\)

= \(\frac{(2 * 2k)}{(2k + 3k)}\) + \(\frac{(3 * 3k)}{(2k - 3k)}\) + \(\frac{(2k)^2}{((2k)^2 - (3k)^2 )}\)

= \(\frac{4k}{5k}\) + \(\frac{9k}{(-k)}\) + \(\frac{4k^2}{(4k^2- 9k^2 ) }\)

= \(\frac{4}{5}\) – 9 - \(\frac{4}{5}\) = -9.

Therefore, C is the correct 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.
The original condition \(a>b>c>d>0\) is equivalent to \(0 < \frac{1}{a} < \frac{1}{b} < \frac{1}{c} < \frac{1}{d}\). The question asks if \(d < 4\). This is equivalent to asking if \(\frac{1}{d} > \frac{1}{4}\).

By condition \(1, \frac{1}{d} > \frac{1}{4}\) since \(\frac{1}{c} < \frac{1}{d}\). So, \(d < 4\). Condition 1) is sufficient.

Condition 2)
Since \(0 < \frac{1}{a} < \frac{1}{b} < \frac{1}{c} < \frac{1}{d}\) and \((\frac{1}{a})+(\frac{1}{b})+(\frac{1}{c})+(\frac{1}{d})=1\), we have \(\frac{1}{a}<\frac{1}{d}\), \(\frac{1}{b}<\frac{1}{d}, \frac{1}{c}<\frac{1}{d}\) and \(\frac{1}{a} +\frac{1}{b} + \frac{1}{c} + \frac{1}{d} < \frac{1}{d} +\frac{1}{d} +\frac{1}{d} + \frac{1}{d} = \frac{4}{d}.\)
Therefore, \(1 < \frac{4}{d}\) and \(d < 4\).
Condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

Note: This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is hard. For CMT4(B) questions, D is most likely to be the answer.

=>

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).

=>

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 \(3\) variables (\(x, y\) and \(z\)) and \(0\) equations, 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.

Conditions 1) & 2)
If \(x = 2, y = 3, z = 5\), then \(x\) and \(z\) are relatively prime, and the answer is ‘yes’.

If \(x = 2, y = 3, z = 2,\) then \(x\) and \(z\) are not relatively prime, and the answer is ‘no’.

Both conditions together are not sufficient, since they don’t yield a unique answer.

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.

=>

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)
Since the sum of \(2, 3, 4, 5\) and \(6\) is \(20\), which is an even number, and \(n=5\) is an odd number, the answer is sometimes ‘no’.
Since the sum of \(1, 2, 3\), and \(4\) is \(10\), which is an even number, and \(n=4\) is an even number, the answer is sometimes ‘yes’.
Condition 1) is not sufficient since it does not yield a unique answer.

Condition 2)
If \(n = 4\), then \([\frac{n}{2}] = 2\) is even, and \(n\) is even.

If \(n = 5\), then \([\frac{n}{2}] = 2\) is also even, but \(n\) is not even.
Condition 2) is not sufficient since it does not yield a unique answer.

Conditions 1) & 2)
Since the sum of \(2, 3, 4, 5\) and \(6\) is \(20\), which is an even number, and \(n=5\) is an odd number such that \([\frac{n}{2}] = 2\) is even, the answer is sometimes ‘no’.

Since the sum of \(1, 2, 3\), and \(4\) is \(10\), which is an even number, and \(n=4\) is an even number such that \([\frac{n}{2}] = 2\) is even, the answer is sometimes ‘yes’.

Conditions 1) & 2) together are not sufficient, since they do not yield a unique answer.

Therefore, E is the answer.
Answer: E

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.

=>

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 \(1\) 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)
\(n\) is a prime factor of \(21 = 3*7\) and \(n\) is \(3\) or \(7\).
Since it does not give a unique answer, condition 1) is not sufficient.

Condition 2)
If \(n\) is a factor of \(49 = 7^2\), then \(n\) is \(1, 7\) or \(49\).
Since it does not give a unique answer, condition 2) is not sufficient.

Conditions 1) & 2)
The unique integer satisfying both conditions is \(n = 7.\)
Both conditions are sufficient, when taken together.

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.

=>

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.

The expression from the question is equivalent to \(\frac{xy}{( x^2 + xy + y^2)}\) as shown below, which is the same as condition 1):
\(\frac{( x^2y – xy^2 )}{( x^3 – y^3 )}\)
\(= \frac{xy(x-y)}{(x-y)(x^2+xy+y^2)}\)
\(= \frac{xy}{( x^2 + xy + y^2 )}\)
Thus, condition 1) is sufficient.

Condition 2)
If \(x = 3, y = 1\), then \(\frac{( x^2y – xy^2 )}{( x^3 – y^3 )} = \frac{( 9 – 3 )}{( 27 – 1)} = \frac{6}{26} = \frac{3}{13}.\)
If \(x = -3, y = 1\), then \(\frac{( x^2y – xy^2 )}{( x^3 – y^3 )} = \frac{( 9 + 3 )}{( - 27 – 1)} = \frac{-12}{28} = \frac{-3}{7}.\)
Since it does not yield a unique solution, 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Since we have \(3\) variables (\(x, y\), and \(z\)) and \(0\) equations, E is most likely 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)

We have \(x + 2y + 3z = 4\) and \(2x + 3y + 4z = 5.\)

When we subtract twice the second equation from three times the first equation, we have
\(3(x + 2y + 3z) - 2(2x + 3y + 4z) = 3(4) - 2(5)\)

\(3x + 6y + 9z - 4x - 6y - 8z = 12 - 10\)

\(-x + z = 2\)

\(z = x + 2.\)

Substituting \(z = x + 2\) into the first equation gives us:

\(x + 2y + 3z = 4\)

\(x + 2y + 3(x + 2) = 4\)

\(x + 2y + 3x + 6 = 4\)

\(4x + 2y = -2\)

\(2y = -4x - 2\)

\(y = -2x – 1\)

\(x < 2y < 3z\) is equivalent to \(x < 2(-2x – 1) < 3(x + 2)\), or \(x < -4x – 2 < 3x + 6\).

Then we have \(x < \frac{-2}{5}\) from \(x < -4x – 2, \)because:

\(x < -4x – 2\)

\(5x < -2\)

\(x < \frac{-2}{5}.\)

We also have \(x > \frac{-8}{7}\) from \(-4x - 2 < 3x + 6\), because:

\(-4x - 2 < 3x + 6\)

\(-7x < 8\)

\(x > \frac{-8}{7}\) (the inequality sign changes direction since we divided by a negative)

Thus \(\frac{-8}{7} < x < \frac{-2}{5}\) and \(x = -1\) since \(x\) is an integer.

Since both conditions together yield a unique solution, they are sufficient.

Since this question is an integer 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)

\(x = -2, y = 1, z = \frac{4}{3}\) and \(x = -3, y = 0, z = 1\) are solutions.
Condition 2)

\(x = -2, y = 0, z = \frac{9}{4}\) and \(x = -1, y = 0, z = \frac{7}{4}\) are solutions.

Since condition 2) does not yield a unique solution, it is not sufficient.

Therefore, C is the answer.
Answer: C

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.

=>

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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.
Since we put \(x\) after \(y\), we have \(S = 100y + x.\)

And we have \(13 ≤ x ≤ 9\) and \(13 ≤ x ≤ y ≤ 99\) since \(x\) and \(y\) are two-digit numbers.

Since we have \(3\) variables (\(x, y\), and \(S\)) and \(1\) equation, C is most likely 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)
We can rearrange the equation \(S - (y - x) = 4289\) from condition 1) to get \(S = 4289 + (y - x).\) Substituting \(y - x = 27\) from condition 2) we get \(S = 4289 + 27 = 4316.\) Then \(y = 43\) and \(x = 16\).

Since both conditions together yield a unique solution, they are sufficient.

Since this question is an integer 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)
We have many possible pairs of \(x\) and \(y\) from \(y – x = 27.\)

Since condition 1) does not yield a unique solution, it is not sufficient

Condition 2)
Substituting \(S = 100y + x\), into \(S – (y-x)\) we get \(100y + x – (y - x) = 99y + 2x = 4289.\)

If we make \(y = 10a + b\) and \(x = 10 + c\) where \(1 ≤ a ≤ 9, 0 ≤ b ≤ 9\) and \(3 ≤ c ≤ 9.\)

Then we have \(S = 99y + 2x = 990a + 99b + 20 + 2c = 4289\) or \(990a + 99b + 2c = 4269.\)

When we substitute integers between \(1\) and \(9\) for the variable a one by one, we notice \(a = 4\) is the unique value in order to have units digits \(b\) and \(c\).

Then we have \(99b + 2c = 4269 – 3960 = 309.\)

When we substitute integers between \(1\) and \(9\) for the variable \(b\) one by one, we notice \(b = 3\) is the unique value in order to have a units digit \(c\).
Then we have \(c = 6.\)

Thus the boy’s age is \(16\) and his father’s age is \(43.\)

Since condition 2) yields a unique solution, it is sufficient.

Therefore, B is the answer.
Answer: B

This question is a CMT 4 (A) question: When we easily get C as an answer, consider A and B as an answer.
If the question has both C and B as its answer, then B is an answer rather than C by the definition of DS questions. Also this question is a 50/51 level question and can be solved by using the relationship between the Variable Approach and Common Mistake Type 3 and 4 (A or B).

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.