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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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[GMAT math practice question]

(number properties) a and b are positive integers. What is the remainder when b is divided by 4?

1) if a is divided by 4, the remainder is 3
2) if a^2+b is divided by 4, the remainder is 1

=>

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 (a and b) 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) tells us that a has remainder 3, when it is divided by 4. So, a^2 has remainder 1, when it is divided by 4.
Condition 2) tells us that a^2+b has remainder 1 when it is divided by 4. Since a^2 has remainder 1 when it is divided by 4, b has remainder 0 when it is divided by 4.
Thus, both conditions together 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)
Since there is no information about b in condition 1), it is not sufficient.

Condition 2)
If a = 1 and b = 4, then b has remainder 0 when it is divided by 4.
If a = 4 and b = 1, then b has remainder 1 when it is divided by 4.
Condition 2) is not sufficient since it does not yield a unique answer.

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.
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[GMAT math practice question]

(speed) A river flows with a constant speed of 2 miles per hour. It takes 3 hours for a ship to go 6 miles upstream. How many hours does it take the ship to travel the same distance downstream?

A. 1/4
B. 1/3
C. 1/2
D. 1
E. 3/4

=>

Assume v is the speed of the ship.
Then 6 / ( v – 2 ) = 3 and 2 = v – 2. Thus, v = 4.
The number of hours it takes for the ship to go 6 miles downstream, is 6 / ( v + 2 ) = 6 / ( 4 + 2 ) = 1.

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[GMAT math practice question]

(geometry) ABCD is a square with AB=10. What is the perimeter of the shaded area?

Attachment: 7.25ps.png [ 14.82 KiB | Viewed 482 times ]

A. 10π/3 +10
B. 5π/3 +10
C. 5π/6 +10
D. 5π/3 +5
E. 5π/6 +5

=>

Attachment: 7.29ps.png [ 8.99 KiB | Viewed 482 times ]

Since triangle PBC is equilateral, angle ABP has measure 30° and the length of arc AP is 2π*10*(30/360) = (5/3) π.
Thus, the perimeter of the shaded area is (5/3) π *2 + 10 = (10/3) π + 10.

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[GMAT math practice question]

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

1) n+200 is a perfect square
2) n+292 is a perfect square

=>

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)
If n + 200 = 400 = 20^2, then n = 200.
If n + 200 = 441 = 21^2, then n = 241.
Since condition 1) does not yield a unique answer, it is not sufficient.

Condition 2)
If n + 292 = 400 = 20^2, then n = 108.
If n + 292 = 441 = 21^2, then n = 149.
Since condition 2) does not yield a unique answer, it is not sufficient.

Conditions 1) & 2)
Write n + 200 = a^2 and n + 292 = b^2, for some positive integers a and b. Then
b^2 – a^2 = (n+292)-(n+200) = 92 = 2^2*23 and
(b+a)(b-a) = 2^2*23

Since b + a and b – a have the same parity, which means both b + a and b – a are even or both b + a and b – a are odd, b + a = 46 and b – a = 2.
Solving these equations simultaneously yields a = 22 and b = 24.
Thus n = 24^2 – 292 = 284.
The two conditions are sufficient, when applied together.

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.
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MathRevolution, I agree with chetan2u - question 1 is poorly written. There is no way this would appear on an actual GMAT, there are numerous grammatical errors.

Quote:
On a certain transatlantic crossing, 40 percent all passengers held round-trip tickets. [............] what percent of the ship’s passengers held round-trip tickets?

You can ignore everything in the middle, and get the answer.

Here's how the question was probably trying to be formulated:

On a certain transatlantic crossing, 40 percent of all passengers held round-trip tickets and also took their laptops aboard the ship. If 20 percent of the passengers with round-trip tickets did not take their laptops aboard the ship, what percent of the ship’s passengers held round-trip tickets?
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8425
GMAT 1: 760 Q51 V42
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[GMAT math practice question]

(algebra) What is the value of (a-b)/(a+b) –ab + b/c ?

1) a=bc
2) a=1/2

=>

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)
Plugging in a = bc = 1/2 yields
(a-b)/(a+b) – ab + b/c = (bc-b)/(bc+b) – b^2c + b/c = b(c-1) / b(c+1) – (1/2)c + (bc)/c^2 = (c-1)/(c+1) – 1/2c + 1/(2c^2).
Since we don’t know the value of c, both conditions together don’t yield a unique solution and they are not sufficient.

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.
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1
[GMAT math practice question]

(number properties) n ranges over the positive integers between 100 and 200, inclusive. Find the number of values of 7n+2 which are multiples of 5

A. 18
B. 20
C. 22
D. 24
E. 26

=>

If n = 5k, then 7n + 2 = 7(5k) + 2 = 5(7k) + 2 is not a multiple of 5.
If n = 5k+1, then 7n + 2 = 7(5k+1) + 2 = 5(7k) + 7 + 2 = 5(7k) + 9 = 5(7k+1)+4 is not a multiple of 5.
If n = 5k+2, then 7n + 2 = 7(5k+2) + 2 = 5(7k) + 14 + 2 = 5(7k) + 16 = 5(7k+3)+1 is not a multiple of 5.
If n = 5k+3, then 7n + 2 = 7(5k+3) + 2 = 5(7k) + 21 + 2 = 5(7k) + 23 = 5(7k+4)+3 is not a multiple of 5.
If n = 5k+4, then 7n + 2 = 7(5k+4) + 2 = 5(7k) + 28 + 2 = 5(7k) + 30 = 5(7k+6) is a multiple of 5.
Thus, n has remainder 4 when it is divided by 5.

The possible values of n are 104, 109, …, 199.
The number of possible values of n is (199-104)/5 + 1 = 20.

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[GMAT math practice question]

(algebra) What is the value of (3mr-nt)/(4nt-7mr)?

1) m/n = 4/3
2) r/t= 9/14

=>

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. We should simplify the conditions if necessary.

We rearrange (3mr-nt)/(4nt-7mr) to see if we can write in terms of the ratios m/n and r/t given in the conditions:
(3mr-nt)/(4nt-7mr)
= ( (3mr)/(nt) – (nt/nt) ) / ( 4(nt/nt) – 7mr/nt )
= ( 3(m/n)*(r/t) – 1 ) / ( 4 – 7(m/n)(r/t) )

Now, both conditions 1) & 2) together are sufficient since the simplified question requires only the values of (m/n) and (r/t).

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[GMAT math practice question]

(function) For which value of x will y=ax^2+20x+b have a minimum in the xy-plane?

1) b=10
2) a=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.

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

We can modify the original condition and question as follows:

If a > 0, the function will have a minimum at x = (-20)/(2a) = (-10)/a.
If a < 0, the function has no minimum. So, to answer the question, we need to find the value of a.
Thus, condition 2) is sufficient.

Note: condition 1) cannot be sufficient as it provides no information about the value of a.

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[GMAT math practice question]

(Number) What is the units digit of 3^n?

1) n is a multiple of 4
2) n is a multiple of 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.

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 the conditions if necessary.

The units digits of 3^n for n = 1, 2, 3, 4, … are 3, 9, 7, 1, 3, 9, 7, 1, …
So, the units digits of 3^n have period 4:
They form the cycle 3 -> 9 -> 7 -> 1.
Thus, 3^n has a units digit of 1 if n is a multiple of 4.

Note that 6 is not a multiple of 4.

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[GMAT math practice question]

(number properties) What is the remainder when 1+n+n^2 +…+ n^8 is divided by 5?

1) The remainder when n is divided by 5 is 3
2) n is less than 5

=>

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. We should simplify conditions if necessary.

The easiest way to solve remainder questions is to plug in numbers.
The units digits of 3^n for n = 1, 2, 3, 4, … are 3, 9, 7, 1, 3, 9, 7, 1, …
So, the units digits of 3^n have period 4:
They form the cycle 3 -> 9 -> 7 -> 1.
Thus, if n has remainder 3 when it is divided by 5, 1+n+n^2 +…+ n^8 has the same remainder as 1 + 3 + 9 + 7 + 1 + 3 + 9 + 7 + 1 = 21 when it is divided by 5. It has a remainder of 1 when it is divided by 5.
Condition 1) is sufficient.

Condition 2)
If n = 1, then 1+n+n^2 +…+ n^8 = 9, which has remainder 4 when it is divided by 5.
If n = 3, then 1+n+n^2 +…+ n^8 has remainder 1 when it is divided by 5.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

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[GMAT math practice question]

(Statistics) When playing a coin-tossing game, Tom wins \$5 when the coin lands on heads and loses \$3 when the coin lands on tails.
After tossing the coin 20 times Tom has won a total of \$12. How many times did the coin land on heads?

A. 7
B. 8
C. 9
D. 10
E. 11

=>

Let h be the number of times the coin landed on heads.
Then 20 – h is the number of times the coin landed on tails.
Thus, Tom won 5h – 3(20-h) = 8h – 60 dollars.
Since Tom won \$12, this gives the equation 8h - 60 = 12. Solving this equation yields 8h = 72 and h = 9.

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[GMAT math practice question]

(Distance / Rate Problems) A lake has a circumference of two kilometers. A and B both departed from the same point on the circumference at the same time and walked in opposite directions. They met 30 minutes after their departure. What is the sum of the speeds of A and B?

A. 3 km/hr
B. 4 km/hr
C. 5 km/hr
D. 6 km/hr
E. 7 km/hr

=>

Let a and b be the velocities of A and B, respectively.
Then (30/60)a + (30/60)b = (1/2)a + (1/2)b = 2.
Thus, a + b = 4.

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1
[GMAT math practice question]

(Number properties) m and n are positive integers such that m(n+10) = 75. What is the value of m?

1) n is not less than m
2) m is not a 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 (m and n) and 1 equation, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
We can find two pairs of solutions: m = 1 and n = 65, and m = 3 and n = 15.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
We can find only one pair of solutions: m = 1 and n = 65.
Since condition 2) yields a unique solution, it is sufficient.

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.
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Joined: 16 Aug 2015
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[GMAT math practice question]

(algebra) What is the value of the integer a?

1) x - (2/3)(x-4a) = 7 has a positive integer solution
2) a is positive

=>

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 a) 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 - (2/3)(x - 4a) = 7 is equivalent to 3x – 2(x-4a) = 21 or x = 21 – 8a.
The possible pairs (x,a) are (13,1) and (5,2).
Since both conditions together don’t yield a unique solution, they are not sufficient.

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.
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Joined: 16 Aug 2015
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[GMAT math practice question]

(number properties) A and B are positive integers. G is the greatest common divisor of A and B, and L is the least common multiple of A and B. What is the value of A+B?

1) G/A + G/B = 7/10
2) L=70

=>

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 (A and B) 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)
Suppose A=aG and B=bG for some integers a, b and G, where a and b are relatively prime.

Then
G/A + G/B = 7/10
=> G/(aG) + G/(bG) = 7/10, since A=aG and B = bG
=> bG/(abG) + aG/(abG) = 7/10, taking a common denominator
=> (aG+bG)/(abG) = 7/10
=> (A+B)/L = 7/10
=> (A+B)/70 = 7/10, since L = 70
=> A+B = 49
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)
If A = 2 and B = 5, then A + B = 7.
If A = 6 and B = 15, then A + B = 21.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
If A = 14 and B = 35, then A + B = 49.
If A = 2 and B = 35, then A + B = 37.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

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.
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[GMAT math practice question]

(set) X is the set of positive integer multiples of 3, and Y is the set of positive integer multiples of 7. Define X+Y as {x+y|x ∈X and y ∈Y}. How many elements of X+Y are less than or equal to 21?

A. 4
B. 6
C. 8
D. 11
E. 13

=>

X = { 3, 6, 9, 12, 15, 18, 21, … } and Y = { 7, 14, 21, … }
The following pairs ( x, y ) satisfy x + y <= 21:
(3, 7), (3, 14), (6, 7), (6, 14), (9,7) and (12,7).
There are 6 such pairs, (x,y).

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[GMAT math practice question]

(Probability) A={2, 4, 6, 8, 10} is given. What is the number of subsets of A containing 3 elements?

A. 5
B. 10
C. 12
D. 24
E. 32

=>

The number of subsets is equal to the number of ways ways to choose 3 elements out of 5 elements, which is 5C3 = (5*4*3)/(1*2*3) = 10.

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[GMAT math practice question]

(number properties) What is the value of x?

1) the remainder, when 170 is divided by x, is 2
2) the remainder, when 140 is divided by x, is 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 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Now, 170 = x*a + 2, so 168 = 2^3*3*7 = x*a. Note that x > 2 since the dividend must be greater than the remainder.
So, x is a factor of 168 greater than 2. The possible values of x are 3, 4, 6, …, 168.
Since condition 1) doesn’t yield a unique solution, it is not sufficient.

Condition 2)
Now, 140 = x*b + 4, so 136 = x*b. Note that x > 4 since the dividend must be greater than the remainder.
So, x is a factor of 136 = 23*17 greater than 4. The possible values of x are 8, 17 and 140.
Since condition 2) doesn’t yield a unique solution, it is not sufficient.

Conditions 1) & 2).
When we consider both conditions together, x is a common factor greater than 4 of 168 = 2^3*3*7 and 136 = 2^3*17. The only possible value of x is 8.

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

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.
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Re: Overview of GMAT Math Question Types and Patterns on the GMAT  [#permalink]

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[GMAT math practice question]

(algebra) What is the value of (x+y)(y+z)(z+x)+5?

1) xyz=-3
2) x+y+z=0

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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)
Since x + y + z = 0 from condition 2), x + y = -z, y + z = -x, and z + x = -y. Since condition 1) tells us that xyz = -3, (x+y)(y+z)(z+x)+5 = (-x)(-y)(-z) + 5 = -xyz + 5 = -(-3) + 5 = 8.
Since both conditions together yield a unique solution, they are sufficient.

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.
_________________ Re: Overview of GMAT Math Question Types and Patterns on the GMAT   [#permalink] 10 Sep 2019, 18:46

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# Overview of GMAT Math Question Types and Patterns on the GMAT  