GMAT Quantitative Reasoning Test Guide - Prepare with over 500 GMAT math questions to Get a High GMAT Quantitative Reasoning Score.

Answer Explanation:

Answer: (D)

When |x| > 4, it means that x is either greater than 4 or less than -4 (because the absolute value of any number greater than 4 or less than -4 will be greater than 4).

Let's analyze each statement:

4. x > 4: the inequality |x| > 4 is equivalent to x < -4 or x > 4. Therefore, |x| > 4 does not imply that condition I must be true, since |x| > 4 is true and x > 4 is false for x = -5. 

16. \(x^2 > 16\): This statement is true when x is either less than -4 or greater than 4 because in both cases, \(x^2\) will be greater than 16. Therefore, this statement is true when |x| > 16.

III. |x - 1| > 3: Let's check this statement for both scenarios of x: when x > 4 and when x < -4.

3. a) When x > 4: If x > 4, then |x - 1| > |4 - 1| = |3| = 3. Therefore, this statement is true for this scenario.
4. b) When x < -4: If x < -4, then |x - 1| > |-4 - 1| = |-5| = 5. Therefore, this statement is also true for this scenario.

Since all three statements are true for the given condition |x| > 3, the correct answer is: (D) II, and III only.

Answer Explanation:

Answer: (A)

Let's first find the prime factorization of each of the three numbers.

• The prime factorization of 90 is \(2 \times 3^2 \times 5 \)
• The prime factorization of 196 is \(2^2 \times 7^2 \)
• The prime factorization of 300 is \(2^2 \times 3 \times 5^2 \)

The least common multiple (LCM) is found by multiplying the highest powers of all primes that appear in the factorization of each number. So, the LCM, M, would be:

M = \(2^2 \times 3^2 \times 5^2 \times 7^2\)

Now let's prime factorize each of the options and see which one cannot be made using the prime factors of M:

• The prime factorization of 600 is \(2^3 \times 3 \times 5^2\). This is not a factor of M because it requires three 2's, but M only has two 2's in its prime factorization.
• The prime factorization of 700 is \(2^2 \times 5 \times 7^2\). This is a factor of M as M has at least these many of each prime in its prime factorization.
• The prime factorization of 900 is \(2^2 \times 3^2 \times 5^2\). This is a factor of M as M has at least these many of each prime in its prime factorization.
• The prime factorization of 2100 is \(2^2 \times 3 \times 5^2 \times 7\). This is a factor of M as M has at least these many of each prime in its prime factorization.
• The prime factorization of 4900 is \(2^2 \times 5 \times 7^2\). This is a factor of M as M has at least these many of each prime in its prime factorization.

So, the number that is NOT a factor of M is 600.

Answer Explanation:

Answer: (C)

The difference in years between 2000 and 2006 is 6 years. In 6 years, there are normally 6 * 365 = 2190 days.

However, this range includes two leap years (2000 and 2004), which adds 2 extra days, so the total becomes 2192 days.

We know every 7 days is a cycle, so let’s divide 2192 and find the remainder.

\( 2192 = 7 \times 313 + 1 \)

The remainder when 2192 is divided by 7 (the number of days in a week) is 1.

This means that April 10, 2006, is one day after April 10, 2000.

April 10, 2000, fell on a Monday, so April 10, 2006 is one day after Monday, which is Tuesday.

So, the correct answer is (C) Tuesday.

Answer Explanation:

Answer: (A)

Step 1: Observe the numbers 432 and 486 and try to find their prime factors:

\(432 = 2^4 \times 3^3\)

\(486 = 2^1 \times 3^5\)

Step 2: Now, we can rewrite the given equations using their prime factorization:

\((2^a \times 3^b) = (2^4 \times 3^3)\)

\((2^b \times 3^a) = (2^1 \times 3^5)\)

Step 3: Equate the powers of 2 and 3 on both sides of each equation:

For powers of 2: a = 4 (from equation 1) b = 1 (from equation 2)

For powers of 3: b = 3 (from equation 1) a = 5 (from equation 2)

Step 4: Find the value of (a + b): a + b = 4 + 1 = 5

So, the value of (a + b) is 5.

The correct answer is: (A) 15

Answer Explanation:

Answer: (C)

Given expression: 

\( \frac{\sqrt[3]{0.00027}}{0.01^2} \) 

We can write this as: 

\( \frac{\sqrt[3]{27 \times 10^{-6}}}{(10^{-2})^2} \) 

Simplify further: 

\( \frac{(27^{1/3}) \times 10^{-2}}{10^{-4}} \) 

Calculate the cube root of 27, which is 3: 

\( \frac{3 \times 10^{-2}}{10^{-4}} \) 

Now, use the property of exponents 

\( a^{-m} = \frac{1}{a^m} \) 

to simplify: 

\( = 3 \times 10^{4-2} \) 

Simplify the exponent: \( = 3 \times 10^{2} \) 

And finally, simplify the multiplication: \( = 300 \) 

So, the result of the expression is indeed 300.

Answer Explanation:

Answer: (A)

A pair of linear equations will not have a unique solution if the ratios of their coefficients are equal, which means they are dependent or parallel.

For the two equations, the coefficients are:

For x: 7 in the first equation, k in the second. 

For y: 4 in the first equation, 12 in the second.

The equations will not have a unique solution when the ratios of these coefficients are equal. That is, when:

7/k = 4/12

Solving for k, we find:

k = 7 * (12/4) k = 21

So, the equations will not have a unique solution when k is 21. The answer is (A) 21.

Answer Explanation:

Answer: (B)

A quadratic equation \(ax^2 + bx + c = 0\) has two real and distinct roots if the discriminant \((b^2 - 4ac)\) is greater than 0.

Here, a = 1, b = -8, and c = m.

The discriminant is \((-8)^2 - 41m = 64 - 4m\) .

To have two real and distinct roots, we need 64 - 4m > 0. Solving this inequality gives:

64 > 4m => 16 > m

The highest integer m that satisfies this inequality is 15.

So, the highest integral value of 'm' for which the quadratic equation \(x^2 - 8x + m = 0\) will have two real and distinct roots is 15. The correct answer is (B) 15.

Answer Explanation:

Answer: (B)

Firstly, a discount of y percent reduces the price to \(P(1 - y/100)\).

Secondly, a further discount of 3y percent reduces this to \(P(1 - y/100)(1 - 3y/100)\).

This simplifies to \(P(1 - y/100 - 3y/100 + 3y^2/10000) = P(1 - 0.01y - 0.03y + 0.0003y^2) = P(1 - 0.04y + 0.0003y^2)\)

So, the price of the bicycle after the two successive discounts is represented by the expression \(P(1 - 0.04y + 0.0003y^2)\).

Therefore, the correct answer is (B) \(P(1 - 0.04y + 0.0003y^2)\).

Answer Explanation:

Answer: (D)

Let's denote:

• x as the number of tickets sold at $2.00
• y as the number of tickets sold at $3.00
• z as the number of tickets sold at $4.00

From the problem, we can form the following equations:

1. x + y + z= 800 (Total number of tickets)
2. y=1/4 ∗ 800 = 200 (1/4 of the tickets were sold at $3.00)
3. 2.00x + 3.00y + 4.00z = 2600 (Total sale was $2,600)

We can substitute y from equation 2 into equation 1 

x + 200 + z = 800
x + z = 600

W can substitute y from equation 2 into equation 3:

2.00x + 3.00∗200 + 4.00z = 2600
2.00x + 600 + 4.00z = 2600
2.00x + 4.00z = 2000
x + 2z = 1000

So we get the following two equations

1. x + z = 600
2. x + 2z = 1000

Let's multiply the first equation by 2.00:

1. 2x + 2z = 1200
2. x + 2z = 1000

Now subtract this first equation from the second equation:

\( \frac { \begin{aligned} 2x+2z &= 1200\\ x+2z &= 1000 \end{aligned} } {x = 200} \ - \)

Finally, solve for z by putting x = 200 into the equation “x + z = 600”

200 + z = 600

z=400

So, 400 tickets were sold at $4.00 per seat. Hence, the correct answer is (D) 400.

Answer Explanation:

Answer: (D)

Let's solve each inequality separately:

3x + 6 < 18 

Subtract 6 from both sides: 3x < 12

Divide both sides by 3: x < 4

4x > 28 

Divide both sides by 4: x > 7

Therefore, the correct answer is (D) 4 < x < 7.

Answer Explanation:

Answer: (B)

Let's calculate the expression:

First, we will simplify the expression:

\( (0.04)×(2.5) \over (0.01)×(0.05)^2 \) 

First, we can write the decimals as fractions with powers of 10:

\(0.04 = \frac{4}{100} = 4 \times 10^{-2} \)

\(0.01 = \frac{1}{100} = 1 \times 10^{-2} \)

\(0.05 = \frac{5}{100} = 5 \times 10^{-2} \)

Now, we'll plug in these values into the original expression:

\( (4 \times 10^{-2}) \times (2.5) / ((1 \times 10^{-2}) \times (5 \times 10^{-2})^2) \)

This simplifies to:

\( = (4 \times 10^{-2}) \times (2.5) / ((1 \times 10^{-2}) \times (25 \times 10^{-4})) \)

\( = (4 \times 2.5 \times 10^{-2}) / (25 \times 10^{-6}) \)

Multiply the numbers:

\( = (10 \times 10^{-2}) / (25 \times 10^{-6}) \)

\(= 10 \times 10^{-2} \times 10^{6} / 25 \)

Apply the properties of exponents to combine like bases:

\(= 10 \times 10^{4} / 25 \)

\(= 10,000 / 25 \)

Finally, divide the numbers:

\(= 400 \)

So the result of the calculation is 400.

Answer Explanation:

Answer: (B)

Given:

• Steve deposits $700
• The bank pays 3% of the savings or $20, whichever is lesser, in the first month
• The bank pays a flat $20 interest every subsequent month

First, let's calculate the interest for the first month:

Interest for the first month = Minimum of (3% of $700, $20)
= Minimum of ($21, $20)
= $20

For every subsequent month, the bank pays a flat $20 interest, so for the next three months, he receives 3 * $20 = $60.

Adding the interest earned in all four months, we get:

Total interest = Interest of first month + Interest of next three months
= $20 + $60
= $80

Hence, Steve earned $80 as interest after four months.

So, the correct answer is (B) $80.

Answer Explanation:

Answer: (A)

Let's denote x as the amount of pure syrup needed. We can create the following equation based on the problem statement:

\(0.24(x + 20) = x + 0.05 \times 20 \)

Here, the left side of the equation stands for the amount of syrup in the solution after adding x liters of pure syrup (24% of the total solution's volume). 

The right side is the initial amount of syrup (5% of 20 liters) plus the pure syrup added.

Solving this equation, we get:

\( \begin{aligned} 0.24x - x = 1 - 4.8 \\ 0.76x = 3.8 \end{aligned} \)

x = 3.8 / 0.76 = 5 liters

So, 5 liters of pure syrup need to be added. 

The answer is (A).

Answer Explanation:

Answer: (D)

Firstly, we need to convert the time taken by each pair to work rates:

• Jennifer and Kate's combined rate is 1/(4/3) = 3/4 of a mural per hour.
• Kate and Lucy's combined rate is 1/(3/2) = 2/3 of a mural per hour.
• Jennifer and Lucy's combined rate is 1/(3/4) = 4/3 of a mural per hour.

Secondly, we know that the combined rate of multiple workers working together is the sum of their individual rates. Therefore, we can add up the rates of the pairs:

(3/4) + (2/3) + (4/3) = 33/12 = 11/4

This value equals twice the sum of Jennifer, Kate, and Lucy's rates because each person is counted twice in the sum of the pairs' rates.

To find the combined rate of Jennifer, Kate, and Lucy, we divide 11/4 by 2:

(11/4) / 2 = 11/8

This means that Jennifer, Kate, and Lucy, working together, can paint 11/8 of the mural per hour.

The time it would take for all three to complete the mural together is the reciprocal of their combined rate. Thus, they will be able to complete the mural in 8/11 hours.

Therefore, the correct answer is (D).

Answer Explanation:

Answer: (E)

The standard deviation of a set of numbers is a measure of the amount of variation or dispersion in the set. If you add or subtract a constant value to each number in the set, the standard deviation does not change. This is because while each individual value has changed, their relative position and the dispersion among them remains the same.

So, if d is the standard deviation of n, k, and p, then the standard deviation of n + 1, k + 1, and p + 1 is still d.

Therefore, the answer is (E) d.

Answer Explanation:

Answer: (E)

Firstly, let's understand what the problem is asking. We need to find out the probability that the ship sails. This can happen in two cases: Either Propeller A or Propeller B works, or both work.

Since we know the probability of each propeller working, it may seem like we could just add these probabilities. But this would ignore the possibility of both propellers working at the same time, leading to a miscount. So we need another approach.

A useful method in probability problems is to consider the opposite of what we are looking for - in this case, the ship not being able to sail. This only happens when both propellers malfunction.

The problem tells us that the probability of each propeller working is 3/5. This means the probability of it malfunctioning is 1 - 3/5 = 2/5. Since the propellers' performance is independent, the likelihood that both propellers fail is the product of their individual failure probabilities, i.e., 2/5 * 2/5 = 4/25.

This gives us the probability of the ship not sailing. But we need the probability of the ship sailing, which is the opposite. In probability, the sum of the probabilities of all possible outcomes is always 1. So, the probability of the ship sailing is 1 - the probability of the ship not sailing = 1 - 4/25 = 21/25.

So, the answer is (E): 21/25.

Answer Explanation:

Answer: (C)

The formula for the nth term is q_n = (2_qn-1 - 3)²

If q₆ = 225, then what is the value of q₅?. 

The 6th term q₆ is provided as 225. We can write this as (2q₅ - 3)² = 225.

Solving this equation gives two potential roots for q₅: 2q₅ - 3 = 15 or 2q₅ - 3 = -15.

Solving these two equations gives q₅ = 9 (from the first equation: 2q₅ = 18, so q₅ = 9) and q₅ = -6 (from the second equation: 2q₅ = -12, so q₅ = -6).

As per the given formula q_n= (2_qn-1 - 3)²

, q_n cannot be negative. Therefore, q₅ cannot be -6.

So, q₅ = 9 is the only valid solution.

Hence, the correct answer is (C) 9.