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

In this guide you will learn about GMAT Quantitative Reasoning section, and find over 500 GMAT Quantitative Reasoning practice questions. This page contains everything you need to know and the essential skills for a high GMAT Quantitative Reasoning score.

### The Introduction to the GMAT Quantitative Reasoning Section

The GMAT Quantitative Reasoning section consists of 21 multiple-choice questions that must be answered in 45 minutes. All multiple-choice questions are problem-solving questions. (Note: The new Quantitative Reasoning section DOES NOT include the Data Sufficiency question type; Data Sufficiency will now be included in the new Data Insights Section.)

The GMAT problem-solving questions cover a wide range of topics, including arithmetic, algebra, data analysis, and number properties. Test-takers are required to use their mathematical skills and critical thinking to solve problems efficiently within time constraints. The following categories are what you can expect to see in the GMAT Quantitative Reasoning section: Integer Properties, Factors, Multiples, Exponents, Roots Subcategory Test Frequency Integer Properties and Absolute Values 10% Divisibility, Multiples, and Factors 5% Remainder Problem 3% Exponents and Roots 4% Algebra, Equations, and Functions Subcategory Test Frequency Linear Equations and Quadratic Equations 5% Equation Inequality and Min/Max problems 5% Word Problems - Quadratic Equations and Linear Equations 12% Word Problems - Equation in Multiple Variables 4% Word Problems - Define Functions 6% Rates, Ratios, and Percents Subcategory Test Frequency Arithmetic Operations with decimals, fractions, and percents 4% Word Problems - Percent and Interest 8% Word Problems - Mixture Problems 7% Word Problems - Fraction 4% Word Problems - Work/Rate Problems 5% Statistics, Sets, Combinations, Probability, and Series Subcategory Test Frequency Statistics 5% Sets 4% Combinations 3% Probability 3% Sequences and Series 3%

💡Note: Questions related to Geometry are no longer included in the GMAT Quantitative Reasoning section.

Now, let's look at some example GMAT problem-solving questions associated with each category.

### Integer Properties, Factors, Multiples, Exponents, Roots

In this category, common math topics include:

##### Integer Properties and Absolute Values

Integer properties involve understanding the fundamental properties and characteristics of numbers, such as even/odd, prime/composite, positive/negative, etc.

Absolute values or modules are mathematical expressions that represent the distance of a number from zero on the number line. It is denoted by two vertical bars enclosing the number, such as |x|. The result of the absolute value is always non-negative, as it disregards the sign of the number.

Sample Question Submit

If | x | > 4, which of the following must be true?

1. $$x > 4$$
2. $$x^2 > 16$$
3. $$|x - 1| > 3$$
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III

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:

1. 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.
1. $$x^2 > 16$$: 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.

1. a) When x > 4: If x > 4, then |x - 1| > |4 - 1| = |3| = 3. Therefore, this statement is true for this scenario.
2. 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.

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##### Divisibility, Multiples, and Factors

Divisibility, multiples, and factors are concepts related to the properties of numbers. Divisibility refers to whether one number can be divided by another without leaving a remainder. Multiples are the products of a given number and any whole number, and factors are the numbers that divide a given number without leaving a remainder.

Sample Question Submit

If M is the least common multiple of 90, 196, and 300, which of the following is NOT a factor of M?

(A) 600
(B) 700
(C) 900
(D) 2100
(E) 4900

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.

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##### Remainder Problem

Remainders refer to the value left over after dividing one number by another. You will see some GMAT word problem questions involving dividing and finding the remainder.

Quantitative Reasoning Sample Question Submit

April 10, 2000, fell on a Monday. On which day of the week did April 10, 2006, fall? (Note: 2000 and 2004 were leap years.)

(A) Sunday
(B) Monday
(C) Tuesday
(D) Wednesday
(E) Thursday

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.

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##### Exponents and Square Roots

Exponents, also known as powers, involve raising a base number to a certain exponent or power. The exponent indicates how many times the base number should be multiplied by itself.

A square root is a mathematical operation that determines a value that, when multiplied by itself, results in a given number. In other words, it is the inverse operation of squaring a number. The square root of a non-negative real number 'x' is denoted by the symbol $$\sqrt{x}$$.

Sample Question 1 Submit

For positive integers a and b, if $$2^a \times 3^b = 432$$ and $$2^b \times 3^a = 486$$, what is the value of (a + b)?

(A) 15
(B) 18
(C) 21
(D) 24
(E) 27

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

Sample Question 2 Submit

$$\frac{\sqrt{0.00027}}{(0.01^2)}$$ =

(A) 10
(B) 30
(C) 300
(D) 100
(E) 1000

Given expression:

$$\frac{\sqrt{0.00027}}{0.01^2}$$

We can write this as:

$$\frac{\sqrt{27 \times 10^{-6}}}{(10^{-2})^2}$$

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.

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### Algebra

Typical questions that appear in the GMAT quant section from Algebra include solving linear equations and quadratic equations, solving equation inequality, and algebra word problems. In an algebra word problem question, you are expected to translate what is given in words in the question into algebraic equations, and then proceed to solve them to find the solution.

##### Linear Equations

A linear equation is an equation in which the highest power of the variable is always 1. A linear equation in one variable means there is only one variable in the linear equation. Here are examples of linear equations of one variable:

x + 5 = 10
4(x - 3) = 3(x + 2)

A linear equation in two variables means there are two variables in the linear equation. Here are examples of linear equations in two variables:

x + y = 10
3a + b = 5

GMAT math questions related to linear equations often ask you to solve linear equations.

`Quantitative Reasoning Sample Question Submit

For what values of 'k' will the pair of equations 7x + 4y = 12 and kx + 12y = 30 NOT have a unique solution?

(A) 21
(B) 14
(C) 7
(D) 16
(E) 20

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.

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A quadratic equation is an equation in which the highest power of the variable is always 2. Here is an example of quadratic equations in one variable:

$$x^2 + 3x + 4 =25$$

Sample Question Submit

What is the highest integral value of 'm' for which the quadratic equation $$x^2 - 8x + m = 0$$ will have two real and distinct roots?

(A) 10
(B) 15
(C) 16
(D) 20
(E) 30

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.

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##### Word Problems - Define the Function

A equation in one variable can be used to define a function of that variable. A function is denoted by a letter such as f or g along with the variable in the expression. Function notation provides a short way of writing the result of substituting a value for a variable. Here is an example:

The expression $$x^2 - 4x + 7$$ defines a function f that can be denoted by $$f(x) = x^2 - 4x + 7$$ .

Gmat math questions related to functions only ask you to define a function given by a scenario.

Sample Question Submit

The original price of a certain bicycle is discounted by y percent, and the reduced price is then discounted by 3y percent. If P is the original price of the bicycle, which of the following represents the price of the bicycle after the two successive discounts?

(A) $$P (1 - 0.04y + 0.03y^2)$$
(B) $$P (1 - 0.04y + 0.0003y^2)$$
(C) $$P (1 - 0.4y + 0.0003y^2)$$
(D) $$P (1 - 3y^2)$$
(E) $$P (1 - 4y + 3y^2)$$

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) $$P(1 - 0.04y + 0.0003y^2)$$.

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##### Word Problems - Quadratic Equations and Linear Equations

In this type of question, you are expected to translate what is given in words in the question into linear or quadratic equations, and then proceed to solve them to find the solution.

Sample Question Submit

A concert hall with 800 seats sells tickets at $2.00,$3.00, or $4.00 per seat. On Friday evening, ​1/4​ of the tickets sold were at$3.00 per seat and the total receipts from the sale of 800 tickets was $2,600. How many of the tickets sold were at$4.00 per seat?

(A) 100
(B) 200
(C) 300
(D) 400
(E) 500

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 at4.00 per seat. Hence, the correct answer is (D) 400.

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##### Equation Inequalities and Min/Max Problems

Inequalities involve comparing two quantities or expressions to determine their relative magnitudes. They use symbols such as <, >, ≤, and ≥ to represent the relationships between numbers. Min/Max problems require finding the minimum or maximum value of a function or expression, usually subject to certain constraints or conditions.

Sample Question Submit

What is the range of possible x values given: 3x + 6 < 18, 4x > 28:

(A) 4 < x < 6
(B) x > 7 and x < 0
(C) x > 10 and x < -3
(D) 4 < x < 7
(E) x > 3 and x < 1

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.

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### Rates, Ratios, and Percents

In this category, common math topics include fractions, ratios, decimals, and percents, and questions related to these topics are often one of the following:

##### Arithmetic Operations with Decimals, Fractions, and Percents

Fractions, ratios, and decimals are forms of representing parts of a whole. Fractions express a part of a whole as a quotient of two numbers, ratios show the relative comparison of quantities, and decimals represent fractions using a decimal point and place.

Sample Question Submit

What is the value of $$(0.04)×(2.5) \over (0.01)×(0.05)^2$$ ?

(A) 200
(B) 400
(C) 500
(D) 4000
(E) 5000

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.

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##### Percents and Interest Problems

Percents and interest problems involve calculating percentages, including finding percentages of a whole and solving problems related to interest rates and investments.

Sample Question Submit

Steve deposits money into a unique savings scheme at a local bank. The bank pays interest in a peculiar way. In the first month, the bank offers either 3% of the savings or $20, whichever is lesser. For every month following, the bank pays a flat$20 interest irrespective of the total savings amount. If Steve deposited $700, how much, in dollars, did he accumulate as interest after four months? • radio_button_unchecked (A) 60 • radio_button_unchecked (B) 80 • radio_button_unchecked (C) 81 • radio_button_unchecked (D) 82 • radio_button_unchecked (E) 87 • spellcheck Check Answer & Answer Explanation 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.

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##### Mixture Problems

Mixture problems involve calculating the quantities of different components or substances mixed together to obtain a desired mixture with specific characteristics.

Sample Question Submit

How many liters of pure syrup should be added to a 20-liter solution that is 5% syrup in order to fortify it to a solution that is 24% syrup?

(A) 5
(B) 5.5
(C) 6
(D) 7.5
(E) 10

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.

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##### Work/Rate Problems

Work/Rate problems involve calculating the rate at which a task is completed, typically involving multiple individuals working together or separately to accomplish a task in a given time frame.

Sample Question Submit

Jennifer, Kate, and Lucy decide to collaborate on a mural painting project. Jennifer and Kate, working together, can complete the mural in 4/3 hours. Kate and Lucy can finish the same task in 3/2 hours. Meanwhile, Jennifer and Lucy take 3/4 hours to complete it. How long would it take for Jennifer, Kate, and Lucy to paint the mural together?

(A) 1/2 hour
(B) 7/8 hour
(C) 3/4 hour
(D) 8/11 hour
(E) 2/3 hour

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).

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##### Word Problems - Fraction

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### Statistics, Sets, Combinations, Probability, and Sequences

In this category, common math topics include:

##### Statistics

Statistics involves analyzing and interpreting data, including measures such as mean, median, mode, and range.

Sample Question Submit

If the positive number d is the standard deviation of n, k, and p, then the standard deviation of n + 1, k + 1, and p + 1 is

(A) d+3
(B) d+1
(C) 6d
(D) 3d
(E) d

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.

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##### Sets Problems

Sets problems deal with groups of elements and their relationships. Overlapping sets problems deal with groups of items that share common elements. Test-takers must analyze the relationships between the sets and identify shared and exclusive elements.

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##### Combinations

Combinations refer to the different ways of selecting items from a larger set without considering the order. It is often used to calculate the number of ways to choose a group of objects from a given set, where the order of selection does not matter.

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##### Probability

Probability deals with the likelihood of an event occurring. It involves calculating the chances of different outcomes based on the total number of possible outcomes.

Quantitative Reasoning Sample Question Submit

Consider a ship that has two propellers - Propeller A and Propeller B. Both propellers are important for the functioning of the ship, but the ship can still move even if only one of them is operational. The occurrence of a malfunction in one propeller is independent of the functioning or malfunction of the other. If the chance that each propeller operates correctly is 3 out of 5, what is the probability that the ship will still be able to sail?

(A) 4/25
(B) 9/25
(C) 1/5
(D) 3/5
(E) 21/25

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.

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##### Sequences

Sequences are ordered lists of numbers or objects that follow a specific pattern or rule. Test-takers must identify the pattern and find missing elements in the sequence.

Quantitative Reasoning Sample Question Submit

In a certain sequence, the nth term, qn , is given by the formula qn = (2qn-1 - 3)2
If the 6th term q6 = 225, what is the value of the 5th term, q5? .

(A) 3
(B) 6
(C) 9
(D) 15
(E) 8

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

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

So, q5 = 9 is the only valid solution.

Hence, the correct answer is (C) 9.

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