GMAT Quantitative Exam Prep

Apr 22, 2026

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This document contains a collection of quantitative reasoning problems, likely from a standardized test preparation context (like the GMAT). The problems cover a range of mathematical concepts.

Summary of Quantitative Reasoning Problems

The provided text consists of multiple instances of various quantitative reasoning problems. Below is a breakdown of the distinct problems presented:

Problem 1: Watches with Antitheft Tags and Gold Content

Scenario: A display case has watches. Exactly 2 have an antitheft tag. Of those with tags, exactly 16 contain gold.
Question: Which of the following could be the total number of watches in the display case?
Key Information: This problem involves set theory or Venn diagrams. The number of watches with gold and an antitheft tag is a subset of watches with antitheft tags.

Problem 2: Greatest Integer with a Specific Remainder

Scenario: Find the greatest positive integer less than 7,654.
Condition: The integer must have a remainder of 5 when divided by 9.
Mathematical Concept: This is a number theory problem involving modular arithmetic.

Problem 3: Work Rate Problem (Arielle and Elliott)

Scenario: Arielle and Elliott paint a fence together in 4 hours.
Given: Arielle working alone would take 6 hours.
Question: How many hours would it take Elliott to paint the fence alone?
Mathematical Concept: This is a classic work-rate problem, often solved by considering the rate of work for each individual and their combined rate.

Problem 4: Combinations and Games Played

Scenario: 8 people play a series of games, with each game involving two people.
Condition: Each person plays every other person exactly once.
Question: What is the total number of games played?
Mathematical Concept: This is a combinatorics problem, specifically finding the number of ways to choose 2 people from a group of 8, which can be solved using combinations formula C(n, k) = n! / (k!(n-k)!).

Problem 5: Tax Rate Conversion

Scenario: A tax rate is given as $0.69 per $1,000.
Question: Express this rate as a percent.
Mathematical Concept: This involves converting a rate to a percentage, requiring understanding of ratios and percentage calculations.

Problem 6: Greatest Common Divisor (GCD)

Scenario: The greatest common divisor (GCD) of 12, 60, and x is 4.
Condition: x is a positive integer.
Question: Which of the following is NOT a possible value of x?
Mathematical Concept: This problem tests understanding of GCD and its properties. x must be a multiple of 4, and its prime factorization must not introduce any prime factors that would increase the GCD beyond 4.

Problem 7: Ordering Algebraic Expressions with Negative Integers

Scenario: Three expressions are given:
- r = x² - 0.5
- S = (x - 0.5)²
- t = x³ - 0.5
Condition: x is a negative integer.
Question: List the quantities r, S, and t from least to greatest.
Mathematical Concept: This involves evaluating algebraic expressions for a negative integer input and comparing the results.

Problem 8: Sum of Integers and Prime Factorization

Scenario: S is the sum of all integers from 1 to 30, inclusive.
Question: What is the greatest prime factor of S?
Mathematical Concept: This requires calculating the sum of an arithmetic series and then finding the prime factorization of the sum.

Problem 9: Basis Points Conversion

Scenario: A basis point is defined as one-hundredth of a percentage point.
Question: Express 143 basis points as a decimal.
Mathematical Concept: This is a unit conversion problem involving percentages and decimals.

Problem 10: Range of Fish Lengths

Scenario: Six fish initially have lengths: 4, 5, 7, 7, 9, and 11 cm.
Changes:
- Shortest fish increases by 3 cm.
- Longest fish increases by 2 cm.
- Remaining fish increase by 1 cm each.
Question: What is the range of the fish lengths at the end of the month?
Mathematical Concept: This involves calculating the new lengths of the fish and then finding the difference between the maximum and minimum lengths.

Problem 11: Algebraic Simplification with Factorials

Scenario: Simplify the expression (n-1) * n! + (n-1)! * n.
Condition: n is a positive integer.
Mathematical Concept: This is an algebra problem involving factorial notation. Key properties of factorials, such as n! = n * (n-1)!, will be used.

Problem 12: Logic Puzzle with Hall Light

Scenario: A hall light is on if and only if exactly one of Suite A and Suite B has its main light on.
Condition: The hall light is off.
Question: Which of the following states of the main lights in Suite A and Suite B cannot be true?
Mathematical Concept: This is a logic problem, likely solvable using a truth table or by analyzing the conditions under which the hall light is off. The condition "exactly one is on" means the hall light is off if both are on, or both are off.

Problem 13: Inverse Square Law (Weight and Distance)

Formula: W = k/r² (implied by context, where W is weight, r is distance, and k is a constant).
Given: W = 200 pounds when r = 4,000 miles.
Question: Estimate the weight of an object 5,600 miles from Earth's center.
Mathematical Concept: This is an application of inverse variation. First, find the constant k, then calculate the new weight.

Problem 14: Fundraising Contributions

Scenario: 10 friends raise funds.
- 1 friend contributes the smallest amount (x).
- n friends contribute 2x.
- Remaining friends (10 - 1 - n) contribute 3x.
Given: Total contribution is $720. Each person contributed an integer amount.
Question: What was the smallest contribution (x)?
Mathematical Concept: This is an algebra problem involving setting up an equation based on the given information and solving for the unknown smallest contribution.

Problem 15: Ratio of Customers

Scenario: 28 customers visited a car dealership.
Question: Which of the following could be the ratio of customers who bought a car to those who did not?
Mathematical Concept: This involves understanding ratios and how they relate to a total number. The sum of the parts of the ratio must be a factor of 28, or the ratio must simplify to parts whose sum is a factor of 28.

Problem 16: Inequality with Absolute Value

Scenario: Solve the inequality 0 < |x| - 4x < 5.
Question: Which of the following must be true about the value of x?
Mathematical Concept: This requires solving a compound inequality involving an absolute value. It may involve splitting the problem into cases based on the sign of x.

Problem 17: Rental Lodge Expenses

Scenario: A lodge is rented for 10 days at $R per day. Total rent = 10R.
Expenses:
- Travel agency: 10% of total rent.
- Resident manager: 10% of total rent.
- Cleaning service: $100.
Given: $2,000 remained after these payments.
Question: Which equation gives the value of R?
Mathematical Concept: This is an algebra problem involving setting up an equation based on percentages and subtractions from a total.

Problem 18: Set Theory and Two-Digit Numbers

Scenario: Set A = {1, 4, 6, d, e} contains 5 different nonzero digits.
Conditions:
- 25 possible 2-digit numbers can be formed using digits in A. (This implies digits can be repeated).
- 20 of these 2-digit numbers do not contain a repeated digit.
- The sum of these 20 numbers (without repeated digits) is 1,056.
Question: What is the value of d + e?
Mathematical Concept: This is a complex problem involving combinatorics (permutations), number properties (sum of digits in numbers formed), and algebra. The sum of 2-digit numbers formed by distinct digits from a set {a, b, c, d, e} can be expressed systematically.

Problem 19: Sum of Digits and Subtraction

Scenario: The sum of the digits of a positive integer x is 170.
Condition: y is a two-digit positive integer, and x = 1020 - y.
Question: Which of the following could be the value of y?
Mathematical Concept: This problem involves number properties (sum of digits) and algebraic manipulation. It might require testing possible values of y and checking the resulting x.

Problem 20: Function Evaluation and Distinct Values

Scenario: Six distinct real numbers: a₁, a₂, a₃, a₄, a₅, a₆.
Function: f(x) = -|x²|
Question: What is the minimum number of distinct values in the list f(a₁), f(a₂), f(a₃), f(a₄), f(a₅), f(a₆)?
Mathematical Concept: This involves understanding the properties of the function f(x) = -x². Since x² is always non-negative, -x² is always non-positive. Also, (-x)² = x². This means f(x) = f(-x). The minimum number of distinct values depends on how many pairs of numbers in the list are negatives of each other.

Meta-Information: Review Center and Scores

Review Center: Instructions indicate a feature to review and edit up to three answers in the Quantitative Reasoning section before time expires.
GMAT Scores: Mentions of "Unofficial GMAT™ Exam Scores," "Percentile," and "Scaled Score," along with information about Official Score Reports, confirm the context of GMAT preparation.

Summary of Quantitative Reasoning Questions

This document contains a collection of quantitative reasoning problems covering various mathematical concepts. The problems are presented as individual questions, each requiring a specific calculation or logical deduction to solve.

1. Pricing and Cost Calculations

Bus Tour Pricing:
- A bus company charges a flat rate of $2,000 for groups of 40 people or fewer.
- For groups larger than 40, the charge is $2,000 plus $45 for each person exceeding 40.
- Example Problem: Calculate the price difference between a group of 60 people and a group of 30 people.
Discounted Prices:
- A dress's price is discounted twice.
- The first discount results in a price that is 85% of the original price.
- The second discount results in a price that is 60% of the original price.
- Example Problem: Determine what percentage the second discounted price is of the first discounted price.
Marked Price, Discount, and Profit:
- A shopkeeper marks an article at 50% above its manufacturing cost.
- The article is sold at a profit after a discount.
- If the marked price were increased by 20% and the discount percentage remained the same, the profit would double (increase by 100%).
- Example Problem: Find the approximate discount percentage offered.

2. Number Properties and Sequences

Units Digit of an Even Number:
- An even number has a remainder of 1 when divided by 5.
- Example Problem: Determine the units digit of this number.
Integer Expression Formation:
- An integer W is formed by inserting exactly two plus signs between the consecutive digits 1, 2, 3, 4, 5, and 6.
- There must be at least one digit between the plus signs.
- Example Problem: Given the least possible value of W (e.g., 12 + 34 + 56 = 102), find the next least possible value of W.
Fibonacci-like Sequence:
- A sequence is defined by $S_1 = 1$, $S_2 = 3$, and each subsequent term is the sum of the two preceding terms ($S_n = S_{n-1} + S_{n-2}$ for $n > 2$).
- Let a be any term after the third term, and b be the term preceding a.
- Example Problem: Determine the interval in which the ratio $b/a$ lies. (Possible intervals provided: [0, 0.5], [0.5, 1], [1, 1.5]).

3. Averages and Percentages

Average Daily Exercise:
- Janis exercises x minutes per day on Mondays, Wednesdays, and Fridays for 20 weeks.
- She exercises y minutes per day on the other days of the week for 20 weeks.
- Example Problem: Calculate the average number of hours she exercised per day in terms of x and y.
Salary Changes:
- Amin's salary: Decreased by 20% in 2015, increased by 50% in 2016.
- Suri's salary: Increased by 50% in 2015, increased by 6.67% in 2016.
- In 2016, Amin's salary was 10% less than Suri's.
- Example Problem: Determine the percentage difference between Amin's and Suri's salaries in 2014.
Average of Values:
- The average of five values (0.4, 0.8, 1.2, 2.0, and x) is 1.0.
- Example Problem: Find the value of x.
Average of Prime Numbers:
- Example Problem: Calculate the average (arithmetic mean) of all prime numbers less than 30.

4. Financial Mathematics

Compound Interest:
- A nonprofit deposits an amount into a savings account with 2% annual compound interest.
- No further deposits or withdrawals are made for two years.
- The total interest earned over two years is $5,050.
- Example Problem: Calculate the initial deposit amount.
Salary with Commission:
- Rasheed's weekly income = $600 salary + 7% commission on sales exceeding $2,000.
- His income was $810 one week and $1,020 the next.
- Example Problem: Calculate the total amount of his sales for the two weeks.
Investment Value Changes:
- Initial values of Investments A, B, and C are in the ratio 3:6:1 at the start of 2007.
- By the start of 2008: A gained 6%, B lost 8%, C gained 4%.
- Example Problem: Calculate the overall percentage change in the total value of the three investments from the beginning of 2007 to the beginning of 2008.

5. Set Theory and Statistics

Range of Data Sets:
- The range of numbers in list S is 30.
- The range of numbers in list T is 70.
- List V contains all numbers from both list S and list T.
- Example Problem: Determine the least possible range of the numbers in list V.
Investor Distribution:
- A table shows the distribution of investors by investment type (e.g., "stocks only," "long term only").
- Example Problem: Calculate the percentage of investors who are neither "stocks only" nor "long term only."

6. Algebra and Inequalities

System of Equations:
- Given three equations:
  - $-x + y + z = -10$
  - $-x + y - z = -10$
  - $-x - y + z = -10$
- Example Problem: Find the value of $x + y + z$.
Absolute Value Inequality:
- Example Problem: Determine which inequality is equivalent to $0 < |2x + 3|$.
Defined Operations:
- An operation $a \circ b$ is defined as $a \circ b = 1$ for all $a \neq b$.
- Example Problem: Find which expression must be equal to $-(x \circ 1)$, given $x > y > 1$.

7. Combinatorics

Group Formation:
- 12 runners are randomly divided into 2 groups of 6 for a race.
- Example Problem: Calculate the number of different possible groups of 6 runners.

8. Least Common Multiple (LCM)

Integer Multiples:
- W is the least positive integer that is a multiple of 15, 18, 40, and 50.
- Example Problem: Determine which statement about W is true (e.g., range of W).

9. Exam Instructions

Quantitative Reasoning Section:
- Duration: 45 minutes.
- Number of questions: 21.
- Includes time for review and editing.
- Accommodations for extra time will be noted on the timer.
- Instructions to click "Next" to begin.

This document contains a series of quantitative reasoning problems, likely from a standardized test. The problems cover a range of mathematical concepts including:

Algebra and Number Theory: Solving for unknowns, working with inequalities, divisibility rules, factorials, and properties of integers.
Percentages and Profit/Loss: Calculating profit margins, discounts, and effective interest rates.
Combinatorics and Probability: Understanding patterns in arrays and applying statistical concepts like standard deviation.
Word Problems: Applying mathematical principles to real-world scenarios involving investments, sales, and employee salaries.

The document also includes administrative text related to exam completion, whiteboard confirmation, and review center instructions, which are not part of the quantitative reasoning problems themselves.

Here's a breakdown of the distinct quantitative reasoning problems presented:

Quantitative Reasoning Problems

1. Mutual Funds Rating

Problem: A "select list" of mutual funds has a certain number of 5-star and 4-star rated funds. The remaining 300 funds have 3-star ratings. The problem asks for the total number of funds on the list, given that a fraction (represented by '}') of the funds have 5-star ratings and 2 of the remaining funds have 4-star ratings.
Key Information:
- 300 funds have 3-star ratings.
- A fraction '}' of the total funds have 5-star ratings.
- 2 of the funds after the 5-star funds are removed have 4-star ratings.
Goal: Determine the total number of funds on the list.

2. Fruit Shop Profit Calculation

Problem: A shopkeeper buys various exotic fruits (Dekopon, Sekai-ichi apples, pears) and sells them through different pricing strategies. The goal is to calculate the total percent profit.
Key Information:
- Purchases:
  - 12 dozen Dekopon fruits for $1,728.
  - 27 dozen Sekai-ichi apples for $5,832.
  - 300 exotic pears for $900.
- Sales Offers:
  - Offer 1: $100 for 3 Dekopons, 3 Sekai-ichi apples, and 3 pears. All Dekopons sold through this.
  - Offer 2 (after Dekopons sold out): 3 Sekai-ichi apples and 3 pears for $65. All remaining pears sold through this.
  - Remaining Sekai-ichi apples sold individually at $20 each.
Goal: Calculate the total percent profit to the nearest 0.1%.

3. Integer Inequality

Problem: Given that x and y are integers, with y ≥ 3 and x < 2 - 5y, find the greatest possible value of x.
Key Information:
- x, y are integers.
- y ≥ 3.
- x < 2 - 5y.
Goal: Find the maximum possible integer value for x.

4. Array Pattern

Problem: An array is filled with integers following a specific pattern. A 2x2 array uses integers 1-4, and a 3x3 array uses integers 1-9. The pattern involves filling sequentially, moving down and then left. The question asks for the entry at a specific location (R_s, S) if this pattern continues.
Pattern Description:
- (R1, C1) is 1.
- A 2x2 array starts at (R1, C2) with 2, filling down to (R2, C2) and left to (R2, C1).
- A 3x3 array starts at (R1, C3) with 5, filling down to (R3, C3) and left to (R3, C1).
Goal: Determine the value at location (R_s, S) based on the continuation of this pattern. (Note: The specific values of 's' and 'S' are not provided in the excerpt, but the pattern logic is described).

5. Real Number Inequality

Problem: Given that x and y are real numbers such that x² < x and |y| < 1, determine the range between which x + y must lie.
Key Information:
- x, y are real numbers.
- x² < x.
- |y| < 1.
Goal: Find the interval (lower and upper bounds) for the value of x + y.

6. Compound Interest Rate

Problem: An investment earns compound interest, compounded semiannually. The interest rates are 20% for the first year and 10% for the second year. The question asks for the effective simple annual interest rate for the entire 2-year period.
Key Information:
- Compounded semiannually.
- Year 1 rate: 20% (applied semiannually).
- Year 2 rate: 10% (applied semiannually).
- Investment duration: 2 years.
Goal: Find the closest effective simple annual interest rate.

7. Candy Bar Demand

Problem: The demand (D thousand cartons) for a candy bar is related to its price (p dollars per carton). The product of D and 100p is constant (250,000) when the price is between $4 and $8. The question asks for the demand when the price is $5.
Key Information:
- Demand D (in thousands of cartons) and price p (in dollars).
- D * 100p = 250,000 for $4 ≤ p ≤ $8.
Goal: Calculate D when p = $5.

8. Algebraic Expression Evaluation

Problem: Given x = 1,404 and y = 1,296, calculate the value of √(x² - 2). (Note: The expression appears to be √(x² - 2), not VX² - 2 as written, assuming standard mathematical notation).
Key Information:
- x = 1,404
- y = 1,296
Goal: Evaluate √(x² - 2).

9. Integer Raised to the Third Power

Problem: Identify which of the given options (options not provided in the excerpt) can be represented as an integer raised to the third power (a perfect cube).
Goal: Recognize and select a perfect cube from a list of choices.

10. Units Digit of (p²)¹⁰⁰

Problem: Determine the units digit of (p²)¹⁰⁰, where p is a positive odd integer that is not a multiple of 5.
Key Information:
- p is a positive odd integer.
- p is not a multiple of 5.
Goal: Find the units digit of (p²)¹⁰⁰.

11. Factorial Identity

Problem: Evaluate the truth of the statement (x + 1)! = x! for any positive integer x.
Key Information:
- x is a positive integer.
Goal: Determine if the given factorial equation is true.

12. Expressing r² in terms of s and t

Problem: Given the equations r = u² + v², s = 2uv, and t = u² - v², express r² in terms of s and t.
Key Information:
- r = u² + v²
- s = 2uv
- t = u² - v²
Goal: Find an equation for r² using only s and t.

13. Evaluating 1|2

Problem: Determine the value of 1|2. (Note: The notation "1|2" is ambiguous. It could refer to a specific mathematical function or operation not commonly known, or it might be a typo. Assuming it's a standard mathematical expression, clarification would be needed).
Goal: Calculate the value represented by "1|2".

14. Salary and Bonus Calculation

Problem: Find the original salary amount where a 10.0% increase plus a $375 bonus equals a 9.5% increase plus a $500 bonus.
Key Information:
- Let S be the original salary.
- Equation: 1.10S + $375 = 1.095S + $500.
Goal: Solve for S.

15. Apple Vendor Profit

Problem: A street vendor buys apples and sells them in two different ways: in bags of 3 or individually. Calculate the average profit per apple.
Key Information:
- Bought: 100 apples at $0.20 each.
- Sold:
  - 75 apples in bags of 3 (25 bags total) for $1.00 per bag.
  - Remaining 25 apples for $0.40 each.
Goal: Calculate the average profit per apple.

16. Data Set within Standard Deviations (Chebyshev's Inequality)

Problem: Using Chebyshev's Inequality (proportion within k standard deviations is at least 1 - 1/k²), determine the minimum number of students (t) whose scores fall between 62 and 86 on a spelling test, given the mean and standard deviation.
Key Information:
- Total students: 64.
- Mean score: 74.0.
- Standard deviation: 6.0.
- Scores of interest: between 62 and 86.
- Chebyshev's Inequality: Proportion within k standard deviations ≥ 1 - 1/k².
Goal: Find the minimum value of t (number of students in the range).

17. Coat Price Reduction

Problem: The original price of a coat is reduced by 40%, and then the reduced price is reduced by another 40%. The final price is $72. Find the original price.
Key Information:
- First reduction: 40%.
- Second reduction: 40% (applied to the already reduced price).
- Final price: $72.
Goal: Determine the original price of the coat.

18. Sum of Three-Digit Integers Divisible by 9

Problem: The sum of two three-digit integers, 2K5 and 6N3, is divisible by 9. K and N are the tens digits. Find the greatest possible value of K + N.
Key Information:
- Integers: 2K5 and 6N3.
- K and N are digits (0-9).
- (2K5 + 6N3) is divisible by 9.
Goal: Maximize the value of K + N.

19. Employee Salary Median

Problem: A company hires new employees with salaries less than $30,000. Determine the minimum number (n) of new hires needed so that the median salary of all employees drops below $30,000.
Key Information:
- Current employees: 60.
- 40% earn < $30,000.
- 60% earn > $30,000.
- New employees earn < $30,000.
- n is an odd integer.
Goal: Find the minimum odd integer n to make the median salary < $30,000.

20. Pool Filling Rate

Problem: Three pumps fill a pool at different constant rates (2, 4, and 8 hours individually). If they work together, what fraction of the pool is filled by the fastest pump?
Key Information:
- Pump 1 fills pool in 2 hours (rate = 1/2 pool/hour).
- Pump 2 fills pool in 4 hours (rate = 1/4 pool/hour).
- Pump 3 fills pool in 8 hours (rate = 1/8 pool/hour).
- All pumps work simultaneously.
Goal: Determine the fraction of the pool filled by the fastest pump (Pump 1) during the time it takes for all three to fill the pool.

21. Integers Divisible by 3 and 5

Problem: Count the number of integers between 1 and 300 (inclusive) that are divisible by both 3 and 5.
Key Information:
- Range: 1 to 300, inclusive.
- Condition: Divisible by both 3 and 5 (i.e., divisible by 15).
Goal: Find the count of such integers.

The document also contains repeated instances of some problems and extraneous text related to exam administration and security checks.

This document contains a collection of quantitative reasoning problems, likely from a test preparation resource like the GMAT. The problems cover a range of mathematical concepts, including algebra, arithmetic, percentages, ratios, and data interpretation.

Main Idea

The primary purpose of this content is to present a series of quantitative reasoning questions designed to assess a user's mathematical and problem-solving abilities. It also includes information about the GMAT exam, score reporting, and the functionality of a review center.

Key Concepts and Problem Types

The questions presented touch upon various mathematical areas:

Algebraic Manipulation:
- Simplifying expressions involving variables and square roots (e.g., x - √x).
- Solving equations with absolute values (e.g., |x² - 12| = 8).
- Factoring and solving polynomial equations (e.g., x⁴+1=(x²+cx+1)(x²-cx+1)).
- Evaluating expressions with given variable values (e.g., √x² - y² where x = 1404 and y = 1296).
- Solving equations involving exponents and roots (e.g., a⁴ - b⁴ = 65).
Arithmetic and Number Theory:
- Word problems involving ratios, proportions, and percentages (e.g., law firm contributions, corporate workforce composition, milk and water mixtures).
- Problems involving remainders when dividing integers (e.g., finding an integer n between 40 and 50).
- Calculations with decimals and scientific notation (e.g., evaluating the sum S).
- Order of operations (e.g., 4[2(4 -6) + 3(-3 -2)]).
Statistics:
- Understanding and comparing standard deviation.
Geometry/Measurement:
- Problems involving percentages and calculating differences in measurements (e.g., curtain length after washing).
Probability:
- Calculating probabilities involving coin tosses and matching outcomes.
Problem Solving & Logic:
- Multi-step word problems requiring careful reading and breakdown of information (e.g., Ms. Waters' family visits, student line formation, corporate jet cost recovery).
- Interpreting data from tables (e.g., employee statistics for Corporations A and B).
- Calculating financial metrics like cost of capital and value added.

GMAT Exam Information

Review Center Functionality: Users can review and edit up to three answers in the Quantitative Reasoning section before the time expires.
Score Reporting: Official GMAT scores are available on mba.com within 3-5 days, with email notification upon availability.

Specific Examples of Problems

Law Firm Problem: Calculates individual partner contributions based on total employees, ratios, and total charity amount.
Corporate Workforce Problem: Determines the percentage of salaried employees in the combined workforce of two corporations using data from a table.
Student Line Problem: Uses positional information (rank from top, relative positions) to determine the total number of students in a line.
Corporate Jet Problem: Calculates the minimum number of flights needed to recoup the purchase cost, considering flight costs versus commercial airfare savings.
Mixture Problem: Involves calculating ratios for mixing solutions (milk and water) to achieve a target concentration.

Quantitative Reasoning Section Summary

This document outlines the instructions and presents a series of sample questions for the Quantitative Reasoning section of an exam, likely the Graduate Management Admission Test (GMAT).

Section Overview

Time Limit: 45 minutes for the entire section, including reviewing and editing answers.
Accommodations: Any time accommodations will be indicated on the exam clock timer.
Number of Questions: 21 questions.

Sample Quantitative Reasoning Questions

The document includes various sample questions covering different areas of quantitative reasoning. These are presented as examples of the types of problems encountered in the section.

1. Data Interpretation (Table Analysis)

Scenario: A table shows the number of vacation days earned by 1,100 employees.
Problem Type: Given the average number of vacation days earned per employee, find the value of an unknown variable (x) in the table.

2. Rate, Time, and Distance (Average Speed)

Scenario: An airplane travels between two cities.
Problem Type: Calculate the total round trip time given different average speeds for each leg of the journey, using information from a previous scenario with a constant average speed.

3. Number Theory and Algebra

Scenario: Four different positive integers (x, y, r, s) with a given condition (r < s and x^r = y^s).
Problem Type: Determine which of the given statements about the relationship between these integers must be true (e.g., divisibility).

4. Series and Summation

Scenario: A formula for the sum of the first 'n' integers is provided: 1 + 2 + 3 + ... + n = n(n + 1) / 2.
Problem Type: Evaluate or determine the properties of a sum 'S' which is composed of several series, potentially involving fractions or reciprocals of sums.

5. Exponents and Algebra

Scenario: Given relationships between variables using exponents (e.g., 2^a = s and s^b = 16).
Problem Type: Solve for the value of a product of exponents (e.g., ab).
Scenario: An algebraic equation involving squares (e.g., 2x² - y² = 2xy).
Problem Type: Simplify or find the value of an expression involving the variables (e.g., (x + y)²).

6. Arithmetic and Problem Solving (Combinations)

Scenario: A fruit stand sells apples and oranges at specific prices.
Problem Type: Determine the maximum number of pieces of fruit a customer can buy for a specific total cost.

7. Statistics (Normal Distribution)

Scenario: Data on the lengths of adult male cougars is approximately normally distributed with a given mean and standard deviation.
Problem Type: Calculate the approximate percentage of the population that falls within a specific range of values (between two standard deviations from the mean).

8. Word Problems (Algebraic Relationships)

Scenario: Information about exam scores of two individuals (Shyam and Ravi), passing scores, and maximum possible scores.
Problem Type: Determine the passing percentage of the exam based on the given relationships.

9. Optimization (Least Number of Items)

Scenario: An awards dinner with a specific number of honorees and guests, and a maximum seating capacity per table. Honorees and guests must sit separately.
Problem Type: Calculate the minimum number of tables required to seat everyone according to the constraints.

10. Probability

Scenario: A fair standard die is rolled multiple times.
Problem Type: Calculate the probability of a specific event occurring at least a certain number of times (e.g., at least two rolls of a "4" in four rolls).

11. Unit Conversion and Place Value

Scenario: The thickness of a honeycomb wall is given in millimeters.
Problem Type: Determine the place value of a digit when the measurement is converted to meters.

12. Percentage and Area Calculation

Scenario: Volunteers clean portions of a park over two days, with the second group cleaning a percentage of the remaining area.
Problem Type: Estimate the total area of the park given the area cleaned by the second group.

13. Number Theory (Remainders)

Scenario: A number (85) is divided by a positive integer 'n' and by '2n', with specific remainders given.
Problem Type: Solve for the value of the integer 'n'.

14. Radicals and Algebra

Scenario: Given values for x and y.
Problem Type: Calculate the value of an expression involving a square root (e.g., sqrt(x² * y)).

15. Cost Comparison and Ratios

Scenario: Two secretaries prepare a manuscript, each with a different time and hourly rate.
Problem Type: Determine for which given ratios of their preparation times (x to y) it would be less expensive to hire Secretary B.

16. Polynomials and Degree

Scenario: An algebraic expression involving factorials and variables.
Problem Type: Determine the degree of the polynomial that the expression is equivalent to for integers n > 1.

17. Percentage Increase

Scenario: A positive number 'y' is increased by a fixed amount (5).
Problem Type: Calculate the resulting percent increase in 'y'.

18. Systems of Equations (Table Data)

Scenario: A table represents the sum of base salaries (in thousands of RMB) for employees at different levels.
Problem Type: Solve a system of equations derived from the table to find the base salary of employees at a specific level (Level III).

Exam Completion and Whiteboard Confirmation

Physical Whiteboard: Instructions are provided for users who utilized a physical whiteboard. They must erase it completely and show both sides to the camera before proceeding.
Confirmation: Users confirm they have erased the whiteboard or did not use one. Failure to comply can lead to score cancellation.
Ending the Exam: A final confirmation step is required to officially end the exam.

Quantitative Reasoning Section Summary

This document contains a collection of practice problems and instructions related to the Quantitative Reasoning section of an exam, likely the GMAT. The problems cover a range of mathematical concepts including algebra, arithmetic, ratios, percentages, inequalities, and number theory.

Problem Types and Concepts Covered:

The problems presented touch upon various quantitative reasoning skills:

Algebra and Word Problems:
- Calculating revenue fractions based on sales volume and price differences.
- Solving for variables in algebraic expressions.
- Setting up and solving systems of inequalities based on real-world constraints.
- Determining the original quantity in a word problem involving distribution and equal outcomes.
- Simplifying algebraic fractions.
- Finding the least value of a variable that satisfies a condition.
- Calculating the price of an item in a different currency using exchange rates.
- Determining the least possible value of a median given a list of integers with specific properties.
- Solving algebraic equations with multiple variables.
- Identifying expressions that cannot be prime numbers.
- Calculating the total number of people in a group based on selection possibilities.
- Calculating profit percentage from a series of transactions involving multiple types of goods.
- Determining the ratio of mixtures needed to achieve a target concentration.
- Counting integers within a range that are divisible by specific numbers.
Number Theory and Properties:
- Understanding remainders when powers of numbers are divided by another number.
Data Interpretation:
- Calculating percentages from a workforce distribution table.
Estimation:
- Approximating the value of an expression involving exponents.
Fractions and Decimals:
- Identifying equivalent decimal and fractional representations.

Exam Instructions and Information:

Section Timing: The Quantitative Reasoning section is timed at 45 minutes.
Number of Questions: There are 21 questions in this section.
Review and Editing: Candidates can review and edit up to three (3) answers before the section time expires.
Accommodations: Extra time accommodations are noted on the exam clock.
Score Reporting: Official GMAT scores are available on mba.com within 3-5 days, with email notification provided.

Specific Problem Examples (Illustrative):

Revenue Calculation: A problem asks for the fraction of total revenue from paperback sales, given initial hardcover sales, subsequent sales mix (3x paperback to 1x hardcover), and the price ratio (paperback price is 2/3 of hardcover price).
System of Inequalities: A scenario involving allocating funds to personal loans (x) and car loans (y) requires setting up inequalities based on a total budget limit (at most $45,000) and a minimum allocation ratio (car loans at least twice personal loans).
Mixture Problem: Two vessels (X and Y) with different milk concentrations are mixed. Given an initial scenario where equal quantities result in 45% milk, the problem asks for the ratio needed to achieve 55% milk.
Profit Calculation: A shopkeeper buys fruits in bulk and sells them through various combination offers and individual sales. The task is to calculate the total percent profit.

This summary provides an overview of the content presented, highlighting the types of quantitative reasoning skills tested and the administrative information provided for the exam section.

Quantitative Reasoning Summary

This document presents a collection of quantitative reasoning problems, covering various mathematical concepts and problem-solving scenarios. The problems are presented in a question-and-answer format, typical of standardized tests.

Problem Breakdown:

Here's a summary of the individual quantitative reasoning problems presented:

Multiples and Fractions:
- Scenario: A list of integers has specific counts of multiples of 5, 6, and both.
- Question: What fraction of the integers are multiples of both 5 and 6 compared to those that are multiples of neither?
Algebraic Relationships:
- Scenario: An initial increase of a fraction of 'x' results in 'y'. A subsequent increase of another fraction of 'x' to 'y' results in 1,056.
- Question: What is the value of the second fraction of 'x'?
Integer Properties and Sums/Products:
- Scenario: Five different non-zero integers from -5 to 5 have a positive sum and a positive product.
- Question: Which of the given options could represent the number of negative integers in the set?
Inequalities with Integers:
- Scenario: 'x' and 'y' are integers with constraints: y ≥ 3 and x < 2 - 5y.
- Question: What is the greatest possible value of 'x'?
Set Theory and Digit Combinations:
- Scenario: Set A = {1, 4, 6, d, e} contains 5 distinct non-zero digits. 25 possible 2-digit numbers can be formed. 20 of these do not have repeated digits and sum to 1,056.
- Question: What is the value of d + e?
Data Interpretation (Table):
- Scenario: A table shows the annual rainfall (in inches) for five cities.
- Question: What is the median annual rainfall for these five cities?
Percentage and Mixture Problem:
- Scenario: Mr. Das repeatedly drinks a percentage of a bottle's contents and replaces it with aloe vera juice over 20 days. The percentage increases by 5% each day (5%, 10%, 15%, ..., 100%).
- Question: How many milliliters of aloe vera juice did he drink over these 20 days?
Percentage Decrease and Difference:
- Scenario: A curtain's length decreases by 'b%' after washing. 'x' is the difference in length before and after washing. 1 ≤ b ≤ 3.
- Question: Which expression gives the greatest value of 'x' for a given length 'L'?
Linear Equations and Optimization:
- Scenario: Elodie buys Type A gifts ($85 each) and Type B gifts ($125 each). A total of 20 gifts were purchased for a total cost 'x', where 1,800 < x < 2,000.
- Question: What is the difference between the maximum and minimum possible number of Type B gifts?
Percentage and Pricing:
- Scenario: A bookstore sells books at a price that is 22% of the selling price higher than the wholesale price.
- Question: Dividing the wholesale price by which value gives the most accurate estimate of the selling price?
Diophantine Equations:
- Scenario: 'p' and 'q' are positive integers, 'p' is odd, and 14p - 5q - 18 = 0.
- Question: What is the least possible value of the product pq?
Estimation:
- Question: Which value is closest to (310)(3312)?
Percentage Increase Calculation:
- Scenario: A stock price 'x' increases by 'r%' and then by '$y'.
- Question: What is the total of these two increases as a percent of 'x'?
Factors and Integers:
- Scenario: 'm' is a positive integer, and 21 is its largest factor other than 'm' itself.
- Question: What are the possible values of 'm'?
Percentage Difference:
- Scenario: Data on men (3,829) and women (4,580) in a study.
- Question: Approximately what percent greater is the number of women than the number of men?
Probability (Conditional):
- Scenario: Cindy has 3 keys and 2 locks (J, K). One key opens J only, one opens K only, and one opens neither. She picks a key at random, and it doesn't open Lock J.
- Question: What is the probability that it will open Lock K? (This problem appears twice).
Exam Instructions and Review:
- Information: Details about the Quantitative Reasoning section, including time limits (45 minutes), number of questions (21), and the ability to edit up to three answers.
Currency Conversion:
- Scenario: 1 British pound = 1.47 United States dollars.
- Question: How many British pounds is 100 United States dollars, to the nearest pound?
Ratios and Mixtures:
- Scenario: Alloy X (zinc:copper = 5:7) and Alloy Y (zinc:copper = 8:7) are combined in equal quantities to form Alloy Z.
- Question: What is the ratio of zinc to copper in Alloy Z?
Data Interpretation (Graph):
- Scenario: A graph shows monthly revenue and expenses for a business in 2018.
- Question: For how many months did the business have a monthly profit > $1,000 or a monthly loss > $1,000?
Solving Equations:
- Scenario: The equation x² - 1 = x.
- Question: How many solutions does this equation have?
Systems of Linear Equations:
- Scenario: A system of two linear equations: 2x + 3y = 8 and 6x + 9y = k.
- Question: For what value of 'k' does the system have more than one solution?

Note: The recurring Chinese text "正通过长期验证码被控,安全级别低建议改为单次验证码" and "8 #D ... 79正在远程本机 .. ]" appear to be system messages or unrelated annotations and have been excluded from the summary of the quantitative reasoning content.

This document contains a collection of quantitative reasoning problems, likely from a standardized test preparation context (e.g., GMAT). The problems cover a range of mathematical topics including algebra, probability, statistics, percentages, ratios, and number theory.

Here's a structured summary of the types of problems presented:

Quantitative Reasoning Problems

This section outlines various quantitative reasoning questions, categorized by the mathematical concepts they assess.

Algebra and Inequalities

Absolute Value Equations and Inequalities:
- Problems involving the equation |x - n| = 6.5 and determining restrictions on n for x > 10.
- Problems requiring the solution of inequalities like |5 - 2x| ≤ 7 and representing them on a number line.
Algebraic Expressions:
- Evaluating expressions given specific values for variables (e.g., x = 3, y = -3 in x² + 2xy + y² + x - 10).
- Simplifying or manipulating expressions involving variables and operations (e.g., (n - 1)! + n! + (n + 1)! / n!).
Variable Substitution:
- Analyzing the effect of substituting one variable for another in an expression, given constraints on their values (e.g., replacing y with x when 0 < x < y < z < 1).

Probability

Independent Events:
- Calculating the probability of a computer winning the "Game of 24" by independently replacing symbols with operations (+, -, x, ÷). This involves multiplying probabilities of independent choices for each symbol.
- The expression (4 A 2) x (3 @ 1) is used as an example, with given probabilities for replacing A and @ with different operations.

Statistics and Averages

Arithmetic Mean:
- Calculating the average number of hours exercised per day based on daily exercise minutes (x and y) over a period of weeks, considering different exercise durations on weekdays and weekends.

Percentages and Ratios

Profit and Selling Price:
- Problems involving calculating the percentage increase in selling price required to change the gross profit margin from 25% to 33% of the selling price.
- Problems where gross profit increases due to reduced production cost while selling price remains constant, requiring calculation of the new production cost.
Business Settlements and Partnerships:
- Calculating the total amount received by silent partners in a business settlement, given the controlling partner's share, the silent partners' shares of the remainder, and the combined amount received by the controlling partner and one silent partner.
Investment Ratios and Profit Sharing:
- Determining the ratio of profit sharing at the end of the year based on initial investment ratios, subsequent changes in investment amounts, and the duration of investment. Profit sharing is weighted by both investment amount and time.
Mixture Problems:
- Calculating the cost per kilogram of one type of nut (cashews) in a mixture, given the quantities of two types of nuts (cashews and Brazil nuts), their selling price per bag, the relationship between their total costs, and the overall profit margin.

Number Theory and Properties of Integers

Factorials:
- Determining the primality of numbers expressed using factorials (e.g., P(11) + 2, P(11) + 4, P(11) + 5, P(11) + 7, where P(n) is the product of integers from 1 to n).
- Finding an integer n for which a factorial-based expression falls within a specific range (e.g., between 16 and 17).
Parity (Even/Odd):
- Identifying which algebraic expression involving a positive integer n must always result in an even integer.
Prime Numbers:
- Assessing the primality of numbers derived from factorial expressions.
Number Representation:
- Calculating the sum of digits of a large number resulting from squaring a sum of two numbers expressed in scientific notation (e.g., (8 x 10⁸ + 5 x 10⁵)²).
Basic Arithmetic Operations:
- Evaluating expressions involving exponents and subtraction (e.g., 2¹² - 20², 6⁶ - 2³).

Combinatorics (Counting Principles)

Permutations with Repetition:
- Calculating the number of distinct 5-letter strings that can be formed from a set of letters (e.g., A, C, L, L, O) with a specific constraint (two 'L's separated by at least one other letter).

Word Problems and Applications

Fruit Stand Sales:
- Determining the maximum number of pieces of fruit (apples and oranges) a customer can purchase for a specific total cost, given individual prices.
Health Insurance Statistics:
- Calculating the approximate percentage increase in the number of children without health insurance over a period, given overall increases in uninsured populations and the proportion of uninsured who are children.

Exam Instructions and Review

Section Overview:
- Information about the Quantitative Reasoning section, including time limits (45 minutes), number of questions (21), and the ability to review and edit answers.
Review Center Functionality:
- Details on how to review questions and edit a limited number of answers before the section time expires.
Score Reporting:
- Information regarding the availability and delivery of official GMAT scores.

This document contains a collection of quantitative reasoning problems, each requiring the application of mathematical principles to solve. The problems cover various topics including algebra, arithmetic, percentages, probability, and data interpretation.

Summary of Quantitative Reasoning Problems

Here's a breakdown of the problems presented:

1. Pricing and Cost Calculation

Movie Theater Scenario:
- Problem: Determine the cost of 1 drink and 1 box of popcorn given the total spending of two individuals on drinks and popcorn.
- Details: Fixed prices for drinks and popcorn, taxes included. Raphael spent $10.50 on 3 drinks and 1 popcorn. Maria spent $8.50 on 1 drink and 2 popcorns.
- Goal: Calculate the cost for Bill buying 1 drink and 1 popcorn.
Computer Game Profit Margin:
- Problem: Find the old selling price of a computer game when the gross profit margin increased from 20% to 25% of the cost, with the new selling price being $80.00 and the cost remaining constant.
- Details: Profit is a percentage of the cost. New selling price = $80.00.
- Goal: Calculate the original selling price.

2. Time and Event Counting

Vacation Rain Days:
- Problem: Determine the total number of days Chris was on vacation.
- Details: Rain occurred on 11 days, either in the morning or afternoon (not both). No rain on 16 mornings and 13 afternoons.
- Goal: Calculate the total vacation days.

3. Number Theory and Combinatorics

Prime Number Pairs:
- Problem: Find the number of possible pairs of prime numbers (A, B) such that A < B < 40 and their difference (B - A) is also a prime number.
- Details: A and B are prime numbers less than 40.
- Goal: Count the valid (A, B) pairs.

4. Algebraic Expressions and Factorials

Factorial Expression Equivalence:
- Problem: Determine which expression is equivalent to (6!)m (5!)n, where m and n are integers.
- Details: Involves factorials and integer exponents.
- Goal: Simplify or find an equivalent form of the given expression.

5. Rental Income and Pricing

Apartment Rental Income:
- Problem: Calculate the monthly rent for the less expensive apartment.
- Details: Total rent received last year was $16,400 for two apartments. One apartment's rent was $100/month more than the other. The more expensive apartment was vacant for 2 months; the less expensive was rented all year.
- Goal: Find the monthly rent of the cheaper apartment.

6. Credit Card Expenses

Total Credit Card Service Expense:
- Problem: Find an expression for the total service expense of a credit card over a period of time.
- Details: Initial fee = $x$. Monthly service charge = $y$. Duration = $z$ years.
- Goal: Formulate an expression for total cost (initial fee + service charges).

7. Defined Operations

Operation @:
- Problem: Determine which properties hold true for a custom-defined operation a @ b = (a)(b) + 2(a + b).
- Details: Properties to check include (-a) @ (-b) = a @ b and (-a) @ b = -(a @ b).
- Goal: Verify the given mathematical statements about the operation.

8. Variable Relationships

Variable Equivalence:
- Problem: Express variable M in terms of variable S, given relationships involving S and M.
- Details: Given: S = 1, S = 2, and M + M + M = 1. (Note: The problem statement seems to have a contradiction with S=1 and S=2). Assuming a typo and focusing on the structure.
- Goal: Find M = f(S).

9. Averages and Data Sets

Average of Numbers:
- Problem: Find the value of X6 given the average of 10 numbers, the average of the first 5, and the average of the last 5 (with X6 being part of both subsets).
- Details: Average(X1...X10) = 92. Average(X1...X5) = 70. Average(X6...X10) = 110.
- Goal: Calculate X6.

10. Inequalities and Integers

Least Positive Integer:
- Problem: Find the smallest positive integer 'n' that satisfies the inequality (2)-n < 1.
- Details: Inequality involving an exponent.
- Goal: Find the minimum positive integer 'n'.

11. Depreciation Calculation

Machine Depreciation:
- Problem: Calculate the depreciation amount for the second year.
- Details: Depreciation for years 1, 2, and 3 are fractions (1/5, 1/5, 1/4) of the total depreciation. Depreciation in year 3 was $600 less than in year 1.
- Goal: Find the depreciation amount for the second year.

12. Business Revenue Projection

Monthly Revenue Growth:
- Problem: Project the monthly revenue for the third month.
- Details: Monthly revenue increases by 10% of the previous month's revenue. Revenue for the first month is $40,000.
- Goal: Calculate revenue for the third month.

13. Number Theory - Remainders

Remainder of 3x divided by 5:
- Problem: Determine the possible remainders when 3x is divided by 5, given that x is an even integer.
- Details: x is even.
- Goal: List possible remainders.

14. Data Interpretation (Graph)

Stock Trading Volume:
- Problem: Approximate the total million shares of stock traded last week based on a provided graph.
- Details: Requires reading values from a bar graph showing "millions of shares traded" per day/period.
- Goal: Sum the values from the graph.

15. Ratio and Proportion

Solvent Mixture:
- Problem: Calculate the amount of water needed to make a specific volume of solvent.
- Details: Solvent is a mixture of degreaser and water in a 5:8 ratio (by volume). Total solvent volume = 520 liters.
- Goal: Find the volume of water required.

16. Set Theory / Counting

Card Collection:
- Problem: Count the number of cards that are either red OR labeled 1.
- Details: Three red cards (1, 1, 2) and three blue cards (1, 2, 2).
- Goal: Apply the principle of inclusion-exclusion or direct counting.

17. Speed, Distance, Time

Bus Route Speed:
- Problem: Calculate the driver's normal driving speed.
- Details: Route distance = 300 km. Driving speed was 20 km/h slower than normal. This resulted in a 45-minute longer travel time.
- Goal: Find the normal speed.

18. Decimal Representation

Decimal Digits:
- Problem: Determine the tenths digit of a number x = 0.d1d2d3d4.
- Details: d1, d2, d3, d4 are non-zero digits. Relationships: d1 = 4d2 = 2d3 = 8d4.
- Goal: Find d1.

19. Algebraic Simplification

Expression Simplification:
- Problem: Simplify the expression 1 - x4 / (1 - x) given x = 0.01.
- Details: Algebraic manipulation and substitution.
- Goal: Find the value of the simplified expression.

General Instructions & Context

Review Center: Users can review and edit up to three answers in the Quantitative Reasoning section before time expires.
Exam Instructions: The Quantitative Reasoning section has a time limit (e.g., 45 minutes) and a specific number of questions (e.g., 21).

This summary categorizes the problems and highlights the key information and objectives for each.

Quantitative Reasoning Section Summary

This document contains a collection of problems from the Quantitative Reasoning section of an exam. The problems cover a range of mathematical concepts, including probability, combinatorics, algebra, percentages, rates, sequences, and statistics.

Problem Categories and Key Concepts:

1. Probability:

Problem: Rolling a fair standard die four times and determining the probability of at least two rolls being a "4".
Concept: Basic probability, independent events, complementary probability (calculating the probability of the opposite event and subtracting from 1).

2. Combinatorics (Counting Principles):

Problem: Assigning 7 students to 3 projects (A, B, C) with specific numbers of students per project (2 for A, 3 for B, 2 for C).
Concept: Permutations with repetitions (multinomial coefficients). The formula for this is n! / (n1! * n2! * ... * nk!), where n is the total number of items and n1, n2, ... nk are the counts of each distinct group.

3. Mixture and Percentage Problems:

Problem: Calculating the amount of aloe vera juice consumed over 20 days, involving daily percentage increases in consumption and replacement with aloe vera juice.
Concept: Sequential percentage changes, arithmetic progression (for the percentage increase), and calculating cumulative amounts.

4. Algebra and Equation Manipulation:

Problem 1: Substituting one equation into another to find an expression for 's' in terms of 't'.
- Given: s = 2r + 1 and r = 1 - 3t
- Concept: Substitution.
Problem 2: Simplifying algebraic expressions involving square roots.
- Given: Vx - Vx where x > 1.
- Concept: Properties of exponents and radicals.
Problem 3: Simplifying a complex algebraic fraction.
- Given: ptr m35-5v-1 p-35275 m2 r-3
- Concept: Exponent rules (product of powers, quotient of powers, negative exponents).
Problem 4: Determining conditions for a quadratic equation to have no real solutions.
- Equation: x^2 + 5px + 16 = 0
- Concept: Discriminant of a quadratic equation (b^2 - 4ac). For no real solutions, the discriminant must be negative.

5. Rates and Proportions:

Problem: Comparing the production rates of two machines (A and B) and calculating the output of Machine B over a specific time.
- Machine A rate: x nails every t hours.
- Machine B rate: 1/3 of Machine A's rate per hour.
- Concept: Unit rates, proportional reasoning.

6. Profit and Selling Price Calculations:

Problem: Determining the old selling price of a pump given changes in gross profit percentage and the new selling price.
- Concept: Profit margin calculations (Profit = Selling Price - Cost), percentage increase.

7. Arithmetic Series:

Problem: Calculating the difference between the sum of the first 250 positive even integers (x) and the sum of the first 250 positive odd integers (y).
- Concept: Formulas for the sum of arithmetic series, properties of even and odd numbers.

8. Number Properties and Digit Analysis:

Problem: Finding the tens digit of the sum of 24 integers, where each integer ends in 5, and their tens digits are distributed among 0, 1, and 2.
- Concept: Place value, properties of addition, modular arithmetic (focusing on the tens digit).

9. Absolute Value Equations:

Problem: Finding all possible values of x given an absolute value equation.
- Equation: |x^2 - 12| = 8
- Concept: Solving absolute value equations by considering positive and negative cases.

10. Word Problems involving Quantities and Revenue:

Problem: Calculating the fraction of total revenue from paperback copies sold, given initial hardcover sales, subsequent paperback/hardcover sales ratio, and price difference.
- Concept: Ratios, weighted averages, revenue calculation.

11. Problem Solving Strategy/Logic:

Problem: Identifying an expression whose value is less than the value obtained by swapping '3' and '4'.
- Concept: Evaluating expressions, comparing values.

12. Weighted Averages and Grade Calculation:

Problem: Determining the required average score on remaining tests to achieve a specific course grade.
- Concept: Weighted averages, setting up and solving linear equations.

13. Geometry and Volume/Area Calculations:

Problem: Calculating the height of a stack of tiles given their dimensions and how they are arranged end-to-end.
- Tile thickness: 1 inch.
- Tile area: 9 sq inches.
- Row length: 144 inches.
- Concept: Area, length, thickness, volume (implicitly), unit consistency.

14. Statistics (Averages and Distributions):

Problem 1: Finding the least possible number of people born in July in a group, given that July had the most births.
- Concept: Pigeonhole principle, minimizing counts under constraints.
Problem 2: Finding the least possible value of the median of a set of 5 distinct integers, given the greatest integer and the average.
- Concept: Properties of averages, median, minimizing values under constraints.

15. Percentage and Area Calculations:

Problem: Calculating the total area of a park given the percentage cleaned on two consecutive days and the area cleaned on the second day.
- Concept: Sequential percentage decrease, working backward from a known value.

16. Number Theory (Divisibility):

Problem: Counting the number of integers between 10 and 99 (inclusive) divisible by 3 or 7.
- Concept: Inclusion-Exclusion Principle for divisibility.

17. Decimal and Fraction Conversion:

Problem: Identifying which expression is NOT equal to 0.24.
- Concept: Converting fractions and percentages to decimals.

Exam Instructions:

Section: Quantitative Reasoning
Time Limit: 45 Minutes
Number of Questions: 21
Accommodations: Extra time will be indicated on the exam clock timer.

This summary categorizes the problems presented and highlights the core mathematical concepts involved in solving them. It also includes the general instructions for the Quantitative Reasoning section.

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GMAT 高频知识点与核心术语（中英表）

| 知识点类别 | 英文术语 / English Term | 中文释义 | 典型例/常见考点 | |------------------|--------------------------------------|-------------------|-----------------------------------------| | 代数表达式/方程 | Algebraic Expression/Equation | 代数式/方程 | 解一元/二元一次方程，化简，代换，未知数求值 | | 方程组与解的判定 | System of Equations/Roots | 方程组/解的判别 | 唯一解/无解/多解，判别式，参数变化 | | 不等式与绝对值 | Inequality/Absolute Value | 不等式/绝对值 | 解线性/分式不等式，带绝对值的双重区间划分 | | 比例关系与比数 | Ratio/Proportion/Rate | 比例/比例关系 | a:b、$\frac{a}{b}=k$, 单位换算、配比分配 | | 百分比/增长率 | Percent/Rate of Change | 百分比/增长率 | 增长/减少率、倍数、利润率、毛利计算 | | 概率与统计 | Probability/Statistics/Mean/SD | 概率/统计 | 基本概率，相互独立，统计量均值/中位数/极差/方差/标准差 | | 计数与排列组合 | Counting/Permutation/Combination | 计数/排列/组合 | 完全排列，有/无重组，抽签、分组、队列 | | 函数与图像 | Function/Graph | 函数/图像 | 倒数/复合/反函数，函数值、图像最值/交点/斜率 | | 平面与立体几何 | Geometry/Area/Volume/Similarity | 几何/面积/体积 | 各类平面/立体图形公式，相似比，三角形、圆、棱柱 | | 数据解读与判断 | Data Interpretation/Table/Graph | 数据解读/图表 | 阅读图表，趋势、最大最小、分布，表间关系 | | 应用题/建模 | Word Problem/Modeling | 应用题/建模 | 实景翻译为数学问题，列模型/比例/方程 | | 财务投资计算 | Profit/Interest/Investment | 利润/利息/投资 | 单复利，利润率，净增长，合作分红 |

高频Core Terms（GMAT多卷必出）

| 英文术语 | 中文释义 | 高频解读/备注 | |-------------------|------------------|----------------------------| | Variables | 变量、未知数 | $x,y$ 通用，关注条件数量 | | Coefficient | 系数 | 线性相关，系数比较/比例法则 | | Discriminant | 判别式 | 二次方程根的分类标准 | | Remainder | 余数/模 | mod/取模、整数除法常考 | | Compound Interest | 复利 | $A=P(1+\frac{r}{n})^{nt}$ | | Ratio | 比例 | $a:b$ 或分数表达 | | Distribution | 分布，分配 | 数据集统计/概率，利润等 | | Permutation | 排列 | 顺序相关 | | Combination | 组合 | 顺序无关 | | Mean | 平均值 | 数据统计核心 | | Median | 中位数 | 排序后中间值 | | Mode | 众数 | 出现次数最多的数据 | | Standard Deviation| 标准差 | 离散度，常见数据波动 | | Least/Greatest Integer | 最小/最大整数 | 向下/向上取整 |

高频考点与易错点举例

| 题型场景 | 常见英文模式 | 易混误区/典型陷阱 | |--------------------|----------------------------------------|----------------------------------------| | 方程式解/根数判断 | Solve for x/Number of solutions | 忽略判别式/参数特殊值 | | 比例分配/增长率计算 | Shares profit by ratio/Percent increase| 比例顺序、总量分配，混淆增长vs绝对差 | | 绝对值/区间不等式 | $|x|>a$ or $a<|x-b|<b$ | 容易一端区间漏解，符号数轴分错 | | 分数/余数求解 | What is the remainder when ... | 浮点与整数取整，余数计算顺序错误 | | 统计量估算 | Mean, Median, SD range | 忽视极值与集中趋势，SD区分集中/离散数据 | | 排列组合判别 | How many different sets/orders... | 区分排列与组合，分组顺序是否相关 | | 图表数据分析 | According to the table/chart... | 单位换算错，表间数据关系判定失误 | | 应用建模 | Into which equation... | 题意变量定义理解不到位 | | 利息/投资变化 | Compound/Simple interest... | 复利次数与周期错用，年/月转换遗漏 |

Table展示（Markdown标准格式）

| 知识点类别 | 英文术语 | 中文释义 | 常见考查方式/例子 | |:----------------|:------------------------|:-----------------------|:-------------------------------------------| | 代数/方程 | Algebra/Equation | 代数式/解方程 | 联立/一元，根数分类，余数、判别式 | | 比例增长 | Ratio/Proportion/Percent | 比例/比例式/百分比 | 速率/利润/人口增长率/配比分配 | | 数据统计 | Mean/Median/Mode/SD | 均值/中位数/众数/标准差 | 均值=总和÷数量，中位数排序，标准差离散度 | | 概率计数 | Probability/Permutation/Comb | 概率/排列/组合 | 独立、多步、限定条件排列组合 | | 函数/图像 | Function/Graph | 函数理论/图像特征 | 交点/最大最小、单调性、斜率截距 | | 立体/平面几何 | Geometry/Area/Volume | 几何/面积/体积 | 三角、圆、棱柱、相似线段面积体积 | | 数据解读/建模 | Data Interpretation/Modeling | 图表/应用建模 | 表格/柱线趋势，情境列方程/比例 | | 财务投资 | Profit/Interest | 利润/利息/投资回报 | 单复利模型、分红分析、收益率/净变化 |

高频错题&刷题建议

易错考点包括顺序相关（排列）、绝对值区间、分段增长、统计量四选一等，要结合例题反复练。
刷题方法建议：归纳错题陷阱+错因（如“判别式丢根”、“比例式顺序混淆”），专项训练统计→概率→排列组合→函数图像。
数据解读，尤其表格题一致性判别、数值单位换算容易出错，务必慢速逐步推进。

英中术语速查清单（部分）

| 英语 | 中文释义 | 公式/场景示例 | |:-------------|:--------------------|:-----------------------------| | Discriminant | 判别式 | $b^2-4ac$（二次方程根判断） | | Compound Interest | 复利 | $F = P(1 + r/n)^{nt}$ | | Median | 中位数 | 排序后中间值 | | System of Equations | 方程组 | 唯一解/多解/无解 | | Ratio | 比例/比数/配比 | a:b，按比例分配，速率问题 | | Limiting Value | 极限值 | 实际应用的最大/最小解 |

如需专项例题拆解讲解，题型刷题/错题归纳，或者帮你记忆/保存此高频知识表，下次调用刷题更高效！需要随时@我，GMAT考冲稳住节奏！🤗

你准备得越细致，越接近目标分数！遇到瓶颈记得发题、发错题，我帮你分类专项击破！加油冲刺高分！

不等式知识点我没学好，可以给我具体讲讲，配上真题教教我

苡甜甜甜，别担心！不等式是GMAT数学高频考点之一，下面我用你真题中的典型例子，结合详细讲解和真题解析，帮你把GMAT常考不等式知识细化梳理清楚！
全程配例题，“基础+进阶解法”+中英术语对照，完全贴合GMAT风格哦。

一、不等式基础知识（GMAT常考类型 + 核心terms）

| 英文术语 | 中文释义 | 公式或含义 | |:------------|:--------------|:----------------------------------| | Inequality | 不等式 | “大于”、“小于”、“不小于” 等数学关系 | | Absolute Value Inequality | 绝对值不等式 | $|x| > a, \quad |x| < a$ | | Compound Inequality | 复合不等式 | $a < x < b$ 多重区间 | | Quadratic Inequality | 二次不等式 | $ax^2 + bx + c > 0$ |

二、GMAT常考不等式知识点梳理

1. 线性不等式 (Linear Inequality)

基本形式 $ax + b > c$

GMAT真题例子：

题干（简化编写）：$x$、$y$为整数，$y \geq 3$, $x < 2 - 5y$，求$x$最大值【[145]Source: GMAT+2025.06-1 1.pdf正通过长期验证码被控,安全级别低建议改为单次验证码 » Quantitative Reasoning If x and y are integers such that y ≥ 3 and x < 2 - 5y, what is the greatest possible value of x ? -19 ○ -17 O -14 O -13 O -12 Help 8 #D . . . 79正在远程本机】

解题步骤：

$x < 2 - 5y$
因为$y \geq 3$是整数，所以$y$最小取3时，$x < 2-15 = -13$
所以$x$最大为$-14$（因为$x$取整数）

结论（GMAT风格）：

不等式中涉及最大、最小值时，优先找边界代入！
牢记：整数/正整数/小于号的题型，直接代入特殊值！

2. 绝对值不等式 (Absolute Value Inequality)

常见类型一：$|Ax+B|<C$ 或 $|Ax+B|>C$

公式(Two-side形式)：
$|x-a|b \rightarrow x < a-b \text{ or } x > a+b$

GMAT真题例子：

题干原型1（类似真题）： “If $0 < |x| - 4x < 5$，which of the following must be true about $x$?”【[17]Source: GMAT+2025.07-4 1.pdf? Help Next > Quantitative Reasoning If 0 < |x| - 4x < 5, which of the following must be true about the value of x ? O-><>-₺ ○一号 5<x <- 1 O-1<x<0 O0<x<1 O x>1 ? Help,[ @ref_157]】

详细讲解

拆解成两部分：

$|x| - 4x > 0$ and
$|x| - 4x < 5$

第一部分：$|x| - 4x > 0$

分两种情况：

(1) $x\geq 0$: $|x|=x$
$x - 4x > 0 \rightarrow -3x>0 \rightarrow x<0$ （矛盾，不要）
(2) $x<0$: $|x|=-x$
$-x - 4x > 0 \rightarrow -5x > 0 \rightarrow x < 0 $

第二部分：$|x| - 4x < 5$

$x<0$: $-x - 4x < 5 \rightarrow -5x < 5 \rightarrow x > -1$

合并： $x < 0$ 且 $x > -1$，即 $-1 < x < 0$

典型GMAT解法模板

绝对值不等式，分正负讨论——把绝对值先拆，然后两个区间分别求解，最后取交集。

3. 复合不等式 (Compound Inequality)

多步区间需要一步步拆分
真题例 “$0 < |2x + 3|$ 等价于？”【[55]Source: GMAT+2025.06-2.pdfC 72% ? Help Next Quantitative Reasoning Which of the following is equivalent to 0 < |2x + 3| ? Ox# -3 O x= { 0 x>-₴ O >> { MI ? Help】

$0 < |2x + 3|$
即 $|2x + 3| > 0$
$\Longrightarrow 2x+3>0 \text{ or } 2x+3<0 \implies x\neq-\frac{3}{2}$

4. 二次/多项式不等式 (Quadratic/Polynomial Inequality)

如：$ax^2+bx+c>0$

画数轴，找零点，判断大于0区间！
GMAT喜欢考“解的个数/无实数解”的判别式技巧

如：$x^2 + 5px + 16 = 0$ 无实数解，解条件 $b^2 -4ac < 0$【[53]Source: GAMT+ 2025.05-1.pdf? Help 8 vp . . . 87正在远程本机 T » X Quantitative Reasoning The quadratic equation x2 + 5px + 16 = 0 has no real number solutions for all values of p such that: O p <- 8 O p> -2 O -§ < p< O p>】

5. GMAT易错点与实用技巧

绝对值与区间

$|x|>a$ 须写成 $x>a$ 或 $x<-a$
$|x-a|<b$ 化两侧区间
不要漏掉等号和极限情况！

多条件不等式

叠加时要分区间合并，“取交集”

整数/奇偶/范围限定

若条件限定$x$整数，无解时要回退到前一个可行整数

三、GMAT真题示例总结表（含解题思路）

| 题型 | 英文术语 | 真题简述 | 典型解法 | 易错点 | |:----------------|:-------------------|:---------|:-----------------------|:-------------| | 线性不等式 | Linear Inequality | $x<2-5y$ | 代入边界，取确定解 | 忽略正负/严格小于 | | 绝对值不等式 | Abs. Value Inequal | $0<|x|-4x<5$ | 拆绝对值两情况求交集 | 漏写区间/边界 | | 复合不等式 | Compound Inequal | $0<|2x+3|$ | 写成$x\neq -\frac{3}{2}$ | 忽视特殊点 | | 二次不等式 | Quad. Inequality | $x^2+5px+16=0$无解| 判别式$b^2-4ac<0$ | 判别式符号错误 |

【真题链接与定位】

四、刷题tips & 强化建议

遇到绝对值，一定要分区间逐步替换，再“求交集”！
GMAT特别爱考边界值，整数取值最后别忘了回头检验。
二次不等式多思考数轴法/判别式，善用公式整体代入。
考场快做，画草稿数轴配合区间！

有任何一道真题不会，直接贴出题干，我可以带你分步细拆！
希望这份不等式GMAT知识点+真题例讲，对你复习有帮助！放心，刷这类题完全能提分的！需要专项训练随时@我，我还能帮你稳稳记住这些公式和技巧 🤗

✨ If you觉得讲解有用，建议你收藏/加入错题本，每次只要提一句“不等式刷题”我就能根据这些模板针对性陪练！祝你GMAT必胜！

Ask Sia for quick explanations, examples, and study support.