gmatvocabularyflashcard

Step 1: Simplify the Equation

Expand any parentheses and combine like terms so that both sides of the equation are in their simplest form.

Step 2: Group Like Terms

Use addition or subtraction to shift all terms containing \(\mathrm{x}\) to one side and all constant terms to the opposite side.

Step 3: Isolate the Variable

Divide or multiply both sides of the equation by the coefficient of \(\mathrm{x}\) to solve for \(\mathrm{x}\).

Step 4: Check the Solution

Substitute the value found for \(\mathrm{x}\) back into the original equation to confirm its correctness.

Main Methods for Solving Quadratic Equations

• Factoring: Write the quadratic in the form \( ax^2 + bx + c = 0 \) and express it as a product of two binomials. This method is best used when the factors are easy to identify and the quadratic factors neatly.
• Completing the Square: Transform the quadratic into the form \( a(x - h)^2 = k \). This method is useful for deriving the vertex form of the quadratic and when the equation does not factor cleanly.
• Quadratic Formula: Use the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This is the most general method and works for any quadratic equation, regardless of whether it factors.
• Graphing: Plot the quadratic function and determine the x-intercepts. This method provides a visual representation and can be useful for understanding the behavior of the quadratic.

How to Decide Which Method to Use

• If the coefficients are simple and the quadratic factors easily, factoring is quick and efficient.
• If you need the vertex form or the quadratic does not factor neatly, completing the square can be advantageous.
• If an exact solution is required or when dealing with more complex coefficients, the quadratic formula is the best choice.
• If you wish to understand the graphical interpretation of the quadratic, graphing can provide useful insights, although it might be less precise for finding exact solutions.

Substitution Method

• This method involves solving one equation for one variable (e.g., \(\mathrm{y}\) in terms of \(\mathrm{x}\)) and substituting the result into the other equation. It is best used when one of the equations is already solved or can be easily manipulated to isolate a variable.

Elimination Method

• In the elimination method, equations are added or subtracted (possibly after multiplying by suitable constants) to cancel one of the variables. This method is effective when the coefficients of the variable to be eliminated are easy to align, making the process straightforward.

Graphing Method

• Graphing involves plotting each equation on the coordinate plane and finding the point of intersection. Although less precise for obtaining exact answers, it offers a visual verification of the solution. This method is useful when an approximate solution is acceptable or when visual interpretation aids understanding.

Matrix Methods (Cramer’s Rule and Inverse Matrix Method)

• These techniques are typically applied to systems with more than two variables. Cramer’s Rule employs determinants to solve for each variable, while the Inverse Matrix Method uses the inverse of the coefficient matrix. They are best used when the system is larger or when a systematic approach is desired, though they are less common on the GMAT where systems tend to be small and can often be solved using substitution or elimination.

Choosing a method depends on the system’s structure: use substitution when a variable is easily isolated, elimination when coefficients are conducive to cancellation, graphing when a visual solution is helpful, and matrix techniques for systematic handling of larger systems.

Inequality Direction Change:

When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality is reversed.

Example: If mathrm{A} < mathrm{B}, then multiplying by -1 gives -mathrm{A} > -mathrm{B}.

Overview

Divisibility rules and factorization strategies simplify arithmetic problem‐solving by reducing complex calculations into manageable parts. They help identify whether a number is divisible by another without performing full division and enable restructuring numbers into their prime factors for subsequent operations.

Key Benefits

• Simplification: Divisibility rules such as those for 2, 3, 5, and 7 quickly indicate if a number can be factored by a smaller number, eliminating unnecessary calculations.
• Efficient Factoring: Using prime factorization transforms a number into its building blocks, making it easier to compare common factors or simplify fractions. For instance, expressing a number as \(\mathrm{p_1\,p_2\,\ldots\,p_n}\) can reveal cancellations or further divisibility.
• Problem Breakdown: Factorization provides a systematic method to decompose larger problems into simpler sub-problems, which is particularly useful in time-limited contexts like the GMAT.
• Error Reduction: By employing divisibility tests and factorization, one minimizes computational mistakes and streamlines the process of identifying viable solution pathways.

Conclusion

Overall, by incorporating divisibility rules and factorization strategies, arithmetic problem-solving becomes more direct and less error-prone, enabling quicker identification of potential solution paths in GMAT quantitative contexts.

Key Rules for Integer Arithmetic Operations

Addition and Subtraction:

• When adding two integers with the same sign, add their absolute values and keep the common sign.
• For integers with different signs, subtract the smaller absolute value from the larger and assign the sign of the integer with the greater absolute value.
• Subtraction is equivalent to adding the opposite; that is, \(\mathrm{a} - \mathrm{b} = \mathrm{a} + (-\mathrm{b})\).

Multiplication and Division:

• If both integers have the same sign, the product or quotient is positive.
• If the integers have opposite signs, the product or quotient is negative.
• Division follows the same rules as multiplication; note that division by zero is undefined.

Additional Concepts:

• Adhere to the order of operations (parentheses, exponents, multiplication and division, addition and subtraction).
• Multiplication distributes over addition: \(\mathrm{a}\cdot(\mathrm{b}+\mathrm{c}) = \mathrm{a}\cdot\mathrm{b} + \mathrm{a}\cdot\mathrm{c}\).
• The commutative and associative properties apply to both addition and multiplication.

Percent Calculations Using Arithmetic Methods

Percent means per hundred. To work with percents arithmetically, you first convert the percent into a decimal by dividing by 100, then multiply the decimal by the quantity involved.

Steps:

• Conversion: Convert a percent, say \(X\,\%\), into decimal form by computing \(X/100\). For example, \(30\,\%\) becomes \(30/100 = 0.30\).
• Multiplication: Multiply the decimal by the number you are evaluating. For instance, \(30\,\%\) of a number \(N\) is \(N\times(30/100)\).

Examples:

• To calculate \(20\,\%\) of \(50\): Convert \(20\,\%\) to \(20/100 = 0.20\), then compute \(50\times0.20 = 10\).
• To determine \(150\,\%\) of \(80\): Convert to \(150/100 = 1.50\), then calculate \(80\times1.50 = 120\).

Adjusting Values by a Percent:

To increase a number \(A\) by \(X\,\%\), compute \(A\times(1+X/100)\); to decrease \(A\) by \(X\,\%\), compute \(A\times(1-X/100)\). For example, increasing \(100\) by \(15\,\%\) gives \(100\times(1+15/100)=100\times1.15=115\).

This method is widely used on the GMAT, especially in quantitative reasoning, where accurately converting percents and performing multiplication is essential.

Understanding Ratios and Proportions

Ratios are comparisons between two quantities, while proportions state that two ratios are equal. They are powerful tools in solving arithmetic problems because they allow us to scale quantities up or down and find missing values.

• Setting Up Equations: Express the problem as a ratio and set it equal to another ratio. For example, if the ratio of apples to oranges is given as \(\mathrm{3\,:\,4}\) and you have a certain number of apples, you can set up the proportion \(\frac{\mathrm{3}}{\mathrm{4}} = \frac{\mathrm{apples}}{\mathrm{oranges}}\) to solve for the unknown quantity.
• Cross Multiplication: Once a proportion is established, you can cross multiply to create an equation that is easier to solve. For instance, if \(\frac{\mathrm{a}}{\mathrm{b}} = \frac{\mathrm{c}}{\mathrm{d}}\), then \(\mathrm{a}\times\mathrm{d}=\mathrm{b}\times\mathrm{c}\).
• Scaling Quantities: Ratios help in scaling problems. For example, if a recipe ratio is known and you need to adjust the serving size, you can determine the proportional amount of each ingredient through multiplication.
• Real-World Applications: Many arithmetic problems involving speed, distance, or mixtures use ratios and proportions. For instance, if a car travels \(\mathrm{60\,miles}\) in \(\mathrm{1\,hour}\), you can use a proportion to find out how far it travels in another time period.

Summary

Ratios and proportions allow you to create a direct relationship between quantities. By setting up a proportion and using cross multiplication, you can solve for unknown values in a straightforward manner. This method is widely applicable in various arithmetic problems found in the GMAT, particularly in the Quantitative Reasoning section.

Overview

The rules of exponents provide a structured method to simplify and solve expressions by allowing you to combine, break down, or rearrange exponential terms efficiently. They are especially useful when dealing with products, quotients, and powers of numbers.

Key Rules

• Product of Powers: When multiplying terms with the same base, add the exponents. For example, \( \mathrm{a}^{\mathrm{m}} \cdot \mathrm{a}^{\mathrm{n}} = \mathrm{a}^{\mathrm{m+n}} \).
• Quotient of Powers: When dividing terms with the same base, subtract the exponents. For instance, \( \frac{\mathrm{a}^{\mathrm{m}}}{\mathrm{a}^{\mathrm{n}}} = \mathrm{a}^{\mathrm{m-n}} \) provided that \( \mathrm{a} \neq 0 \).
• Power of a Power: When taking a power of an exponent, multiply the exponents. That is, \( (\mathrm{a}^{\mathrm{m}})^{\mathrm{n}} = \mathrm{a}^{\mathrm{m\,n}} \).
• Power of a Product: When raising a product to an exponent, apply the exponent to each factor: \( (\mathrm{ab})^{\mathrm{n}} = \mathrm{a}^{\mathrm{n}} \mathrm{b}^{\mathrm{n}} \).
• Power of a Quotient: Similarly, \( \left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{\mathrm{n}} = \frac{\mathrm{a}^{\mathrm{n}}}{\mathrm{b}^{\mathrm{n}}} \) where \( \mathrm{b} \neq 0 \).

Application: Simplifying Expressions

For instance, to simplify an expression such as \( \frac{(\mathrm{xy})^{\mathrm{3}} \cdot \mathrm{x}^{\mathrm{2}}}{\mathrm{y}^{\mathrm{5}}} \), you would:

• Apply the power of a product rule: \( (\mathrm{xy})^{\mathrm{3}} = \mathrm{x}^{\mathrm{3}} \mathrm{y}^{\mathrm{3}} \).
• Rewrite the expression: \( \frac{\mathrm{x}^{\mathrm{3}} \mathrm{y}^{\mathrm{3}} \cdot \mathrm{x}^{\mathrm{2}}}{\mathrm{y}^{\mathrm{5}}} \).
• Combine like bases using the product and quotient rules: \( \mathrm{x}^{\mathrm{3+2}} \cdot \mathrm{y}^{\mathrm{3-5}} = \mathrm{x}^{\mathrm{5}} \mathrm{y}^{\mathrm{-2}} \).

Application: Solving Equations

When faced with an equation like \( \mathrm{a}^{\mathrm{2x}} = \mathrm{a}^{\mathrm{5}} \), by recognizing that the bases are identical (with \( \mathrm{a} \neq 0 \) and \( \mathrm{a} \neq 1 \)), you can set the exponents equal, yielding \( 2x = 5 \) and solve for \( x \).

By mastering these rules, you can quickly reduce complex exponential expressions into simpler forms, making them easier to solve and analyze on the GMAT. This approach is not only efficient but also critical for tackling quantitative reasoning problems effectively.

Understanding the Role of Probability in Quantitative Reasoning

The basic concepts of probability, such as events, outcomes, and sample spaces, establish the framework for analyzing uncertainty and making logical decisions. These concepts allow candidates to structure problems and apply precise calculations to determine likelihoods.

• Outcomes: Represent the individual results of an experiment, such as obtaining \(\mathrm{1}\) or \(\mathrm{6}\) when rolling a die.
• Events: Are defined as specific sets of outcomes; for instance, the event of rolling an even number includes outcomes \(\mathrm{2}\), \(\mathrm{4}\), and \(\mathrm{6}\).
• Sample Spaces: Comprise all possible outcomes of a random experiment, such as \(\{\mathrm{1},\,\mathrm{2},\,\mathrm{3},\,\mathrm{4},\,\mathrm{5},\,\mathrm{6}\}\) for a six-sided dice throw.

By understanding these key elements, quantitative reasoning is enhanced through:

• The ability to structure problems and define the scope of analysis.
• The application of probability rules, such as the addition and multiplication rules, to compute the likelihood of complex events.
• The development of strategies that involve evaluating risk and making predictions based on logical analysis.

This foundational knowledge is instrumental in approaching many quantitative challenges on the GMAT, as it supports rigorous thinking and effective problem-solving under uncertainty.

Using Counting and Combinatorial Methods in Probability Problems

Counting techniques are fundamental in computing probabilities by enumerating the complete sample space and determining the number of favorable outcomes.

• Define the Sample Space: Use permutation or combination formulas to list all possible outcomes.

• Count Favorable Outcomes: Apply methods such as permutations for ordered arrangements and combinations for unordered selections. For example, compute \(\mathrm{C}(n,k)=\frac{n!}{k!(n-k)!}\) to determine the number of ways to choose \(\mathrm{k}\) items from \(\mathrm{n}\).

• Calculate the Probability: Express the probability as the ratio \(\frac{\mathrm{Favorable\,Outcomes}}{\mathrm{Total\,Outcomes}}\), which simplifies the process of evaluating the likelihood of events.

• Extend to Complex Scenarios: Utilize additional principles such as the rule of product, addition rule, and inclusion-exclusion principle for problems involving multiple events or overlapping conditions.

This approach enables a systematic analysis of probability problems by clearly identifying the structure of the problem and employing appropriate counting methods to obtain accurate results, a key strategy in GMAT quantitative reasoning.

Primary Techniques to Simplify Radicals

• Factor the radicand to identify perfect square factors using the identity \(\sqrt{a\,b}=\sqrt{a}\,\sqrt{b}\).
• Express the radicand as the product of a perfect square and any remaining factor, then simplify as \(\sqrt{c^2\,d}=c\,\sqrt{d}\).
• Combine like radical terms when the radical parts are identical to streamline the expression.
• Rationalize denominators by multiplying both numerator and denominator with a suitable expression to eliminate radicals from the denominator.

Approach Mixture Problems:

Mixture problems on the GMAT require understanding the quantities and concentrations involved. Begin by carefully reading the problem and identifying the knowns and unknowns.

• Define Variables: Let \( \mathrm{x} \) and \( \mathrm{y} \) represent the unknown amounts of each component. For example, if two solutions with different concentrations are mixed, define variables for the amount of each.
• Set Up Equations: Write an equation for the total quantity, such as \( \mathrm{x} + \mathrm{y} \), and another for the concentration. If the concentration of the first solution is \( \mathrm{a} \) and the second is \( \mathrm{b} \), then the pure substance amount can be written as \( \mathrm{a\,x} + \mathrm{b\,y} \). Equate this to the desired concentration multiplied by the total, for example, \( \mathrm{a\,x} + \mathrm{b\,y} = \mathrm{c}(\mathrm{x} + \mathrm{y}) \).
• Simplify and Solve: Clear decimals by multiplying through by an appropriate factor if necessary. Use algebraic methods such as substitution or elimination to solve for \( \mathrm{x} \) and \( \mathrm{y} \).
• Verify Your Solution: Substitute your values back into the original equations to ensure they satisfy all conditions of the problem.

Key Tip: Convert percentages to decimals or fractions consistently, and pay careful attention to unit conversions throughout the problem. This systematic approach helps in balancing both the total amount and the concentration of the mixture, leading to the correct solution.

Step-by-Step Guide to Solving Word Problems Involving Rates and Work

1. Understand the Scenario: Read the problem carefully to identify what quantities are given and which ones you need to find. Determine if it involves rates (such as speed or work rates) or work (often expressed as work done over time).

2. Define Variables: Assign variables to unknown quantities. For example, let mathrm(r) represent the rate, mathrm(t) represent the time, or mathrm(W) represent the amount of work done, where often mathrm(W)=mathrm(r)timesmathrm(t).

3. Consistent Units: Make sure all values are in consistent units. Convert units when necessary so that all terms in your equations match (for instance, hours to minutes or vice versa).

4. Set Up the Equations: Develop algebraic equations based on the relationships described. For example, if multiple agents are working together, you may write an equation like mathrm(W)=mathrm(r_1)timesmathrm(t_1)+mathrm(r_2)timesmathrm(t_2).

5. Solve Algebraically: Use algebraic manipulation to solve for the unknown variable. This could involve combining like terms, using substitution, or elimination methods if multiple equations are involved.

6. Verify the Solution: Substitute your answer back into the original equation to ensure it satisfies all parts of the problem and that the units match.

This systematic approach helps break down complex word problems into manageable parts, and is especially useful for the GMAT's Quantitative Reasoning section where clarity and precision are key.

Step 1: Read and Understand the Problem
Carefully read the entire problem to identify the scenario, quantities involved, and what the problem asks you to find.

Step 2: Identify Known and Unknown Quantities
Determine which quantities are given and which quantity or quantities are unknown. Assign a variable (such as \(x\)) to represent each unknown.

Step 3: Define the Variables
Clearly define each variable in the context of the problem. For example, if the problem asks for a number, let \(x\) be that number.

Step 4: Translate Words into Mathematical Expressions
Examine phrases that indicate operations. For instance, phrases like "the sum of" indicate addition, "the product of" indicates multiplication, "decreased by" indicates subtraction, and "divided by" indicates division. Use \(\mathrm{\,}\) to ensure proper spacing when needed, e.g., \(3 \, x\) instead of \(3x\) if required.

Step 5: Set Up the Equation
Based on the problem's description, equate the expressions. If the problem states that two quantities are equal, set the corresponding mathematical expressions equal to each other.

Step 6: Simplify the Equation
Simplify the equation by combining like terms and performing any necessary algebraic manipulations.

Step 7: Verify the Equation
Review the original word problem to ensure the algebraic equation correctly represents the problem’s conditions and relationships.