Ratios and proportions on the GMAT
59 flashcards covering Ratios and proportions on the GMAT for the GMAT Quantitative section.
Ratios compare two or more quantities by showing their relative sizes, such as 2:3 meaning for every 2 parts of one item, there are 3 parts of another. Proportions extend this by stating that two ratios are equal, allowing you to solve for unknowns in equations. These concepts are fundamental in math because they help model relationships in everyday scenarios, like mixing ingredients or scaling recipes, making them essential for logical problem-solving.
On the GMAT Quantitative section, ratios and proportions appear in problem-solving and data sufficiency questions, often involving mixtures, rates, or percentages. Common traps include confusing ratios with actual quantities or overlooking simplifications, which can lead to incorrect answers. Focus on practicing how to set up equations from word problems and use cross-multiplication for proportions to ensure accuracy.
Remember to always simplify ratios before proceeding.
Terms (59)
- 01
Ratio
A ratio compares two or more quantities, showing their relative sizes, and is often written as a fraction or with a colon, like a:b.
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Proportion
A proportion is an equation stating that two ratios are equal, such as a/b = c/d, which is used to solve for unknown values.
- 03
Simplifying Ratios
Simplifying a ratio means reducing it to its lowest terms by dividing all parts by their greatest common divisor, making it easier to work with.
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Equivalent Ratios
Equivalent ratios are pairs of ratios that express the same relationship between quantities, such as 2:3 and 4:6 after simplification.
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Part-to-Part Ratio
A part-to-part ratio compares one part of a whole to another part, like the ratio of boys to girls in a class.
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Part-to-Whole Ratio
A part-to-whole ratio compares a specific part of a whole to the entire whole, such as the ratio of apples to total fruits.
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Cross-Multiplication
Cross-multiplication is a method used in proportions to solve for an unknown by multiplying the numerator of one ratio by the denominator of the other.
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Direct Proportion
In direct proportion, as one quantity increases, the other increases at a constant rate, meaning their ratio remains constant.
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Inverse Proportion
In inverse proportion, as one quantity increases, the other decreases such that their product remains constant, like speed and time for a fixed distance.
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Setting Up a Proportion
Setting up a proportion involves writing an equation with two equal ratios based on the relationships described in a word problem.
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Ratio in Mixtures
In mixtures, ratios describe the relative amounts of ingredients, such as the ratio of acid to water in a solution, to determine concentrations.
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Alligation Method
The alligation method is a shortcut for finding the ratio in which two or more ingredients at different prices or concentrations must be mixed to produce a desired result.
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Work Rates and Ratios
Work rates and ratios compare the speeds at which individuals or machines complete tasks, allowing calculation of combined efforts or time required.
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Speed and Distance Ratios
Speed and distance ratios relate how speed, time, and distance interact, such as using ratios to solve problems involving multiple trips.
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Percentage as a Ratio
A percentage expresses a ratio out of 100, making it useful for comparing parts to a whole in problems involving discounts or growth.
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Ratios in Similar Triangles
In similar triangles, ratios compare corresponding sides, allowing calculation of unknown lengths based on proportional relationships.
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Dividing Quantities in a Ratio
Dividing quantities in a ratio means splitting a total amount according to the given ratio parts, like dividing $100 in a 2:3 ratio.
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Combining Ratios
Combining ratios involves merging two or more ratios into a single ratio, often by finding a common multiple of the parts.
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Common Multiples in Ratios
Common multiples in ratios help find equivalent forms or combine ratios by identifying multiples that align the parts.
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Greatest Common Divisor for Ratios
The greatest common divisor is used in ratios to simplify them by dividing all terms by this largest number that divides evenly into each.
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Trap: Confusing Ratio with Fraction
A common trap is treating a ratio exactly like a fraction without considering the context, which can lead to errors in interpretation.
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Trap: Incorrect Cross-Multiplication
An error in cross-multiplication occurs when proportions are not set up correctly, resulting in solving the wrong equation.
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Example: Simple Ratio Problem
For example, if the ratio of red to blue marbles is 3:2 and there are 15 red marbles, you can find the number of blue marbles by setting up and solving a proportion.
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Example: Proportion Word Problem
For instance, if 2 apples cost $1, how much do 5 apples cost? Set up the proportion 2/1 = 5/x and solve for x to find the answer.
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Ratio of Averages
The ratio of averages compares the average values of two groups, which must be calculated carefully to avoid misrepresenting the overall ratio.
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Ratios with Variables
Ratios with variables express relationships using letters, like a:b, and require solving equations to find specific values.
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Solving for Unknown in Ratio
Solving for an unknown in a ratio involves using algebra to isolate the variable, often by cross-multiplying in a proportion.
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Multiple Ratios
Multiple ratios handle scenarios with more than two quantities, such as a:b:c, requiring consistent scaling across all parts.
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Chain of Ratios
A chain of ratios links multiple ratios together, like a:b and b:c, to find relationships between non-directly compared quantities.
- 30
Ratio and Proportion in Percentages
Ratios and proportions help solve percentage problems, such as finding what percentage one quantity is of another by setting up a proportion.
- 31
Dilution and Concentration Ratios
Dilution and concentration ratios describe how mixing solutions changes their strengths, using ratios to calculate final concentrations.
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Harmonic Mean in Ratios
The harmonic mean is used in ratios involving rates, like average speeds, calculated as the reciprocal of the average of reciprocals.
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Mean Proportional
The mean proportional is a number that represents the geometric mean between two quantities in a proportion, such as x in a/x = x/b.
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Continued Proportion
A continued proportion is a sequence of three or more numbers where each pair forms a proportion, like a:b = b:c.
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Fourth Proportional
The fourth proportional is the number that completes a proportion when three are given, found by cross-multiplying.
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Third Proportional
The third proportional is the number that makes two given numbers proportional, such as x where a:b = b:x.
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Ratio in Data Sufficiency
In data sufficiency questions, ratios help determine if given statements provide enough information to solve for unknowns in proportion problems.
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Strategy for Ratio Questions
A strategy for ratio questions is to first express all parts in terms of a common variable, then use the ratio to set up equations.
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Identifying Ratios in Word Problems
Identifying ratios in word problems involves looking for phrases like 'for every' or 'the ratio of' to extract the comparative relationships.
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Converting Ratios to Fractions
Converting ratios to fractions simplifies calculations, such as turning 3:4 into 3/4, while maintaining the proportional relationship.
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Ratios and Probability
Ratios can represent probabilities by comparing favorable outcomes to total outcomes, helping solve chance-related problems.
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Ratios in Sequences
Ratios in sequences, like geometric sequences, show the common ratio between terms, used to find subsequent or missing values.
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Advanced: Ratios with Exponents
In advanced problems, ratios with exponents involve comparing powers, such as in growth rates, requiring manipulation of exponential expressions.
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Ratio of Areas
The ratio of areas of similar figures is the square of the ratio of their corresponding sides, useful in geometry problems.
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Ratio of Volumes
The ratio of volumes of similar solids is the cube of the ratio of their corresponding edges or heights.
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Trap: Overlooking Units in Ratios
A trap is overlooking units in ratios, which can lead to incorrect comparisons if quantities are not in the same units.
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Example: Mixture Ratio Problem
For example, to mix paints in a 2:3 ratio of red to blue for 10 liters total, calculate each color's amount as parts of the whole.
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Ratio in Percent Change
Ratios help calculate percent change by comparing initial and final values, setting up proportions for growth or decline.
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Scaling Ratios
Scaling ratios means multiplying all parts by the same factor to adjust quantities while maintaining the original relationship.
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Inverse Ratios
Inverse ratios flip the terms, like turning a:b into b:a, which is useful in problems involving reciprocals.
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Ratio and Division of Profits
In business problems, ratios divide profits among partners based on their investments, proportional to their contributions.
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Trap: Assuming Equal Parts
A common trap is assuming a ratio implies equal parts, when it actually specifies relative sizes that may not be equal.
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Example: Work Rate Ratio
For instance, if A completes a job in 4 hours and B in 6 hours, their work rate ratio is 3:2, helping find how long they take together.
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Ratio in Algebraic Expressions
Ratios in algebraic expressions compare variables, like x:y, and require solving systems to find actual values.
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Proportional Reasoning
Proportional reasoning involves using ratios to make predictions or comparisons, a key skill for solving real-world GMAT problems.
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Trap: Misinterpreting Worded Ratios
A trap is misinterpreting worded ratios, such as confusing 'A is twice B' with a specific ratio format.
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Example: Speed Ratio Problem
For example, if two cars travel in a 3:4 speed ratio and cover 150 km together in 2 hours, use proportions to find individual distances.
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Ratio Simplification with Decimals
Simplifying ratios with decimals involves converting to fractions first, then reducing, to handle non-integer values accurately.
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Advanced: Ratios in Functions
In advanced problems, ratios appear in functions, like direct variation where y = kx, representing a constant ratio.