Ratios and proportions
60 flashcards covering Ratios and proportions for the SAT Math section.
Ratios are a way to compare two or more quantities, showing how they relate to each other. For instance, a ratio of 3:4 means that for every 3 parts of one thing, there are 4 parts of another. Proportions extend this by setting two ratios equal, like 3/4 = 6/8, which allows you to solve for unknown values in problems involving scaling, mixtures, or rates. Understanding ratios and proportions is essential for everyday math and helps build problem-solving skills.
On the SAT Math section, ratios and proportions show up in word problems, such as dividing quantities or calculating rates, as well as in geometry questions involving similar figures. Common traps include misinterpreting ratios as fractions or forgetting to simplify them, which can lead to incorrect setups. Focus on practicing these questions to master cross-multiplication, check for proportional relationships, and avoid errors in unit conversions. Always simplify ratios first to make solving easier.
Terms (60)
- 01
Ratio
A ratio compares two quantities by division, often written as a fraction or with a colon, and expresses the relative size of the parts.
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Proportion
A proportion is an equation that states two ratios are equal, allowing you to solve for an unknown value by cross-multiplying.
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Equivalent ratios
Equivalent ratios are pairs of ratios that represent the same relationship between quantities, found by multiplying or dividing both parts by the same number.
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Simplifying ratios
Simplifying ratios involves dividing both parts of the ratio by their greatest common divisor to express it in its lowest terms.
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Part-to-part ratio
A part-to-part ratio compares one part of a whole to another part, such as the ratio of boys to girls in a class.
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Part-to-whole ratio
A part-to-whole ratio compares a part of a whole to the entire whole, like the ratio of apples to total fruit in a basket.
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Rate
A rate is a ratio that compares two quantities of different units, such as speed in miles per hour.
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Unit rate
A unit rate is a rate simplified so that the denominator is 1, making it easier to compare, like dollars per item.
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Solving a proportion
Solving a proportion means finding the unknown value in an equation of two equal ratios, typically using cross-multiplication.
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Cross-multiplication method
Cross-multiplication is a technique for solving proportions where you multiply the numerator of one ratio by the denominator of the other and set them equal.
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Direct proportion
In direct proportion, two quantities increase or decrease at a constant rate, so their ratio remains the same as one changes.
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Inverse proportion
In inverse proportion, as one quantity increases, the other decreases such that their product remains constant.
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Constant of proportionality
The constant of proportionality is the fixed value that relates two directly proportional quantities in an equation like y = kx.
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Scale factor
A scale factor is the ratio by which a figure is enlarged or reduced, used in similar figures to relate corresponding sides.
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Similar figures
Similar figures are shapes with the same shape but not necessarily the same size, where corresponding sides are proportional and angles are equal.
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Ratio of perimeters
The ratio of perimeters of similar figures equals the ratio of their corresponding sides.
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Ratio of areas
The ratio of areas of similar figures is the square of the ratio of their corresponding sides.
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Ratio of volumes
The ratio of volumes of similar three-dimensional figures is the cube of the ratio of their corresponding sides.
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Percent as a ratio
Percent expresses a ratio per hundred, such as 25% meaning 25 per 100 or a ratio of 25:100.
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Ratio tables
Ratio tables are organized lists that show equivalent ratios by multiplying or dividing each part by the same number.
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Extending ratios
Extending ratios means creating equivalent ratios that go beyond the original by multiplying each part by the same value.
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Finding missing values in ratios
Finding missing values in ratios involves using the proportion to set up an equation and solve for the unknown quantity.
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Word problems with ratios
Word problems with ratios require identifying the relationship between quantities and setting up a proportion to solve for unknowns.
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Mixture problems
Mixture problems involve combining substances with different ratios or concentrations and using proportions to find the resulting mixture.
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Allotment problems
Allotment problems use ratios to divide a total amount into parts, such as splitting money based on given shares.
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Speed and distance ratios
Speed and distance ratios relate how distance, speed, and time are proportional, often solved using the formula distance = speed × time.
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Time and work ratios
Time and work ratios compare how workers complete tasks at different rates, using proportions to find total time for combined efforts.
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Converting ratios to fractions
Converting ratios to fractions means expressing a ratio like a:b as a fraction a/b, which can then be simplified.
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Adding ratios
Adding ratios involves converting them to fractions with a common denominator and then adding, though ratios themselves aren't directly added.
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Subtracting ratios
Subtracting ratios requires converting to equivalent fractions and subtracting, often used in comparing differences.
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Multiplying ratios
Multiplying ratios means scaling each part by the same factor to create an equivalent ratio.
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Dividing ratios
Dividing ratios involves dividing each part by the same number to simplify or find a related ratio.
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Ratios with decimals
Ratios with decimals compare quantities that include decimal numbers, simplified by multiplying through by a power of ten if needed.
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Ratios with fractions
Ratios with fractions express comparisons where the quantities themselves are fractions, requiring common denominators for simplification.
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Common multiples in ratios
Common multiples in ratios help find equivalent ratios by multiplying each part by a common factor.
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Least common multiple in ratios
The least common multiple is used in ratios to find the smallest equivalent ratio when combining or comparing.
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Greatest common divisor in ratios
The greatest common divisor is divided into each part of a ratio to simplify it to its lowest terms.
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Proportional relationships
Proportional relationships are pairs of variables where one is a constant multiple of the other, forming a straight line through the origin on a graph.
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Graphs of proportional relationships
Graphs of proportional relationships are straight lines passing through the origin, with the slope representing the constant of proportionality.
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Slope as a ratio
Slope is a ratio of the change in y to the change in x on a line, indicating the rate of change between two variables.
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Unit price
Unit price is the cost per single item, calculated as a unit rate to compare prices of different package sizes.
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Best buy comparisons
Best buy comparisons use unit rates to determine which product offers the most value for the price.
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Dilution ratios
Dilution ratios express how a solution is weakened by adding more solvent, using proportions to calculate concentrations.
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Concentration ratios
Concentration ratios compare the amount of solute to the total solution, often as parts per million or percentage.
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Map scales
Map scales are ratios that relate distances on a map to actual distances on the ground, such as 1:100,000.
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Model scales
Model scales are ratios used in replicas to show how dimensions of the model compare to the real object.
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Shadow problems
Shadow problems use similar triangles and ratios to find heights or distances based on proportional shadows.
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Sine ratio
The sine ratio in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
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Cosine ratio
The cosine ratio in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
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Tangent ratio
The tangent ratio in a right triangle is the ratio of the length of the opposite side to the adjacent side.
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Pythagorean theorem in ratios
The Pythagorean theorem relates the ratios of sides in a right triangle, where the square of the hypotenuse equals the sum of the squares of the other two sides.
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Strategy for setting up proportions
To set up proportions, identify the two ratios from the problem that are equal and ensure corresponding parts are in the same positions.
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Common trap: Confusing ratio with fraction
A common trap is treating a ratio as a fraction without considering the context, which can lead to errors in interpretation.
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Example: Simplifying 4:8
Simplifying 4:8 involves dividing both parts by 4, resulting in 1:2.
For instance, if there are 4 red marbles and 8 blue marbles, the simplified ratio is 1:2.
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Formula: Proportion equation
The proportion equation is a/b = c/d, where cross-multiplying gives ad = bc.
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Worked example: Solving 2/3 = x/6
To solve 2/3 = x/6, cross-multiply to get 26 = 3x, so 12 = 3x, and divide both sides by 3 to find x=4.
This shows how to find the missing value in a proportion.
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Advanced: Inverse proportion in real world
In real-world scenarios, inverse proportion occurs when quantities like speed and time for a fixed distance vary oppositely.
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Ratio of differences
The ratio of differences compares the changes in two quantities, often used in rates of change.
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Harmonic mean for rates
The harmonic mean is used for averaging rates when the distance is constant, calculated as the reciprocal of the average of reciprocals.
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Checking for proportionality
Checking for proportionality involves verifying if the ratio of one pair of values equals the ratio of another pair.