Ratios and proportions
50 flashcards covering Ratios and proportions for the ACT Math section.
Ratios compare two or more quantities, showing their relative sizes, like saying 3:4 means for every 3 parts of one thing, there are 4 parts of another. For instance, a recipe might call for a 2:1 ratio of flour to sugar, meaning twice as much flour as sugar. Proportions take this further by equating two ratios, such as 2/3 = 4/6, which helps solve real-world problems like scaling up a recipe or figuring out travel times based on speed.
On the ACT Math section, ratios and proportions often show up in word problems involving rates, mixtures, or similar figures, testing your ability to set up and solve equations. Common traps include misreading the problem, like confusing parts with totals, or errors in cross-multiplying. Focus on identifying the key relationships, simplifying ratios first, and double-checking units to avoid mistakes. Always practice with varied problems to build speed and accuracy.
Terms (50)
- 01
Ratio
A ratio compares two or more quantities by division, often expressed as a fraction or with a colon, and is used to show relative sizes or amounts in problems involving comparisons.
- 02
Proportion
A proportion is an equation that states two ratios are equal, allowing you to solve for unknown values by setting up and cross-multiplying equivalent fractions.
- 03
Simplifying a ratio
Simplifying a ratio means dividing both parts by their greatest common divisor to express it in its lowest terms, making it easier to work with in equations.
- 04
Equivalent ratios
Equivalent ratios are pairs of ratios that represent the same relationship between quantities, found by multiplying or dividing both parts of a ratio by the same number.
- 05
Cross-multiplication
Cross-multiplication is a method used to solve proportions by multiplying the numerator of one fraction by the denominator of the other and setting them equal.
- 06
Solving a proportion
Solving a proportion involves isolating the unknown variable by cross-multiplying and then dividing, which is essential for word problems involving rates or scales.
- 07
Rate
A rate is a ratio that compares two quantities of different units, such as speed in miles per hour, and is used to describe how one quantity changes relative to another.
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Unit rate
A unit rate is a rate simplified so that the denominator is 1, like dollars per item, helping to compare values efficiently in shopping or travel problems.
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Direct proportion
In direct proportion, two quantities increase or decrease at a constant ratio, meaning one is a constant multiple of the other, as seen in scaling recipes.
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Inverse proportion
Inverse proportion occurs when one quantity increases as another decreases at a constant product, such as speed and time for a fixed distance.
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Constant of proportionality
The constant of proportionality is the fixed value that relates two directly proportional quantities, found by dividing one by the other in an equation.
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Scale factor
A scale factor is the ratio by which a figure is enlarged or reduced, used in geometry to determine corresponding lengths in similar shapes.
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Similar figures
Similar figures are shapes with the same shape but different sizes, where corresponding sides are proportional and angles are equal, common in map problems.
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Ratio of perimeters
The ratio of perimeters of similar figures equals the ratio of their corresponding sides, allowing you to find perimeter changes based on scale.
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Ratio of areas
For similar figures, the ratio of areas is the square of the ratio of their corresponding sides, which is key for problems involving squares or circles.
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Ratio of volumes
The ratio of volumes for similar three-dimensional figures is the cube of the ratio of their corresponding edges, used in problems with solids like cubes.
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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, distinct from part-to-whole ratios.
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Part-to-whole ratio
A part-to-whole ratio compares a part of a group to the entire group, like the fraction of red marbles in a bag, often used in probability contexts.
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Mixture problems
Mixture problems involve combining substances with different concentrations in a given ratio to achieve a desired mixture, solved using proportions.
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Work rate problems
Work rate problems use ratios to determine how long it takes for workers or machines to complete tasks together, based on their individual rates.
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Setting up proportions from words
Setting up proportions from word problems requires identifying equivalent ratios in the scenario, such as comparing distances and times for constant speed.
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Converting ratios to fractions
Converting ratios to fractions involves expressing the ratio as a single fraction, which simplifies solving equations or comparing values.
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Dividing quantities in a ratio
Dividing quantities in a ratio means splitting a total amount into parts that match the given ratio, like dividing $100 in a 2:3 ratio.
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Geometric mean
The geometric mean is the square root of the product of two numbers, often used in proportions to find a middle value between two extremes.
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Percentage as a ratio
A percentage expresses a ratio per hundred, allowing conversion between percentages and fractions for problems involving discounts or interest.
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Speed
Speed is a rate that compares distance traveled to time taken, expressed as distance per unit time, and is solved using proportions in motion problems.
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Distance in proportions
Distance in proportions relates to speed and time through the formula distance equals speed times time, set up as a ratio for unknown values.
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Scale drawings
Scale drawings use a ratio to represent real-world sizes on paper, such as 1:100 for maps, requiring proportions to find actual measurements.
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Maps and scales
Maps and scales involve using a given ratio to convert between map distances and real distances, essential for navigation or architecture problems.
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Extended proportions
Extended proportions involve more than two ratios set equal, like a:b = b:c, used to find patterns in sequences or chains of relationships.
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Ratios with variables
Ratios with variables require setting up equations where unknowns represent parts of the ratio, then solving for the variable using algebra.
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Common trap: Confusing ratio with fraction
A common trap is treating a ratio as a simple fraction without considering the context, which can lead to errors in word problems involving parts.
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Common trap: Forgetting to simplify
Forgetting to simplify ratios can result in incorrect comparisons or solutions, as unsimplified ratios may not reveal equivalent relationships.
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Common trap: Incorrect cross-multiplication
Incorrect cross-multiplication in proportions, such as multiplying wrong terms, often causes errors in solving for unknowns.
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Joint variation
Joint variation describes how one quantity varies directly as the product of two or more other quantities, expressed as proportions in advanced problems.
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Inverse variation example
In inverse variation, as one quantity increases, the other decreases proportionally, like the time taken for a job when more workers are added.
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Direct variation equation
The direct variation equation is y = kx, where k is the constant of proportionality, used to model relationships like cost and quantity.
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Proportions in recipes
Proportions in recipes adjust ingredient amounts based on serving size, using ratios to scale up or down without changing the mixture.
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Example: Simplifying 4:8
Simplifying the ratio 4:8 involves dividing both numbers by 4, resulting in 1:2, which is the lowest terms for comparison.
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Example: Solving 2/3 = x/9
To solve the proportion 2/3 = x/9, cross-multiply to get 29 = 3x, so 18 = 3x, and x = 6, showing how to find the unknown.
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Example: Unit rate of 120 miles in 2 hours
The unit rate for 120 miles in 2 hours is 60 miles per hour, found by dividing distance by time and simplifying.
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Example: Direct proportion with cost
If 2 apples cost $1, then the cost is directly proportional to the number of apples, so 4 apples cost $2, based on the ratio.
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Example: Inverse proportion with workers
If 2 workers take 6 hours to build a wall, then 3 workers take 4 hours, as the time decreases inversely with the number of workers.
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Example: Scale factor of 2 for triangles
If a triangle is scaled by a factor of 2, its sides double, so a side of 3 units becomes 6 units in the similar triangle.
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Example: Ratio of areas for squares
For two squares with side ratios 2:3, the area ratio is (2:3)^2 or 4:9, meaning areas scale with the square of the sides.
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Example: Mixture of solutions
Mixing 2 liters of 10% solution with 3 liters of 20% solution creates a new mixture; using proportions, the combined concentration is calculated.
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Example: Dividing $50 in 3:2 ratio
Dividing $50 in a 3:2 ratio means splitting it into parts where the first is 3/5 of total and the second is 2/5, so $30 and $20.
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Example: Geometric mean of 4 and 9
The geometric mean of 4 and 9 is the square root of 36, which is 6, used in proportions to find a balanced middle value.
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Example: Speed proportion
If a car travels 100 miles in 2 hours, the speed is 50 mph; for 150 miles at the same speed, it takes 3 hours, solved via proportion.
- 50
Example: Scale drawing of a room
A scale drawing of a room at 1:50 means a 5-meter wall is drawn as 0.1 meters, using the ratio to convert measurements.