Word problems
58 flashcards covering Word problems for the ACT Math section.
Word problems are math questions that present real-world scenarios in everyday language, requiring you to translate descriptions into equations or formulas to find a solution. For example, you might read about a train traveling at a certain speed and need to calculate its arrival time. This skill tests your ability to connect abstract math concepts to practical situations, which is essential for problem-solving in academics and daily life.
On the ACT Math section, word problems appear frequently across topics like algebra, geometry, and data analysis, often mixed with multiple-choice questions that demand quick interpretation. Common traps include overlooking key details, such as units of measurement, or setting up equations incorrectly due to misreading the problem. Focus on identifying the given information, defining variables clearly, and working step by step to avoid errors and ensure accurate answers.
Always underline the question being asked before solving.
Terms (58)
- 01
Translating words to equations
In word problems, translating words to equations means converting phrases like 'twice as much as' into mathematical expressions, such as 2x, to set up an equation that can be solved for the unknown.
- 02
Setting up systems of equations
For word problems involving multiple variables, setting up systems of equations requires writing two or more equations based on the relationships described, then solving them simultaneously to find the values of the variables.
- 03
Age problems
Age problems involve setting up equations based on the ages of people at a current time or in the future, often using the difference in ages as a constant to solve for unknown ages.
- 04
Mixture problems
Mixture problems require calculating the amount of a substance in a combined solution, such as mixing two alloys, by setting up equations that balance the total quantity and the concentration of the components.
- 05
Work rate problems
Work rate problems involve determining how long it takes for individuals or machines to complete a task together, using rates like jobs per hour to set up equations for combined work.
- 06
Distance, rate, and time formula
The formula distance equals rate times time is used in word problems to relate how far something travels, its speed, and the duration, allowing you to solve for any one variable when the others are known.
- 07
Percent increase and decrease
In word problems, percent increase or decrease calculates the change in a quantity relative to its original value, such as finding a new price after a discount by multiplying the original by (1 + or - the percentage).
- 08
Ratio and proportion
Ratio and proportion in word problems involve setting up fractions that represent relationships between quantities, like mixing ingredients, and solving for unknowns by cross-multiplying.
- 09
Scale factors in word problems
Scale factors are used in word problems involving models or maps to relate actual measurements to scaled ones, such as finding real distances from a map by multiplying the scaled distance by the factor.
- 10
Area word problems
Area word problems require applying formulas for shapes like rectangles or triangles to scenarios, such as calculating the area of a garden based on given dimensions and constraints.
- 11
Volume word problems
Volume word problems involve using formulas for three-dimensional shapes, like cylinders or cones, to solve for capacity or fill rates in real-world contexts.
- 12
Probability word problems
Probability word problems calculate the likelihood of events, such as drawing a certain card, by dividing the number of favorable outcomes by the total possible outcomes based on the scenario.
- 13
Mean in word problems
In word problems, the mean is the average of a set of numbers, calculated by summing the values and dividing by the count, often used to find missing data points in a dataset.
- 14
Linear functions in word problems
Linear functions in word problems model straight-line relationships, like cost versus quantity, by writing equations in the form y = mx + b to predict outcomes based on inputs.
- 15
Quadratic equations in word problems
Quadratic equations in word problems arise from scenarios like projectile motion or area maximization, where you set up an equation of the form ax² + bx + c = 0 and solve using factoring or the quadratic formula.
- 16
Inequalities in word problems
Inequalities in word problems involve setting up and solving expressions like x > 5 to represent constraints, such as budget limits, and finding the range of possible values.
- 17
Break-even point
The break-even point in word problems is the level of production or sales where costs equal revenue, calculated by setting up an equation and solving for the quantity that makes profit zero.
- 18
Simple interest formula
The simple interest formula, interest equals principal times rate times time, is used in word problems to calculate earnings or amounts owed over a period without compounding.
- 19
Arithmetic sequences
Arithmetic sequences in word problems involve a series where each term increases by a constant difference, such as finding the nth term using the formula an = a1 + (n-1)d to solve for totals.
- 20
Pythagorean theorem applications
In word problems, the Pythagorean theorem is applied to right triangles to find missing side lengths, such as the distance between two points on a coordinate plane.
- 21
Similar triangles in word problems
Similar triangles in word problems use the fact that corresponding sides are proportional to solve for unknown lengths, often in scenarios involving shadows or scaled drawings.
- 22
Units conversion trap
A common trap in word problems is forgetting to convert units, like mixing miles and kilometers, which can lead to incorrect answers unless all measurements are standardized.
- 23
Misreading rates
Misreading rates in word problems, such as confusing speed with time, can cause errors, so it's essential to clearly identify what each rate represents before setting up equations.
- 24
Strategy: Drawing diagrams
Drawing diagrams for word problems helps visualize relationships, like sketching a triangle for a geometry scenario, making it easier to identify variables and equations.
- 25
Strategy: Defining variables
Defining variables in word problems means assigning letters to unknowns, such as letting x be the number of items, to clearly structure and solve the resulting equations.
- 26
Strategy: Checking units
Checking units in word problems ensures consistency, like verifying that both sides of an equation are in the same units, such as hours and minutes, to avoid calculation mistakes.
- 27
Direct variation
Direct variation in word problems means one quantity is a constant multiple of another, expressed as y = kx, and is used to solve for unknowns when the constant is known.
- 28
Uniform motion problems
Uniform motion problems involve objects moving at constant speeds, requiring equations like distance = speed × time to determine meeting points or travel times.
- 29
Coin problems
Coin problems set up equations based on the total value and number of coins, such as solving for the number of quarters and dimes that add up to a certain amount.
- 30
Consecutive integers
Consecutive integers in word problems are numbers in sequence, like n, n+1, n+2, and are used to set up equations for sums or products as described.
- 31
Geometric probability
Geometric probability in word problems calculates the chance of an event based on areas or lengths, such as the probability a point in a region falls in a specific sub-region.
- 32
Expected value
Expected value in word problems is the average outcome of a random event, calculated by summing each outcome multiplied by its probability, often for games or decisions.
- 33
Slope as rate of change
In word problems, slope represents the rate of change, like miles per hour in a distance-time graph, allowing you to interpret how one quantity changes with another.
- 34
Intercepts in word problems
Intercepts in word problems, such as the y-intercept in a cost equation, represent initial values or fixed costs, helping to understand the context of linear models.
- 35
Vertex of a parabola
The vertex of a parabola in word problems, like in projectile motion, is the maximum or minimum point, found using the formula for quadratics to determine peak height or time.
- 36
Sine ratio in triangles
In word problems involving right triangles, the sine ratio is opposite over hypotenuse, used to find missing sides or angles in scenarios like ladder problems.
- 37
Assuming constant speed trap
A common trap is assuming constant speed when it varies, so word problems require checking if rates change and adjusting equations accordingly.
- 38
Strategy: Working backwards
Working backwards in word problems starts from the desired outcome and reverses steps to find the initial conditions, useful for complex sequences or investments.
- 39
Strategy: Making a table
Making a table in word problems organizes data into rows and columns, helping to spot patterns or relationships needed to set up equations.
- 40
Percentages in mixtures
Percentages in mixtures calculate the concentration of solutions, such as finding the final percentage when combining different strengths of acid.
- 41
Inverse variation
Inverse variation in word problems means one quantity decreases as another increases, following y = k/x, and is used for scenarios like time and workers.
- 42
Number problems with digits
Number problems with digits involve setting up equations based on place values, like the sum of digits in a two-digit number equaling a certain value.
- 43
Combinations in word problems
Combinations in word problems calculate the number of ways to select items without regard to order, using the formula C(n, k) = n! / (k!(n-k)!), for scenarios like committees.
- 44
Exponential growth
Exponential growth in word problems models quantities that increase by a fixed percentage, like populations, using equations like A = P(1 + r)^t to predict future values.
- 45
Shadow problems
Shadow problems use similar triangles to relate object heights and shadow lengths, solving for unknowns like the height of a flagpole based on proportions.
- 46
Median in word problems
In word problems, the median is the middle value in an ordered list, used to find or verify the middle score in a dataset with an odd number of entries.
- 47
Mode in word problems
The mode in word problems is the most frequent value in a dataset, helping to identify the most common outcome in scenarios like survey results.
- 48
Compound interest
Compound interest in word problems calculates growth over time with periodic compounding, using the formula A = P(1 + r/n)^(nt), to find future amounts.
- 49
Optimization problems
Optimization problems involve maximizing or minimizing a quantity, like profit, by setting up and solving equations with constraints from the scenario.
- 50
Absolute value in word problems
Absolute value in word problems represents distances or differences, such as the distance between two numbers, and requires solving equations like |x - a| = b.
- 51
Permutations in word problems
Permutations in word problems calculate the number of ways to arrange items in order, using the formula P(n, k) = n! / (n-k)!, for scenarios like scheduling.
- 52
Rate of change in functions
Rate of change in function word problems interprets the derivative-like concept for linear or quadratic functions, showing how outputs vary with inputs in real contexts.
- 53
Area of a triangle formula
The area of a triangle formula, (1/2)base × height, is applied in word problems to calculate spaces, like the area of a lot, based on given dimensions.
- 54
Two pipes filling a tank
In work rate problems, two pipes filling a tank involves adding their rates to find the combined time, such as one pipe filling in 4 hours and another in 6 hours.
- 55
Mixing solutions example
An example of mixing solutions is combining 10 liters of 20% acid with 20 liters of 50% acid to find the concentration of the mixture, solved by setting up a percentage equation.
- 56
Law of sines in triangles
The law of sines in word problems relates sides and angles of non-right triangles, using a/sin A = b/sin B, to solve for missing parts in scenarios like navigation.
- 57
Cosine ratio in triangles
In word problems, the cosine ratio is adjacent over hypotenuse in right triangles, used to find angles or sides in contexts like inclined planes.
- 58
Strategy: Using substitution
Using substitution in word problems simplifies systems of equations by solving one equation for a variable and plugging it into another to reduce complexity.