Age problems
49 flashcards covering Age problems for the GMAT Quantitative section.
Age problems are a common type of algebra word problem that involve calculating the ages of people at different points in time. They typically present scenarios where you need to determine someone's current age, past age, or future age based on relationships like "twice as old as" or "in 5 years." For instance, you might solve for how old two people are now if one was a certain age when the other was born. These problems build skills in setting up equations from everyday situations, which is essential for logical reasoning in math.
On the GMAT Quantitative section, age problems appear in problem-solving and data-sufficiency questions, often mixed with other algebra topics to test your ability to interpret wordy descriptions accurately. Common traps include misreading timelines, such as confusing "x years ago" with "in x years," or failing to account for age differences consistently. Focus on identifying key variables, like current ages, and translating the problem into clear equations to avoid errors in setup or calculation.
Always draw a simple timeline to track ages visually.
Terms (49)
- 01
What is an age problem?
An age problem is a type of word problem on the GMAT that involves the ages of people at different points in time, requiring the setup of equations to find unknown ages based on relationships like differences or ratios.
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Current age
Current age refers to a person's age at the present time, which is typically the starting point for setting up equations in age problems.
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Age in the future
Age in the future is calculated by adding the number of years from now to a person's current age, often used to express relationships like one person being twice as old as another in a specific year.
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Age in the past
Age in the past is determined by subtracting the number of years ago from a person's current age, helping to model situations where ages were related at an earlier time.
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Constant age difference
The constant age difference is the unchanging gap between two people's ages over time, which remains the same regardless of the year and is key for forming equations.
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Setting up variables for ages
Setting up variables for ages involves assigning letters, like x for one person's current age and y for another's, to translate word problems into algebraic equations.
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Age difference equation
An age difference equation expresses the fixed gap between ages, such as x - y = 5, where x and y are the current ages of two people.
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Ratio of current ages
The ratio of current ages is the proportion of one person's age to another's at the present, like 2:1 if one is twice as old, which can be written as x/y = 2.
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Ratio of ages in the future
The ratio of ages in the future compares what two people's ages will be after a certain number of years, requiring equations like (x + t)/(y + t) = ratio.
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Systems of equations for ages
Systems of equations for ages involve multiple equations based on different conditions, such as current ratios and future equalities, solved simultaneously to find unknown ages.
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Word problem translation
Word problem translation means converting statements about ages into mathematical equations, for example, 'Alice is twice as old as Bob' becomes x = 2y.
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Common trap: Confusing timelines
A common trap in age problems is mixing up past, present, and future timelines, leading to incorrect equations, so always clearly define the time frame.
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Ages at a specific event
Ages at a specific event refer to how old people were or will be at a particular time, requiring adjustments from current ages to fit the scenario.
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Double the age in x years
Double the age in x years describes a situation where one person's age after x years is twice another's, leading to an equation like (x + a) = 2(b + x).
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Half the age x years ago
Half the age x years ago means one person's age at that time was half of another's, such as (a - x) = 0.5(b - x).
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Sum of ages
The sum of ages is the total of two or more people's ages at a given time, often used in problems where this total equals a specific number.
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Ages forming a ratio
Ages forming a ratio indicate that the ages of people are in a proportional relationship, like 3:2, which must hold true at the specified time.
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Solving for unknown ages
Solving for unknown ages involves using algebraic methods to find the values of variables representing ages after setting up the correct equations.
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Trap: Assuming proportional changes
A trap is assuming that age differences change over time, but they do not; only the ages themselves increase at the same rate.
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Using substitution in age problems
Using substitution in age problems means solving one equation for a variable and plugging it into another to simplify and find the solution.
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Using elimination in age problems
Using elimination in age problems involves adding or subtracting equations to eliminate a variable, making it easier to solve for the remaining ones.
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Ages and birth years
Ages and birth years relate through the formula age = current year - birth year, which can help verify solutions in complex problems.
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Two people age problem
A two people age problem typically involves relationships between just two individuals' ages, requiring one or two equations to solve.
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Three people age problem
A three people age problem includes relationships among three individuals, often needing a system of three equations.
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Future age equality
Future age equality occurs when two people's ages will be the same in a certain number of years, leading to an equation like x + t = y + t, which simplifies differences.
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Past age multiple
Past age multiple means one person's age was a multiple of another's in the past, such as x - t = 2(y - t).
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Average age
Average age is the mean of two or more people's ages, calculated by dividing the sum of ages by the number of people, and may be given in problems.
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Ages in arithmetic sequence
Ages in arithmetic sequence mean the ages form a pattern with a constant difference, like siblings' ages increasing by a fixed amount.
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Identifying key information
Identifying key information in age problems means noting clues like current ages, differences, ratios, or future conditions to determine what equations to form.
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Avoiding algebraic errors
Avoiding algebraic errors in age problems requires double-checking equation setup and solution steps, as small mistakes can lead to incorrect ages.
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Checking solutions
Checking solutions involves plugging found ages back into the original problem to ensure they satisfy all given conditions.
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Example: John is twice Mary's age
In this example, if John is twice as old as Mary now, set x as John's age and y as Mary's, so x = 2y, and use additional info to solve.
If John was 10 when Mary was born, then x = 2y and x - y = Mary's age at John's birth, solving gives x=20, y=10.
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Example: In 5 years, Alice will be twice Bob's age
This example sets up (Alice's current age + 5) = 2(Bob's current age + 5), using variables to find the values.
If Alice is 15 now, then 15 + 5 = 2(Bob's age + 5), so 20 = 2(Bob's age + 5), Bob's age is 7.5.
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Trap: Misinterpreting 'was'
A trap is misinterpreting 'was' as referring to current age instead of past, so always adjust for the time specified.
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Relative age definition
Relative age is the comparison between two or more people's ages, such as one being older by a certain amount or in a specific ratio.
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Age after t years formula
The formula for age after t years is current age plus t, used to express future ages in equations.
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Age t years ago formula
The formula for age t years ago is current age minus t, essential for problems involving past relationships.
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Ages with multiple conditions
Ages with multiple conditions involve satisfying more than one relationship, like a ratio now and equality later, requiring multiple equations.
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Ages in families
Ages in families often deal with relationships like parents and children, where differences are typically generational.
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Strategy: Draw a timeline
Drawing a timeline helps visualize age problems by marking current time, past events, and future points to clarify relationships.
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One versus two variables
Deciding on one versus two variables depends on whether the problem can be solved with a single unknown or needs multiple for complex relationships.
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Ages and their product
Ages and their product might be given or required, such as the product of two ages equaling a number, adding another equation.
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Ages and fractions
Ages and fractions involve scenarios where one age is a fraction of another, like one-third older, translated into equations.
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Verifying with real-world sense
Verifying with real-world sense means ensuring solved ages are positive and logical, as negative ages indicate an error.
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Practice tip: Plug in numbers
Plugging in numbers means testing possible ages in the problem to check which satisfy the conditions, useful for verification.
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Advanced: Quadratic from ages
In advanced cases, age problems might lead to quadratic equations if relationships involve squares or products of ages.
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Sub-concept: Birthdays and ages
Birthdays and ages consider that ages increase on birthdays, but GMAT problems typically use whole years for simplicity.
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Strategy for equal ages
The strategy for equal ages involves setting future or past ages equal in an equation, like x + t = y + t for current equality.
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Trap: Forgetting constant difference
A trap is forgetting that age differences remain constant, which can lead to incorrect assumptions about changing gaps.