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Age problems on the GRE can be surprisingly tricky! But mastering this specific question type is a great way to boost your Quant score. In this article, we’ll consider all aspects of GRE age problems that you could possibly encounter on the exam, mirroring the technique of topical studying just as it is used in our self-study GRE course.
Here are the topics we’ll cover:
- A Model Age Problem
- Translating from Words to Math
- Requisite Algebra Skill: The Substitution Method
- The Star of the Show: Age Problems
- Summary
- What’s Next?
Let’s start by looking at a GRE model age problem.
A Model Age Problem
The GRE problems on age are generally similar to each other. In nearly all questions, each type of age question gives two facts about the relationship between the ages of two people. These two facts are somewhat different from each other, but they give you enough information to create two equations about the ages of the two individuals. This will lead to your being able to determine the ages of the two individuals and thus, you’ll arrive at the correct solution.
Let’s consider a classic age problem and take a step-by-step approach to solving it.
Minerva is 18 years older than Nicole. In two years, Minerva will be 3 times as old as Nicole. How old is Minerva now?
Solution:
Step 1: Define the variables for the current ages.
It is common to use capital letters for the variables, using the first initials of the individuals’ names.
M = Minerva’s current age
N = Nicole’s current age
Step 2: Express the current relationship between the two individuals’ ages (if appropriate).
Here, we are told that Minerva is currently 18 years older than Nicole. Thus, if we were to add 18 to Nicole’s age, we would get Minerva’s age. This is our first equation, and we express it as:
M = N + 18 (equation 1)
Note that a problem might instead give you information about the sum, difference, or even product of their current ages. You would create an appropriate equation to show this relationship.
Step 3: Express each age as a future or past age, depending on the wording of the problem.
In our example, the problem wants to compare the two girls’ ages two years from now. Thus, we can state the following:
M + 2 = Minerva’s age in two years
N + 2 = Nicole’s age in two years
Step 4: Express the relationship between the ages (in the future or past) in one equation.
We are told that in two years, Minerva’s age will be three times Nicole’s. Thus, we create the equation:
M + 2 = 3(N + 2) (equation 2)
Step 5: Combine equations (1) and (2) to solve for the two variables.
This step requires that you use either the substitution method or the combination method for solving a system of two linear equations. We will cover the substitution technique in detail later in this article.
Here, we have the two equations:
M = N + 18 (equation 1)
M + 2 = 3(N + 2) (equation 2)
We can substitute N + 18 from equation 1 for M in equation 2:
(N + 18) + 2 = 3(N + 2)
N + 20 = 3N + 6
14 = 2N
N = 7
We see that Nicole is currently 7 years old. Using equation 1, we calculate Minerva’s age:
M = N + 18 (equation 1)
M = 7 + 18
M = 25
The question asked us to state Minerva’s current age, and we see that she is 25 years old.
TTP PRO TIP:
Use a 5-step procedure for solving nearly all age word problems.
Two Comments about Solving Age Problems
- Make sure you answer the question that is being asked! In the question above, we were tasked with calculating Minerva’s age. One of the common traps in GRE age problems is that you answer the wrong age. Had we been careless, we might have stopped after determining Nicole’s age of N = 7, and we would have gotten the problem wrong, not because of math but because of carelessness.
- As you are learning age problems, it is important to double-check that your answer is correct, not by looking up the correct answer in the answer key, but rather by double-checking that your answers make sense. For example, we calculated that Nicole is 7 years old and Minerva is 25 years old. The problem stated that in two years, Minerva will be three times Nicole’s age. Let’s see if this holds true for the two ages we have calculated:
In two years, Minerva will be 25 + 2 = 27 years old.
In two years, Nicole will be 7 + 2 = 9 years old.
We see that in two years, Minerva will indeed be three times Nicole’s age, since 27 = (3)(9). Thus, we know that our calculations are correct.
TTP PRO TIP:
Make sure you answer what the question asks you to find!
Before we move on to solving more age problems, let’s look at two critical skills we must develop, the first of which is translating from words to math.
Translating from Words to Math
Mention “word problems” to anyone, and you’ll probably hear the response, “I hate word problems!” We all have encountered word problems that were confusing, tricky, or both. One of the best strategies for overcoming our negative relationship with word problems, and specifically age problems, is to get used to the wording in common age problems and figure out what it means.
Let’s look at several examples:
Practice 1 and 2: Translating from Words to Math (Adding)
(1) John is three years older than Sally.
Let’s let J = John’s age and S = Sally’s age. We know that Sally is the younger of the two, so we must ADD 3 to her age to get to John’s age.
J = S + 3
(2) Winston is 5 years younger than Charles.
Let’s let W = Winston’s age and C = Charles’ age. Since Winston is the younger of the two, we must ADD 5 years to Winston’s age to get Charles’ age.
W + 5 = C
TTP PRO TIP:
When dealing with addition in age relationships, determine who is the younger person and then add the difference in the two ages to the younger person’s age.
Practice 3: Translating from Words to Math (Multiplying)
(3) Henri’s age is 5 times Masunori’s age.
Let’s let H = Henri’s age and M = Masunori’s age. Since Masunori is the younger individual, we multiply his age by 5 to get Henri’s age.
H = 5M
TTP PRO TIP:
When dealing with age problems where one person’s age is a multiple of another person’s age, we determine the younger person’s age and apply the multiplier to that age.
Practice 4 and 5: Translating from Words to Math (Multiple Operations)
(4) Byron’s age is 3 years more than twice Tyrell’s age.
Let’s let B = Byron’s age and T = Tyrell’s age. Since Tyrell is the younger individual, we multiply his age by 2 and add 3.
B = 2T + 3
If you were unsure of whether to add or subtract 3, it might be helpful to instead pretend that Tyrell’s age is a particular number. Let’s use 10. So, since Byron is 3 years more than twice Tyrell’s age, we first multiply Tyrell’s age by 2, to get 20. Now, do we add 3 to 20 or subtract it? The key is that Byron is 3 MORE than twice Tyrell’s age. So we would ADD 3 to twice Tyrell’s age. In our numerical example, we would have 2(10) + 3 = 23. Using numbers instead of variables sometimes makes the problem easier to express.
In this example, using numbers helps us see that B = 2T + 3 makes sense.
(5) Melek’s age is 5 years less than 4 times Christina’s age.
Let’s let M = Melek’s age and C = Christina’s age. Christina is the younger person, so we multiply her age by 4. In this case, we would also SUBTRACT 5 from 4C. So we can say this:
M = 4C – 5
Again, it might help if we let Christina be 10 years old. First, 4 times her age would give us 40. Now, the problem states that Melek is 5 years LESS THAN that age of 40. So we know that we need to perform subtraction. Thus, it makes sense that M = 4C – 5.
TTP PRO TIP:
It may be helpful to use a “pretend” number for one person’s age in determining how to express more complicated age relationships.
Requisite Algebra Skill: The Substitution Method
Nearly all GRE word problems involving age will yield two linear equations, often referred to as a system of two linear equations. If you don’t know how to efficiently solve these two equations, you will have spent a lot of time determining the equations but wasted it because of your inability to get the actual solution to the problem!
Luckily, the substitution method is not terribly difficult.
When we’re dealing with two equations, the goal of the substitution method is to isolate one variable in one of the equations and then substitute that variable into the other equation. Let’s practice with an example.
Substitution Method Practice 1
In the system of linear equations below, determine the value of n.
m = 4n (equation 1)
m + n = 15 (equation 2)
Solution:
Looking at our two equations, we see that m is already isolated in equation 1. So, we can substitute 4n (from equation 1) for m (in equation 2). This gives us:
4n + n = 15
5n = 15
n = 3
Let’s look at another practice question that is a bit more challenging.
Substitution Method Practice 2
In the given system of linear equations, what is the value of y?
10x + y = 14 (equation 1)
3x + 4y = -18 (equation 2)
Solution:
We can isolate y in equation 1 by subtracting 10x from both sides of the equation:
10x + y = 14 (equation 1)
y = 14 – 10x
Now, we can use the substitution method to substitute 14 – 10x for y in equation 2. This gives us:
3x + 4(14 – 10x) = -18
3x + 56 – 40x = -18
-37x = -74
x = 2
We have solved for x, but the question asks for the value of y. Substituting 2 for x in equation 1, we have:
10(2) + y = 14
20 + y = 14
y = -6
KEY FACT:
We can use the substitution method to solve a system of linear equations.
The Star of the Show: Age Problems
We have learned to translate from words to math, and we’ve reviewed the substitution method to solve a system of linear equations. With these two skills in hand, we can approach virtually any age problem with confidence! Let’s try several examples.
Example Question 1: Basic Age Question
Tatiana is 23 years older than her daughter Astrid. In 6 years, Tatiana will be twice as old as Astrid. What is Tatiana’s current age?
- 17
- 23
- 29
- 40
- 63
Solution:
Let’s use the 5-step procedure to solve this age problem.
Step 1: Define the variables for the current ages.
Let’s let T = Tatiana’s current age and A = Astrid’s current age.
Step 2: Express the current relationship between the two individuals’ ages (if appropriate).
We know that Tatiana is 23 years older than Astrid, so we have:
T = A + 23 (equation 1)
Step 3: Express each age as a future or past age, depending on the wording of the problem.
In 6 years, Tatiana will be (T + 6) years old, and Astrid will be (A + 6) years old.
Step 4: Express the relationship between the ages (in the future or past) in one equation.
In 6 years, the relationship between their ages will be that Tatiana’s age is twice Astrid’s age. Thus, we have:
T + 6 = 2(A + 6) (equation 2)
Step 5: Combine equations (1) and (2) to solve for the two variables.
We now have two equations and two unknowns. We can use the substitution method to substitute A + 23 for T into equation 2, obtaining:
(A + 23) + 6 = 2(A + 6)
A + 29 = 2A + 12
17 = A
We see that Astrid is 17 years old, so we know that Tatiana is 17 + 23 = 40 years old.
Answer: D
Example Question 2: Sum of Their Ages
Jamal is 12 years older than Louie. Five years ago, the sum of their ages was 28. What is the positive difference between their current ages?
- 12
- 13
- 20
- 25
- 26
Solution:
Step 1: Define the variables for the current ages.
We let J = Jamal’s current age and L = Louie’s current age.
Step 2: Express the current relationship between the two individuals’ ages (if appropriate).
Jamal is 12 years older than Louie. Thus, we have the equation:
J = L + 12 (equation 1)
Step 3: Express each age as a future or past age, depending on the wording of the problem.
Five years ago, Jamal was (J – 5) years old, and Louie was (L – 5) years old.
Step 4: Express the relationship between the ages (in the future or past) in one equation.
The relationship between their ages 5 years ago was that the sum was 28. Thus, we have:
(J – 5) + (L – 5) = 28
J – 5 + L – 5 = 28
J + L = 38 (equation 2)
Step 5: Combine equations (1) and (2) to solve for the two variables.
We substitute L + 12 for J (from equation 1) in equation 2:
(L + 12) + L = 38
2L = 26
L = 13
We see that Louie is 13 years old. We can substitute L = 13 into equation 2 to find Jamal’s age.
J + L = 38
J + 13 = 38
J = 25
Jamal is currently 25 years old, and Louie is 13 years old. Thus, the difference between their ages is 25 – 13 = 12.
Answer: A
Example Question 3: Two Past Age Relationships
Two years ago, Kelly’s age was 3 times Carmen’s age. Seven years ago, Kelly’s age was 5 times Carmen’s age. How old is Carmen now?
- 6
- 12
- 28
- 32
- 40
Solution:
Step 1: Define the variables for the current ages.
Let’s let K = Kelly’s current age and C = Carmen’s current age.
Step 2: Express the current relationship between the two individuals’ ages (if appropriate).
In this case, we do not know the relationship between their current ages. Instead, we have two relationships between prior ages. So we skip Step 2 and move on to Step 3.
Step 3: Express each age as a future or past age, depending on the wording of the problem.
We have two separate relationships about their past ages. Let’s express each.
We know that 2 years ago, Kelly was (K – 2) years old and Carmen was (C – 2) years old.
We also know that 7 years ago, Kelly was (K – 7) years old, and Carmen was (C – 7) years old.
Step 4: Express the relationship between the ages (in the future or past) in one equation.
Two years ago, we know that at that time, Kelly was 3 times Carmen’s age. So we have:
K – 2 = 3(C – 2)
K -2 = 3C – 6
K = 3C – 4 (equation 1)
Similarly, 7 years ago, Kelly was 5 times Carmen’s age. We can express this as:
K – 7 = 5(C – 7)
K – 7 = 5C – 35 (equation 2)
Step 5: Combine equations (1) and (2) to solve for the two variables.
We can substitute 3C – 4 for K into equation 2:
(3C – 4) – 7 = 5C – 35
3C – 11 = 5C – 35
24 = 2C
C = 12
Answer: B
KEY FACT:
Most age problems can be solved by using the 5-step process presented here.
Example Question 4: An Algebraic Age Question
Penelope is 26 years old. Her daughter Tala is 4 years old. In how many years will Penelope be twice as old as her daughter?
- 10
- 12
- 14
- 16
- 18
Solution:
This problem is solved by using traditional algebra instead of the 5-step procedure we have used up to this point.
Let x = the number of years to add to Penelope’s age and her daughter’s age. So, we know that in x years, Penelope will be (26 + x) years old, and Tala will be (4 + x) years old. The relationship between those ages is that Penelope will be twice her daughter’s age.
26 + x = 2(4 + x)
26 + x = 8 + 2x
18 = x
The calculated value of 18 tells us that in 18 years, Penelope will be 26 + 18 = 44 years old, and Tala will be 4 + 18 = 22 years old. This verifies that 18 is the correct answer.
Answer: E
Alternate Solution:
One of the shortcuts for solving age problems on GRE is to backsolve. This means that you can test each of the answer choices to see if the numbers line up with the statements made in the question.
For this question, we would test each answer choice, as follows:
Answer choice A: Is 26 + 10 twice 4 + 10? Is 36 twice 14? No.
Answer choice B: Is 26 + 12 twice 4 + 12? Is 38 twice 16? No.
Answer choice C: Is 26 + 14 twice 4 + 14? Is 40 twice 18? No.
Answer choice D: Is 26 + 16 twice 4 + 16? Is 42 twice 20? No.
Answer choice E: Is 26 + 18 twice 4 + 18? Is 44 twice 22? Yes!
We see that answer choice E satisfies the information from the question stem. Backsolving is not an efficient technique for solving age word problems on GRE, but if you don’t know how to solve the problem any other way, backsolving allows you to get the correct answer.
Answer: E
Summary
In this article, we have covered skills and techniques for solving age word problems on the GRE.
The first skill is to be able to translate from words into math so that equations can be created. The second skill is to know how to solve a system of linear equations using the substitution method.
The most common types of GRE word problems with age can be solved with a 5-step procedure:
- Step 1: Define the variables for the current ages.
- Step 2: Express the current relationship between the two individuals’ ages (if appropriate).
- Step 3: Express each age as a future or past age, depending on the wording of the problem.
- Step 4: Express the relationship between the ages (in the future or past) in one equation.
- Step 5: Combine equations (1) and (2) to solve for the two variables.
You will NOT encounter any age questions of the following types:
- Rate problems with age on GRE
- Ratio problems with age on GRE
- Work problems with age on GRE
- Mixture problems with age on GRE
- Simultaneous equations with age on GRE
- Age-related time problems on GRE
With the techniques and examples presented in this article, you should have no problems with solving GRE quantitative reasoning age problems.
What’s Next?
Now that you feel comfortable with GRE Quantitative strategies for age problems, it’s important to see how this fits into the bigger picture of GRE test preparation. You might want to read about techniques for getting faster at answering GRE math questions. These techniques will help you blend technical knowledge with test-taking skill to increase your GRE score!
