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Age Problems

Age word problems.

Every now and then, we encounter word problems that require us to find the relationship between the ages of different people. Age word problems typically involve comparing two people’s ages at different points in time, i.e. at present, in the past, or in the future.

This lesson is divided into two parts. Part I involves age word problems that can be solved using a single variable while Part II contains age word problems that need to be solved using two variables .

Let’s get familiar with age word problems by working through some examples.

PART I: Age Word Problems Solvable with One Variable

Example 1: Tanya is 28 years older than Marcus. In 6 years, Tanya will be three times as old as Marcus. How old is Tanya now?

In this problem, we are only asked to find Tanya’s current age. However, the problem also gave us a lot of other information which can be overwhelming. To help us organize the important details, let’s create a table to list what we know so far.

Since we are only given details about their current ages and what they will be 6 years from now, we’ll go ahead and gray out the Past column.

You may notice that Tanya’s current age is defined using the age of Marcus. However, Marcus’s present age is currently unknown. So let’s express Marcus’s age using the variable [latex]x[/latex]. Since Tanya is 28 years older than Marcus , then Tanya’s present age must be [latex]x+28[/latex].

Next, let’s fill in the Future column which will consist of their ages in 6 years. All we have to do is add 6 to Tanya and Marcus’s present or current ages. Therefore, we have:

• Tanya: [latex]\left( {x + 28} \right) {\color{red}+ 6} = x + 34[/latex]
• Marcus: [latex]x {\color{red}+ 6}[/latex]

Now that our table is filled out, we can go ahead and create our equation based on the information provided. The problem states the following:

In 6 years , Tanya will be three times as old as Marcus.

Here we are trying to find the relationship between their ages in the future. We can simply say that,

Tanya’s age in 6 years = 3( Marcus’s age in 6 years )

With that in mind, we can easily construct our equation.

Our next step now is to solve for [latex]x[/latex]. But before that, remember that our problem is asking us to find Tanya’s current age. Since Tanya’s age is defined using Marcus’s current age (which is [latex]x[/latex]), we have to find his age first in order to determine what Tanya’s present age is.

Now that we have the value for [latex]x[/latex], let’s find out what Tanya and Marcus’s current ages are. We can do this by simply replacing the [latex]x[/latex]’s with [latex]8[/latex].

CURRENT AGES (present)

• Marcus: [latex]x = {\textbf{8}}[/latex] years old
• Tanya: [latex]x + 28 = {\color{red}8} + 28 = {\textbf{36}}[/latex] years old

Going back to the problem’s question, how old is Tanya now?

Answer: Tanya is 36 years old.

Answer Check:

At this point, we are confident that our answer is correct. But, how can we be 100% sure? Well, it’s always a good idea especially in math, to check our answers so we’re certain that we got the correct values.

For this problem, we can simply verify if our answer makes our future statement true. Do you remember this statement?

In 6 years, Tanya will be three times as old as Marcus.

We know the present ages of Marcus and Tanya which are [latex]8[/latex] and [latex]36[/latex], respectively. Hence in 6 years, Marcus will be [latex]14[/latex] years old while Tanya will be [latex]42[/latex] years old.

So, will Tanya be three times as old as Marcus in 6 years? The answer is Yes .

Example 2: Bruce is 4 years younger than Hector. Twenty years ago, Hector’s age was 13 years more than half the age of Bruce. How old are they now?

By just reading the problem, we can already tell that there is a great deal of information that we have to sort through and that this problem includes a fraction. Most students easily get lost in all the given information, let alone solving equations that involve fractions. But, don’t fret! As long as you stick with the basic principles and steps on how to solve age word problems, you’ll be fine.

Right now, we don’t know Bruce or Hector’s current age. But since Bruce’s age is expressed in relation to Hector’s age, then our unknown variable will be based on Hector’s age. In other words,

• Let [latex]{\textbf{\textit{h}}} =[/latex] Hector’s age
• [latex]{\textbf{\textit{h} – 4}} =[/latex] Bruce’s age, since he is 4 years younger than Hector

Let’s organize all these important data into a table. We’re only given details about their present and past (20 years ago) ages so we’ll gray out the Future column.

Twenty years ago, both Bruce and Hector were 20 years younger so we’ll subtract 20 from each of their present ages.

• Bruce: [latex]\left( {h – 4} \right) {\color{red}- 20} = h – 24[/latex]
• Hector: [latex]h {\color{red}- 20}[/latex]

Our table is now ready so we can proceed to create our equation. As you can see under the Past column, we were able to create algebraic expressions for Bruce and Hector’s ages 20 years ago. But our problem also told us that,

Twenty years ago , Hector’s age was 13 years more than half the age of Bruce.

Since Hector’s age 20 years ago is also 13 years more than half of Bruce’s age, we can take these two algebraic expressions and set them equal to each other, to create an equation.

Hector’s age 20 years ago = [latex]\Large{1 \over 2}[/latex]( Bruce’s age 20 years ago )[latex]+ 13[/latex]

We’re now ready to solve for the unknown variable, [latex]h[/latex].

Therefore, Hector’s present age is [latex]{\textbf{42}}[/latex] years old.

On the other hand, you may recall that Bruce’s current age is: [latex]h – 4[/latex]. Since [latex]h = 42[/latex], then Bruce’s current age is [latex]42 – 4 = {\textbf{38}}[/latex].

So, how old are they now?

Answer: Hector is 42 years old and Bruce is 38 years old .

The final step is to check our answers by substituting the unknown values into our original equation to verify if each side of the equation equals the other.

Great! Our answer checks. This just showed us that if we take Bruce’s age twenty years ago, which is 18, and divide it in half, we get 9. Adding 13 to that ([latex]9 + 13[/latex]), we get 22 which was Hector’s age twenty years ago.

Therefore, we are able to confirm that twenty years ago when Hector was 22 years old and Bruce was 18 years old, Hector’s age was 13 years more than half the age of Bruce.

Example 3: Stella is 13 years younger than Kwame. Nine years from now, the sum of their ages will be 43. Find the present age of each.

This problem is a little different from our previous two examples as we are given the sum of their ages in 9 years. But right off the bat, we can see that Stella’s age is defined in terms of Kwame’s age. Therefore, we’ll select a variable to represent Kwame’s current age. In this instance, let’s use “[latex]k[/latex]”.

• Let [latex]{\textbf{\textit{k}}} =[/latex] Kwame’s age
• [latex]{\textbf{\textit{k} – 13}} =[/latex] Stella’s age, since she is 13 years younger than Kwame

Nine years from now, both Kwame and Stella will be 9 years older. So we’ll simply add 9 to their present ages above to show their future ages.

• Kwame: [latex]k {\color{red}+ 9}[/latex]
• Stella: [latex]\left( {k – 13} \right) {\color{red}+ 9} = k – 4[/latex]

Let’s complete our table.

Now that we have the algebraic expressions for both their ages in 9 years, we can add these expressions to create our equation. We were given the following details:

Nine years from now , the sum of their ages will be 43 .

So we have,

Checking back at our table, [latex]k[/latex] stands for Kwame’s age. But since our problem asked us to find the current ages for both, let’s do a little bit more solving.

• Kwame: [latex]k = {\textbf{19}}[/latex] years old
• Stella: [latex]k – 13 = {\color{red}19} – 13 = {\textbf{6}}[/latex] years old

Answer: Kwame is 19 years old and Stella is 6 years old .

Let’s now verify if indeed the sum of Kwame and Stella’s ages in 9 years will be 43.

• Kwame’s age in 9 years: [latex]k + 9 = {\color{red}19} + 9 = {\textbf{28}}[/latex]
• Stella’s age in 9 years: [latex]k – 4 = {\color{red}19} – 4 = {\textbf{15}}[/latex]

Perfect! The total of their ages nine years from now is 43 so our answers are correct.

Example 4: Mr. Cook is 34 years old. His son is 22 years younger than him. In how many years will Mr. Cook’s age be 24 years less than three times as old as his son?

We already know their current ages, so before we delve any further, let’s start filling in our table.

Note that since the son is 22 years younger than Mr. Cook, we subtracted 22 from 34 to get his son’s current age, [latex]34 – {\color{red}22} = 12[/latex].

This problem is unique because it’s not asking us for their ages at a certain point in time like usual. Instead, it asks us to find out the number of years when Mr. Cook’s age will meet a certain relationship with his son’s age in the future.

But at this point, we don’t know how long it will take for Mr. Cook to be 24 years less than three times as old as his son. So, let’s assign the unknown variable “[latex]x[/latex]” to stand for the number of years then add [latex]x[/latex] to both of their current ages to create algebraic expressions that will represent how old they will be after [latex]x[/latex] years.

Since Mr. Cook’s age after [latex]x[/latex] number of years ([latex]x + 34[/latex]) will also be 24 years less than three times as old as his son , we can set these two algebraic expressions equal to each other, thus creating our equation.

Now that we have our equation, let’s solve for [latex]x[/latex].

As you may recall, [latex]x[/latex] stands for the number of years from now that will take for Mr. Cook to be 24 years less than three times as old as his son. Therefore,

Answer: In 11 years , Mr. Cook’s age will be 24 years less than three times as old as his son.

To check if our answer is correct, we must first find out how old will Mr. Cook and his son be in 11 years. Substituting the value of [latex]x[/latex] which is 11 into our algebraic expressions, we get:

• Mr. Cooks’s age in 11 years: [latex]x + 34 = {\color{red}11} + 34 = {\textbf{45}}[/latex]
• Son’s age in 11 years: [latex]x + 12 = {\color{red}11} + 12 = {\textbf{23}}[/latex]

So in 11 years, Mr. Cook will be 45 years old while his son will be 23 years old.

This time, I’ll leave it up to you to verify if indeed during that time, his age of 45 years old will be 24 years less than three times as old as his son. If it meets the condition, then our answer is correct.

Example 5: The sum of one-fifth of Annika’s age four years ago and half of her age in six years is 33. How old is she now?

Compared to our previous exercises, this problem only involves one person. Also, instead of comparing the ages of two people at a certain point in time, we will be comparing Annika’s ages at different points in time, i.e. 4 years ago and in 6 years.

We don’t know Annika’s current age so let’s select the variable [latex]{\textbf{\textit{a}}}[/latex] to represent this unknown value. We’ll use this variable as well to create algebraic expressions that will stand for her past and future ages.

• Let [latex]{\textbf{\textit{a}}} =[/latex] Annika’s current age
• [latex]{\textbf{\textit{a} – 4}} =[/latex] Annika’s age 4 years ago
• [latex]{\textbf{\textit{a} + 6}} =[/latex] Annika’s age 6 years from now

Our problem also told us that if we add [latex]\Large{1 \over 5}[/latex] of Annika’s age 4 years ago and [latex]\Large{1 \over 2}[/latex] of her age 6 years from now , the sum is 33 .

With this information, it’s easy for us to write our equation.

Our next step is to solve for the unknown variable, [latex]a[/latex].

So, how old is Annika now?

Answer: Annika is currently 44 years old.

As I mentioned before, it’s always a good practice to verify if you got the correct answer. To start, let’s find out what Annika’s past and future ages are.

• Annika’s age 4 years ago : [latex]a – 4 = {\color{red}44} – 4 = {\textbf{40}}[/latex]
• Annika’s age 6 years from now : [latex]a + 6 = {\color{red}44} + 6 = {\textbf{50}}[/latex]

Now that we know how old she was 4 years ago and how old she’ll be in 6 years, we’ll plug in these values into our original equation to see if both sides of the equation equal each other.

And they did! We were able to prove that the sum of [latex]\Large{1 \over 5}[/latex] of Annika’s age 4 years ago and [latex]\Large{1 \over 2}[/latex] of her age 6 years from now is indeed 33.

PART II: Age Word Problems Solvable with Two Variables

Example 6: The sum of Aaliyah and Harald’s ages is 28. Four years from now, Aaliyah will be three times as old as Harald. Find their present ages.

Neither Aaliyah nor Harald’s age is expressed in terms of the other. So for this problem, we will be using more than one variable to represent the unknown values. To start,

• Let [latex]{\textbf{\textit{a}}}[/latex] be Aaliyah’s age
• Let [latex]{\textbf{\textit{h}}}[/latex] be Harald’s age

Since they will be 4 years older in the next 4 years, we simply have to add 4 to their current ages to represent their future ages.

Looking back at our problem, there are two significant statements that can help us find our answers.

1) The sum of Aaliyah and Harald’s ages is 28.

From this statement, we can create the equation below:

2) Four years from now, Aaliyah will be three times as old as Harald.

Meanwhile, the statement above can be translated into the following equation:

We now have two equations to solve.

• Equation 1: [latex]a + h = 28[/latex]
• Equation 2: [latex]a + 4 = 3(h + 4)[/latex]

First, we’ll use equation 1 to solve for [latex]a[/latex].

Next, we’ll replace [latex]a[/latex] with [latex]28 – h[/latex] in equation 2 .

Perfect! We are able to find the values for both our unknown variables, [latex]a[/latex] and [latex]h[/latex], which also stand for the present ages for Aaliyah and Harald. So we have,

• Aaliyah’s present age: [latex]a = 28 – h = 28 – {\color{red}5} = {\textbf{23}}[/latex]
• Harald’s present age: [latex]h = {\textbf{5}}[/latex]

Answer: Currently, Aaliyah is 23 years old while Harald is 5 years old.

I’ll leave it up to you to check if our answers are correct. But as you can see, even with just using mental computation, we can already tell that the sum of Aaliyah and Harald’s ages is 28 ([latex]23 + 5 = 28[/latex]) which makes our first statement true. You may further check our answers by plugging in the values of [latex]a[/latex] and [latex]h[/latex] into equation 2 to verify if the left side of the equation equals the right, thus making our second statement true as well.

Example 7: The sum of the ages of Jaya and Nadia is three times Nadia’s age. Seven years ago, Jaya was three less than four times as old as Nadia. How old are they now?

This problem is similar to our previous example. However, for this one, we are not given the exact number for the sum. We first have to find out each of their current ages so we can determine what the sum is.

• Let [latex]{\textbf{\textit{y}}}[/latex] be Jaya’s age
• Let [latex]{\textbf{\textit{n}}}[/latex] be Nadia’s age

We then need to subtract 7 from their current ages to represent how old they were seven years ago.

Now that we’ve organized our data, let’s go through the significant statements given in our problem and translate each into an equation.

1) The sum of the ages of Jaya and Nadia is three times Nadia’s age.

2) Seven years ago, Jaya was three less than four times as old as Nadia.

Therefore, our two equations are:

• Equation 1: [latex]y + n = 3n[/latex]
• Equation 2: [latex]y – 7 = 4(n – 7) – 3[/latex]

Let’s first focus on equation 1 and solve for [latex]y[/latex].

Now we’ll solve for [latex]n[/latex] using the value of [latex]y[/latex] from equation 1. We’ll do this by replacing [latex]y[/latex] with [latex]2n[/latex] in equation 2 .

Taking the values of [latex]y[/latex] and [latex]n[/latex], we have:

• Jaya’s present age: [latex]y = 2n = 2({\color{red}12}) = {\textbf{24}}[/latex]
• Nadia’s present age: [latex]n = {\textbf{12}}[/latex]

So, going back to our problem. How old are they now?

Answer: Jaya is 24 years old and Nadia is 12 years old.

To check our answers, we’ll replace the values of [latex]y[/latex] and [latex]n[/latex] in equation 1 and equation 2. Again, I’ll leave it up to you to solve both equations and verify if each side of the equation equals the other. Once you’re done with your solutions, you’ll see that we are able to prove that both statements from our problem are true.

Example 8: The difference between the ages of Penelope and her son, Zack, is 34. In six years, Penelope will be four times as old as Zack’s age two years ago. How old are they now?

It’s easy to get lost in all the information given so we’ll focus first on assigning variables that will stand for the unknown values.

• Let [latex]{\textbf{\textit{p}}}[/latex] be Penelope’s current age
• Let [latex]{\textbf{\textit{z}}}[/latex] be Zack’s current age

One thing that’s unique about this problem is that it involves three different points in time. We are given not only the relationship between Penelope and her son’s age in the present time but also how their ages in 6 years are related to their ages two years ago.

To show this, we’ll subtract 2 from their ages now for their ages 2 years ago then add 6 to their current ages for their ages 6 years later .

Great! We now have variables and algebraic expressions to represent Penelope and Zack’s current ages as well as their ages in the past and in the future. Moving forward, let’s go through the important details given in the problem and create an equation from each statement.

1) The difference between the ages of Penelope and her son, Zack, is 34 .

Remember that Penelope is Zack’s mother so she’s definitely older than him. Therefore, we are subtracting Zack’s age from Penelope’s age to find the difference.

2) In six years, Penelope will be four times as old as Zack’s age two years ago.

Here are our two equations:

• Equation 1: [latex]p – z = 34[/latex]
• Equation 2: [latex]p + 6 = 4(z – 2)[/latex]

Let’s now work on equation 1 to solve for [latex]p[/latex].

Next, we’ll replace [latex]p[/latex] with [latex]34 + z[/latex] in equation 2 then solve for [latex]z[/latex].

• Penelope’s current age: [latex]p = 34 + z = 34 + ({\color{red}16}) = {\textbf{50}}[/latex]
• Zack’s current age: [latex]z = {\textbf{16}}[/latex]

How about we replace the unknown values in our table and also find out what their past and future ages are?

Going back to our original question, how old are they now?

Answer: Penelope is currently 50 years old while her son, Zack, is 16 years old.

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Chapter 7: Factoring

7.9 Age Word Problems

One application of linear equations is what are termed age problems. When solving age problems, generally the age of two different people (or objects) both now and in the future (or past) are compared. The objective of these problems is usually to find each subject’s current age. Since there can be a lot of information in these problems, a chart can be used to help organize and solve. An example of such a table is below.

Example 7.9.1

Joey is 20 years younger than Becky. In two years, Becky will be twice as old as Joey. Fill in the age problem chart, but do not solve.

• The first sentence tells us that Joey is 20 years younger than Becky (this is the current age)
• The age change for both Joey and Becky is plus two years
• In two years, Becky will be twice the age of Joey in two years

Using this last statement gives us the equation to solve:

B + 2 = 2 ( B − 18)

Example 7.9.2

Carmen is 12 years older than David. Five years ago, the sum of their ages was 28. How old are they now?

• The first sentence tells us that Carmen is 12 years older than David (this is the current age)
• The second sentence tells us the age change for both Carmen and David is five years ago (−5)

Filling in the chart gives us:

The last statement gives us the equation to solve:

Five years ago, the sum of their ages was 28

[latex]\begin{array}{rrrrrrrrl} (D&+&7)&+&(D&-&5)&=&28 \\ &&&&2D&+&2&=&28 \\ &&&&&-&2&&-2 \\ \hline &&&&&&2D&=&26 \\ \\ &&&&&&D&=&\dfrac{26}{2} = 13 \\ \end{array}[/latex]

Therefore, Carmen is David’s age (13) + 12 years = 25 years old.

Example 7.9.3

The sum of the ages of Nicole and Kristin is 32. In two years, Nicole will be three times as old as Kristin. How old are they now?

• The first sentence tells us that the sum of the ages of Nicole (N) and Kristin (K) is 32. So N + K = 32, which means that N = 32 − K or K = 32 − N (we will use these equations to eliminate one variable in our final equation)
• The second sentence tells us that the age change for both Nicole and Kristen is in two years (+2)

In two years, Nicole will be three times as old as Kristin

[latex]\begin{array}{rrrrrrr} N&+&2&=&3(34&-&N) \\ N&+&2&=&102&-&3N \\ +3N&-&2&&-2&+&3N \\ \hline &&4N&=&100&& \\ \\ &&N&=&\dfrac{100}{4}&=&25 \\ \end{array}[/latex]

If Nicole is 25 years old, then Kristin is 32 − 25 = 7 years old.

Example 7.9.4

Louise is 26 years old. Her daughter Carmen is 4 years old. In how many years will Louise be double her daughter’s age?

• The first sentence tells us that Louise is 26 years old and her daughter is 4 years old
• The second line tells us that the age change for both Carmen and Louise is to be calculated ([latex]x[/latex])

In how many years will Louise be double her daughter’s age?

[latex]\begin{array}{rrrrrrr} 26&+&x&=&2(4&+&x) \\ 26&+&x&=&8&+&2x \\ -26&-&2x&&-26&-&2x \\ \hline &&-x&=&-18&& \\ &&x&=&18&& \end{array}[/latex]

In 18 years, Louise will be twice the age of her daughter.

For Questions 1 to 8, write the equation(s) that define the relationship.

• Rick is 10 years older than his brother Jeff. In 4 years, Rick will be twice as old as Jeff.
• A father is 4 times as old as his son. In 20 years, the father will be twice as old as his son.
• Pat is 20 years older than his son James. In two years, Pat will be twice as old as James.
• Diane is 23 years older than her daughter Amy. In 6 years, Diane will be twice as old as Amy.
• Fred is 4 years older than Barney. Five years ago, the sum of their ages was 48.
• John is four times as old as Martha. Five years ago, the sum of their ages was 50.
• Tim is 5 years older than JoAnn. Six years from now, the sum of their ages will be 79.
• Jack is twice as old as Lacy. In three years, the sum of their ages will be 54.

Solve Questions 9 to 20.

• The sum of the ages of John and Mary is 32. Four years ago, John was twice as old as Mary.
• The sum of the ages of a father and son is 56. Four years ago, the father was 3 times as old as the son.
• The sum of the ages of a wood plaque and a bronze plaque is 20 years. Four years ago, the bronze plaque was one-half the age of the wood plaque.
• A man is 36 years old and his daughter is 3. In how many years will the man be 4 times as old as his daughter?
• Bob’s age is twice that of Barry’s. Five years ago, Bob was three times older than Barry. Find the age of both.
• A pitcher is 30 years old, and a vase is 22 years old. How many years ago was the pitcher twice as old as the vase?
• Marge is twice as old as Consuelo. The sum of their ages seven years ago was 13. How old are they now?
• The sum of Jason and Mandy’s ages is 35. Ten years ago, Jason was double Mandy’s age. How old are they now?
• A silver coin is 28 years older than a bronze coin. In 6 years, the silver coin will be twice as old as the bronze coin. Find the present age of each coin.
• The sum of Clyde and Wendy’s ages is 64. In four years, Wendy will be three times as old as Clyde. How old are they now?
• A sofa is 12 years old and a table is 36 years old. In how many years will the table be twice as old as the sofa?
• A father is three times as old as his son, and his daughter is 3 years younger than his son. If the sum of all three ages 3 years ago was 63 years, find the present age of the father.

Answer Key 7.9

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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"Age" Word Problems

Age Probs Diophantus

What are "age" word problems?

"Age" type word problems are those which compare two persons' ages, or one person's ages at different times in their lives, or some combination thereof.

Here's an example from my own life:

Content Continues Below

Age Word Problems

• In January of the year 2000, I was one more than eleven times as old as my son Will. In January of 2009, I was seven more than three times as old as him. How old was my son in January of 2000?

Obviously, in "real life" you'd have walked up to my kid and asked him how old he was, and he'd have proudly held up three grubby fingers, but that won't help you on your homework.

Here's how you'd figure out his age, if you'd been asked the above question in your math class:

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First, I'll need to name things and translate the English into math.

Since my age was defined in terms of Will's, I'll start with a variable for Will's age. To make it easy for me to remember the meaning of the variable, I will pick W to stand for "Will's age at the start, in the year 2000". Then Will's age in 2009, being nine years later, will be W + 9 . So I have the following information:

Will's age in 2000: W

Will's age in 2009: W + 9

My age was defined in terms of the above expressions. In the year 2000, I was "eleven times Will's age in the year 2000, plus one more", giving me:

my age in 2000: 11(W) + 1

My age in 2009 was also defined in terms of Will's age in 2009. Specifically, I was "three times Will's age in 2009, plus seven more", giving me:

my age in 2009: 3(W + 9) + 7

But I was also nine years older than I had been in the year 2000, which gives me another expression for my age in 2009:

my age in 2009: [ 11(W) + 1 ] + 9

My age in 2009 was my age in 2009. This fact means that the two expressions for "my age in 2009" must represent the same value. And this fact, in turn, allows me to create an equation — by setting the two equal-value expressions equal to each other:

3(W + 9) + 7 = [11(W) + 1] + 9

Solving, I get:

3W + 27 + 7 = 11W + 1 + 9

3W + 34 = 11W + 10

34 = 8W + 10

Since I set up this equation using expressions for my age, it's tempting to think that 3 = W stands for my age. But this is why I picked W to stand for "Will's age"; the variable reminds me that, no, 3 = W stands for Will's age, not mine.

And this is exactly what the question had asked in the first place. How old was Will in the year 2000?

Will was three years old.

Note that this word problem did not ask for the value of a variable; it asked for the age of a person. So a properly-written answer reflects this. " W = 3 " would not be an ideal response.

What are the steps for solving an age-based word problem?

The important steps for solving an age-based word problem are as follows:

• Figure out what is defined in terms of something else
• Set up a variable for that "something else" (labelling it clearly with its definition)
• Create an expression for the first time frame, and then
• Increment the expressions by the required amount (in the example above, this increment was nine years) to reflect the passage of time.

Don't try to use the same variable or expression to stand for two different things! Since, in the above, W stands for Will's age in 2000, then W can not also stand for his age in 2009. Make sure that you are very explicit about this when you set up your variables, expressions, and equations; write down the two sets of information as two distinct situations.

• Currently, Andrei is three times Nicolas' age. In ten years, Andrei will be twelve years older than Nicolas. What are their ages now?

Andrei's age in defined in terms of Nicolas' age, so I'll pick a variable for Nicolas' age now; say, " N ". This allows me to create an expression for Andrei's age now, which is three times that of Nicolas.

Nicolas' age now: N

Andrei's age now: 3N

In ten years, they each will be ten years older, so I'll add 10 to each of the above for their later ages.

Nicolas' age later: N + 10

Andrei's age later: 3N + 10

But I am also given that, in ten years, Andrei will be twelve years older than Nicolas. So I can create another expression for Andrei's age in ten years; namely, I'll take the expression for Nicolas' age in ten years, and add twelve to that.

Andrei's age later: [N + 10] + 12

Since Andrei's future age will equal his future age, I can take these two expressions for his future age, set them equal (thus creating an equation), and solve for the value of the variable.

3N + 10 = [N + 10] + 12

3N + 10 = N + 22

2N + 10 = 22

Okay; I've found the value of the variable. But, looking back at the original question, I see that they're wanting to know the current ages of two people. The variable stands for the age of the younger of the two. Since the older is three times this age, then the older is 18 years old. So my clearly-stated answer is:

Nicolas is 6 years old.

Andrei is 18 years old.

• One-half of Heather's age two years from now plus one-third of her age three years ago is twenty years. How old is she now?

This problem refers to Heather's age two years into the future and three years back in the past. Unlike most "age" word problems, this exercise is not comparing two different people's ages at the same point in time, but rather the same person's ages at different points in time.

They ask for Heather's age now, so I'll pick a variable to stand for this unknown; say, H . Then I'll increment this variable in order to create expressions for "two years ago" and "two years from now".

age two years from now: H + 2

age three years ago: H − 3

Now I need to create expressions, using the above, which will stand for certain fractions of these ages:

The sum of these two expressions is given as being " 20 ", so I'll add the two expressions, set their sum equal to 20 , and solve for the variable:

H / 2 + 1 + H / 3 − 1 = 20 H / 2 + H / 3 = 20 3H + 2H = 120 5H = 120 H = 24

Okay; I've found the value of the variable. Now I'll go back and check my definition of that variable (so I see that it stands for Heather's current age), and I'll check for what the exercise actually asked me to find (which was Heather's current age). So my answer is:

Heather is 24 years old.

Note: Remember that you can always check your answer to any "solving" exercise by plugging that answer back into the original problem. In the case of the above exercise, if Heather is 24 now, then she will be 26 in two years, half of which is 13 ; three years ago, she would have been 21 , a third of which is 7 . Adding, I get 13 + 7 = 20 , so my solution checks.

• In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is 68 . How old is each one now?

The grandfather's age is defined in terms of Miguel's age, so I'll pick a variable to stand for Miguel's age. Since they're asking me for current ages, my variable will stand for Miguel's current age.

Miguel's age now: m

Now I'll use this variable to create expressions for the various items listed in the exercise.

Miguel's age last year: m − 1

six times Miguel's age last year: 6( m − 1)

Miguel's grandfather's age will, in another three years, be six times what Miguel's age was last year. This means that his grandfather is currently three years less than six times Miguel's age from last year, so:

grandfather's age now: 6( m − 1) − 3

Summing the expressions for the two current ages, and solving, I get:

( m ) + [6( m − 1) − 3] = 68

m + [6 m − 6 − 3] = 68

m + [6 m − 9] = 68

7 m − 9 = 68

Looking back, I see that this variable stands for Miguel's current age, which is eleven. But the exercise asks me for the current ages of bother of them, so:

Last year, Miguel would have been ten. In three more years, his grandfather will be six times ten, or sixty. So his grandfather must currently be 60 −3 = 57 .

Miguel is currently 11 .

His grandfather is currently 57 .

The puzzler on the next page is an old one (as in "Ancient Greece" old), but it keeps cropping up in various forms. It's rather intricate.

URL: https://www.purplemath.com/modules/ageprobs.htm

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Algebra: Age Problems

Related Pages Word Problems Involving Ages Solving Age Word Problems Using Algebra More Algebra Lessons

Age problems are algebra word problems that deal with the ages of people currently, in the past or in the future. The ages of the people are compared and usually the objective would be to find their current age.

Solving Age Problems in Algebra

If the problem involves a single person, then it is similar to an Integer Problem. Read the problem carefully to determine the relationship between the numbers. See example involving a single person .

In these lessons, we will learn how to solve age problems that involve the ages of two or more people.

In this case, using a table would be a good idea. A table will help you to organize the information and to write the equations. This is shown in the following age word problems that involve more than one person.

Age Problems Involving More Than One Person

Example: John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three times as old as Alice. How old is Peter now?

Solution: Step 1: Set up a table.

Step 2: Fill in the table with information given in the question. John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three times as old as Alice. How old is Peter now?

Let x be Peter’s age now. Add 5 to get the ages in 5 yrs.

Write the new relationship in an equation using the ages in 5 yrs.

In 5 years, John will be three times as old as Alice. 2 x + 5 = 3( x – 5 + 5) 2 x + 5 = 3 x

Isolate variable x x = 5 Answer: Peter is now 5 years old.

Example: John’s father is 5 times older than John and John is twice as old as his sister Alice. In two years time, the sum of their ages will be 58. How old is John now?

Step 2: Fill in the table with information given in the question. John’s father is 5 times older than John and John is twice as old as his sister Alice. In two years time, the sum of their ages will be 58. How old is John now?

Let x be John’s age now. Add 2 to get the ages in 2 yrs.

Write the new relationship in an equation using the ages in 2 yrs.

In two years time, the sum of their ages will be 58.

Answer: John is now 8 years old.

Video Lessons - More Examples Age Word Problems

Example: Mary is 3 times as old as her son. In 12 years, Mary’s age will be one year less than twice her son’s age. Find their ages now.

Note that this problem requires a chart to organize the information. The rows of the chart can be labeled as Mary and Son, and the columns of the chart can be labeled as “age now” and “age in 12 years”. The chart is then used to set up the equation.

• Sue is 5 years younger than Brian. In 7 years, the sum of their ages will be 49 years. How old is each now?
• Maria is 10 years older than Sonia. Eight years ago, Maria was 3 times Sonia’s age. How old is each now?

• The sum of the ages of a man and his son is 82 years. How old is each, if 11 years ago, the man was twice his son’s age?
• The sum of the ages of a woman and her daughter is 38 years. How old is each, if the woman will be triple her daughter’s age in 9 years?

• Salman is 108 years old. Jonathan is 24 years old. How many years will it take for Salman to be exactly four times as old as Jonathan?
• Tarush is five times as old as Arman is today. 85 years ago, Tarush was 10 times as old as Arman. How old is Arman today?

Example: Zack is four times as old as Salman. Zack is also three years older than Salman. How old is Zack?

Examples For Practise:

• Soo is 8 years older than Marco. In four years, Soo will be twice as old as Marco. How old is Soo?
• The sum of Abbie’s age and Iris’s age is 42 years old. 11 years ago, Abbie was three times as old as Iris. How old will Abbie be in two years?

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

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How to Deciphering the Puzzle of Time: A Step-by-Step Guide to Solving Age Problems in Mathematics

Age problems in mathematics are a type of word problem where you are asked to determine the ages of people at different times based on given information. These problems often involve setting up and solving linear equations. Here's a step-by-step guide to understanding and solving age problems, particularly focusing on the condition that involves calculating a person's age in the past and future:

Step-by-step Guide to Deciphering Age Problems in Mathematics

Step 1: understand the problem.

• Read Carefully: Carefully read the problem to understand what you are asked to find.
• Identify Key Information: Look for information about the current ages of individuals, their ages at different times in the past or future, and the relationships between these ages.

Step 2: Define Variables

• Choose Variables: Typically, let \(x\) represent the current age of a person of interest. Ensure that you clearly define what \(x\) represents.
• Represent Other Ages: If the problem involves the person’s age in the past or future, express these ages in terms of \(x\). For example, a person’s age \(a\) years ago is \(x−a\), and their age \(a\) years in the future is \(x+a\).

Step 3: Set Up Equations

• Use the Given Information: Based on the problem’s information, create one or more equations. These equations might relate the ages of different people or describe how a person’s age changes over time.
• Be Consistent: Ensure that the ages are consistent with the timeline given in the problem.

Step 4: Solve the Equations

• Manipulate the Equation: Use algebraic methods to solve for \(x\). This might involve simplifying expressions, combining like terms, or using methods like substitution or elimination in the case of multiple equations.
• Find the Age(s): Solve for \(x\) and any other unknowns in your equations.

Step 5: Verify Your Solution

• Check for Accuracy: Substitute your solution back into the original equations to ensure they hold true.
• Ensure Reasonability: Make sure the solution makes sense in the context of the problem. For instance, ages should be realistic and non-negative.

By following these steps methodically, you can solve a variety of age problems, understanding the relationships between ages at different times.

John is currently twice as old as his brother Peter. Five years ago, John was three times as old as Peter. How old are John and Peter now?

• Let Peter’s current age be \(x\) years.
• John’s current age is \(2x\) years.
• Five years ago, Peter was \(x−5\) and John was \(2x−5\).
• The equation from the problem:\(2x−5=3(x−5)\).
• Solve: \(2x−5=3x−15\) leads to \(x=10\).
• John is \(2×10=20\) years old.

Peter is \(10\) years old, and John is \(20\) years old.

A mother is four times as old as her daughter. In \(20\) years, she will be twice as old as her daughter. How old are they now?

• Let the daughter’s age be \(x\).
• The mother’s age is \(4x\).
• In \(20\) years, the daughter will be \(x+20\) and the mother \(4x+20\).
• The future age equation: \(4x+20=2(x+20)\).
• Solve: \(4x+20=2x+40\) leads to \(x=10\).
• The mother is \(4×10=40\) years old.

The daughter is \(10\) years old, and the mother is \(40\) years old.

by: Effortless Math Team about 9 months ago (category: Articles )

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Here are some examples for calculating age in word problems.

Phil is Tom's father. Phil is 35 years old. Three years ago, Phil was four times as old as his son was then. How old is Tom now?

First, circle what it is you must ultimately find— how old is Tom now? Therefore, let t be Tom's age now. Then three years ago, Tom's age would be t – 3. Four times Tom's age three years ago would be 4( t – 3). Phil's age three years ago would be 35 – 3 = 32. A simple chart may also be helpful.

Now, use the problem to set up an equation.

Therefore, Tom is now 11.

Lisa is 16 years younger than Kathy. If the sum of their ages is 30, how old is Lisa?

First, circle what you must find— how old is Lisa? Let Lisa equal x . Therefore, Kathy is x + 16. (Note that since Lisa is 16 years younger than Kathy, you must add 16 years to Lisa to denote Kathy's age.) Now, use the problem to set up an equation.

Therefore, Lisa is 7 years old.

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Age Calculator

The Age Calculator can determine the age or interval between two dates. The calculated age will be displayed in years, months, weeks, days, hours, minutes, and seconds.

Related Date Calculator | Time Calculator

The age of a person can be counted differently in different cultures. This calculator is based on the most common age system. In this system, age increases on a person's birthday. For example, the age of a person who has lived for 3 years and 11 months is 3, and their age will increase to 4 on their next birthday one month later. Most western countries use this age system.

In some cultures, age is expressed by counting years with or without including the current year. For example, a person who is twenty years old is the same age as another person who is in their twenty-first year of life. In one of the traditional Chinese age systems, people are born at age 1 and their age increases up at the Traditional Chinese New Year rather than their birthday. For example, if one baby is born just one day before the Traditional Chinese New Year, 2 days later, the baby will be 2 even though he/she is only 2 days old.

In some situations, the months and day result of this age calculator may be confusing, especially when the starting date is the end of a month. For example, we count Feb. 20 to Mar. 20 to be one month. However, there are two ways to calculate the age from Feb. 28, 2022 to Mar. 31, 2022. If we consider Feb. 28 to Mar. 28 to be one month, then the result is one month and 3 days. If we consider both Feb. 28 and Mar. 31 as the end of the month, then the result is one month. Both calculation results are reasonable. Similar situations exist for dates like Apr. 30 to May 31, May 30 to June 30, etc. The confusion comes from the uneven number of days in different months. In our calculations, we use the former method.

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The authors’ research found that three persistent problems plague DEI initiatives: They do not connect to operational or strategic goals and objectives; they do not include the rank-and-file; and they are often implemented through periodic efforts like annual diversity training that aren’t integrated into day-to-day work processes. Organizations can overcome these problems by using IT in three ways.

Diversity, equity, and inclusion (DEI) programs are under attack. Confronted by high costs, mixed outcomes , unclear organizational benefits , and a political and regulatory backlash , organizations are rolling back their initiatives. Google and Meta, for example, recently reduced investment in their DEI programs and let go of DEI staff.

• Monideepa Tarafdar is Charles J. Dockendorff Endowed Professor at the Isenberg School of Management at the University of Massachusetts Amherst.
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• Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go.
• How to solve math problems step-by-step?
• To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
• Practice Makes Perfect Learning math takes practice, lots of practice. Just like running, it takes practice and dedication. If you want...

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