problem solving age problems

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

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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?

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Breadcrumbs

Age-related problems.

If x = present age of a person x – 3 = age of the person 3 years ago x + 5 = age of the person 5 years from now or 5 years hence  

Note: The difference of the ages of two persons is constant at any time.  

If A = present age of Albert and B = present age of Bryan

Sum of their ages 4 years ago = ( A - 4) + ( B - 4) Sum of their ages 2 years hence = ( A + 2) + ( B + 2) Difference of their ages = A - B  

Example Six years ago, Romel was five times as old as Lejon. In five years, Romel will be three times as old as Lejon. What is the present age of Lejon?  

Solution Click here to expand or collapse this section Let $R$ = present age of Romel $L$ = present age of Lejon  

Six years ago $R - 6 = 5(L - 6)$

$R - 6 = 5L - 30$

$R = 5L - 24$  

Five years from now (in five years) $R + 5 = 3(L + 5)$

$R + 5 = 3L + 15$

$R = 3L + 10$  

Substitute R = 5 L - 24 $5L - 24 = 3L + 10$

$L = 17 \, \text{ yrs old}$           answer  

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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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• Bob's\:age\:is\:twice\:that\:of\:Barry's.\:Five\:years\:ago,\:Bob\:was\:three\:times\:older\:than\:Barry.\:Find\:the\:age\:of\:both.
• Mr\:John\:is\:7\:times\:as\:old\:as\:his\:grandson.In\:3\:years\:time,\:the\:sum\:of\:their\:ages\:will\:be\:102.\:How\:old\:is\:Mr\:John\:now?
• Dan\:is\:twice\:as\:old\:as\:Bob.\:The\:sum\:of\:their\:ages\:seven\:years\:ago\:was\:13.\:How\:old\:are\:they\:now?

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Algebra Age-Related Word Problem Worksheets

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Problem-Solving to Determine Missing Variables

Many of the  SAT s, tests, quizzes, and textbooks that students come across throughout their high school mathematics education will have algebra word problems that involve the ages of multiple people where one or more of the participants' ages are missing.

When you think about it, it is a rare opportunity in life where you would be asked such a question. However, one of the reasons these types of questions are given to students is to ensure they can apply their knowledge in a problem-solving process.

There are a variety of strategies students can use to solve word problems like this, including using visual tools like charts and tables to contain the information and by remembering common algebraic formulas for solving missing variable equations.

Birthday Algebra Age Problem

 Deb Russell

In the following word problem, students are asked to identify the ages of both of the people in question by giving them clues to solve the puzzle. Students should pay close attention to key words like double, half, sum, and twice, and apply the pieces to an algebraic equation in order to solve for the unknown variables of the two characters' ages.

Check out the problem presented to the left: Jan is twice as old as Jake and the sum of their ages is five times Jake's age minus 48. Students should be able to break this down into a simple algebraic equation based on the order of the steps, representing Jake's age as a and Jan's age as 2a : a + 2a = 5a - 48.

By parsing out information from the word problem, students are able to then simplify the equation in order to arrive at a solution. Read on to the next section to discover the steps to solving this "age-old" word problem.

Steps to Solving the Algebraic Age Word Problem

First, students should combine like terms from the above equation, such as a + 2a (which equals 3a), to simplify the equation to read 3a = 5a - 48. Once they've simplified the equation on either side of the equals sign as much as possible, it's time to use the distributive property of formulas to get the variable  a  on one side of the equation.

In order to do this, students would subtract 5a  from both sides resulting in -2a = - 48. If you then divide each side by -2 to separate the variable from all real number in the equation, the resulting answer is 24.

This means that Jake is 24 and Jan is 48, which adds up since Jan is twice Jake's age, and the sum of their ages (72) is equal five times Jake's age (24 X 5 = 120) minus 48 (72).

An Alternate Method for the Age Word Problem

No matter what word problem you're presented with in algebra , there's likely going to be more than one way and equation that's right to figure out the correct solution. Always remember that the variable needs to be isolated but it can be on either side of the equation, and as a result, you can also write your equation differently and consequently isolate the variable on a different side.

In the example on the left, instead of needing to divide a negative number by a negative number like in the solution above, the student is able to simplify the equation down to 2a = 48, and if he or she remembers, 2a is the age of Jan! Additionally, the student is able to determine Jake's age by simply dividing each side of the equation by 2 to isolate the variable a.

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Lesson Age problem for three participants

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Tricks To Solve Age-Based Problems

Problems on ages is an important chapter for SSC/Banking and all other exams. In SSC Prelims 2 to 3 questions are asked and in mains 4 to 5 questions can be asked on this topic. As most of the questions on this topic of this chapter can be solved by analytical thinking. So, it plays a crucial role in the examination. Sometimes the question asked are confusing and complicated, so we have to learn the concept well to easily score marks for these types of questions.

Table of Contents:

1. Problems on ages :- basics and concept

2. Useful tips and tricks to solve the problems of ages

3. Important Formulas

4. Sample Questions – problems on ages

1. Problems on ages – basics and concept          Problems on ages basically consist of information about the ages of two or more persons and a relationship between their ages in the past/present/future.

In the quantitative aptitude section( banking /SSC exams) the problems on ages are kind of brain teasers,  at the very first time when a candidate studies this it may seem to be complex, but it becomes easy when solved step by step. 

In the quantitative aptitude sector problems on ages mostly are asked for 2-3 marks but there may be a possibility of age-based questions being asked as a part of the data sufficiency or data interpretation. So it is very important to clarify the concept to each and every candidate. 

The useful tips given below are for the Candidates who are not much familiar with the concept and tendency to skip the problems on ages or answer them incorrectly. These tips may help you to clear the concept and answers the questions correctly.

I) To read the question carefully and gradually form the equation is the most important thing. It helps you answer the question.

II) Basic things like subtraction, addition, multiplication, and division will help a candidate reach the answer and no complicated calculations are required to answer these questions.

III) Arrange the values correctly and use them as linear equations.

IV) Once the linear equation has been formed, solve the equation to find the answer.

V) The final step is to recheck the answer obtained by placing it in the equation formed to ensure that no error has been made while calculating.

The ‘problems on ages’ is one such topic which is not just asked in the preliminary phase of the examination but questions from this topic may also be asked in the main examination in a rather complex manner.

The formulas given below are related to the problems on ages which may help you to answer the problems quicker and also get a better idea of the concept:-

I) Suppose if the present age of a person is x, then after n years, the age of the person will be (x+n) years.

II) Suppose if the present age of a person is x, then before n years the age of the person will be (x-n) years.

III) Suppose the ratio of two persons is p : q, then their age will be px and qx respectively.

IV) Suppose the present age of a person is x, then  n times the present age will be (x * n) years

V) Suppose the present age of a person is x, then 1/n of the age shall be equal to (x/n) years

These formulas and tricks will help you in solving the questions easily and more efficiently.  

1. Question Five years ago the ratio of the ages of Amit and Neha was 8 : 7. Three years hence, the ratio of their ages will be 12 : 11. what is Neha’s age at present?

A) 13 years

B) 16 years

D) 19 years

E) None of these

Answer:- D Explanation:-

Let the age of Amit and Neha five years ago 8x and 7x respectively.

Amit’s present age = (8x + 5)

Neha’s present age = (7x + 5)

Now, as per the equation,

{(8x + 5) + 3} /{( 7x + 5) + 3} = 12/11                          => (8x + 8)/(7x + 8) = 12/11

=>  88x + 88 = 84x + 96  

=>  4x = 8  

⇒  x = 2.  

 ∴     Neha’s present age = (7x + 5) = (7 × 2 + 5) = 19 years.

Hence, option D is correct.

Amit : Neha =  8 : 7      ( -5)

                     = 12 : 11   (+3)

= ( – 5 + 3 ) = 2

So, Neha’s age = 7 × 2 = 14

Neha’s present age = 14 + 5 = 19 years  

Or, Neha’s age = 11 × 2 = 22

Neha’s present age = 22 – 3 = 19 years

2.Question The present ages of the three friends are in the ratio 3 : 5 : 7. Eight years ago, the sum of their ages was 96. find the sum of the present ages of the first two friends (in years)?

A) 42 years

B) 64 years

C) 70 years

D) 67 years

Answer:- B Explanation:-

Let the present age of three friend’s are = 3x, 5x and 7x

(3x – 8) + (5x – 8) + (7x – 8) = 96

15x – 24 = 96  15x = 120 x = 8

Their present ages are 24 years, 40 years and 56 years respectively. The sum of the present ages of the first two friends = ( 24 + 40) = 64 years Hence, option B is correct.

3. Question The ratio of the father’s age to his son-in-law’s age is 9 : 5. The product of their ages is 1125. The ratio of their ages after five years will be :

Answer – D  Explanation:-

Let the present ages of Father and son-in-law be 9x and 5x respectively.

9x × 5x = 1125       45×2 = 1125        x2 = 25    x = 5.

Required ratio  = (9x + 5) : (5x + 5)  ⇒   50 : 30  ⇒  5 : 3.  

4. Question The total ages of Aman, Nageshwar and Satyam is 96 years. The ratio of their ages before 5 years was 2 : 3 : 4. What is the present age of Aman?

A) 23 years

B) 32 years

C) 21 years

D) 33 years

Answer:- A Explanation:-

Let the ages of Aman, Nageshwar and Satyam 5 years ago be 2x, 3x and 4x years respectively.

So, total of their present ages will be,

(2x + 5) + (3x +5) + (4x + 5) = 96 => 9x + 15 = 96 => 9x = 81 x = 9.

So, the present age of Aman =( 2x + 5 )=( 2 × 9 + 5 )= 23 years.

Hence, option A is correct.

5. Question The ratio of the present ages of two boys is 2 : 3 and six years back, the ratio was 1 : 3. What will be the ratio of their ages after 4 years ?.

Answer :-. B Explanation:-

Assume 2x and 3x be the present age of two friends respectively. 

Then,      2x – 6  = 1 3x – 63 ⇒ 6x – 18 = 3x – 6   ⇒  3x  = 12   ⇒  x = 4.

So,  required ratio = (2x + 4) : (3x + 4)  ⇒  12 : 16  ⇒  3 : 4.  

Hence, option B is correct.

2 : 3    —– ( × 2)  = 4 : 6   (× 2)  = 8 : 12

1 : 3    —– ( × 1)  = 1 : 3  ( × 2)  = 2 : 6

So, the present ages of the two boys 8 years and 12 years respectively.

4 years later their ages will be (8+4) = 12 years and (12+4)= 16 years.

Ratio = 12 : 16 = 3 : 4.

6. Question mother’s age is twice of her daughter. Before ten years, she was twelve times as old as her daughter. What is the sum of the present age of a mother and her daughter together?

A) 33 years

B) 36 years

C) 40 years

D) 26 years

Answer:-.  A Explanation:-

Let, the present age of daughter = p 

Present age of her mother = 2p

10 yr back, the ratio of their ages was = 12: 1

(2p – 10)/(p – 10) = 12 /1 => 2p – 10 = 12p – 120 => 10p = 110 => P = 110/10 = 11

Present age of daughter = 11 years

Present age of mother = 11× 2 = 22 years

Thus, the sum of the present age of mother and her daughter 

= 22+11 yrs = 33 years Here , A is the correct answer. 

Mother : Daughter =  2 : 1     (× 11)  = 22 : 11

                                 = 12 : 1   ( × 1)   = 12 : 1

Present age of Mother = 22 years

Present age of Daughter = 11 years

Thus, Sum of their ages = 22 + 11  = 33 years.

7. Question Amit is 2 years older than Bikash who is twice as old as Chetan. If the sum of the present ages of Amit, Bikash and Chetan be 52, then how old is Bikash?

A) 20 years B) 22 ears C) 15 years D) 13 years E) 21 years

Answer: A Explanation :-

Let the present age of Chetan = x years

So, Bikash’s present age = 2x

And Amit’s present age = 2x + 2

According to the question,

x+2x+2+2x = 52

So, Bikash’s age = 2×10 = 20 years Here ,the correct option is A .

8. Question In a family, of five-person, the total age of the elder brother and younger brother is 56 years and after four years the age of the elder brother will be three times that of the younger brother. What is the age of the two brothers respectively?

A) 12 years, 41 years              B) 15 years, 53 years C) 11 years, 34 years D) 12 years, 44 years E) 21 years, 42 years

Let the present age of the elder brother = x years The present age of the younger brother = y years

According to the question,  x+y = 56     ————-(1)

After 4 years, age of the elder brother = x+4

after 4 years the age of younger brother = y+4

x+4 = 3 (y+4)     ———–(2)

x+4 = 3y + 12

From the equation (1) we get,  x = 56-y

Now putting the value of x in equation 2, we get

(56-y) + 4 = 3y + 12

⇒60 – y = 3y + 12

So, the younger brother’s present age is =12 years

the elder brother’s present age = 56-12 = 44 years

Here the correct answer is D.

9. Question Akash is as much elder than Vijay as he is younger to Ketan and the sum of the ages of Vijay and Ketan is 48 yr, then find the age of Akash.

A) 36 years B) 24 years C) 27 years D) 18 years E) 22 years

Answer:- B Explanation:- Let the present age of Akash = x years and he is younger to Ketan by y years  Then, Ketan’s age = (x + y) years Vijay’s age = (x – y) years

Now, according to the question, Sum of ages of Ketan and Vijay = 48 (x + y) + (x – y) = 48 2x = 48 years x = 24 years

Hence, present age of Akash is 24 yr.

10. Question Your uncle is three times as old as you. 15 yr hence, your uncle will be twice as old as you. What is the sum of the present ages of you and your uncle?

A) 60 years B) 55 years C) 75 years D) 50 years E) none of these

Answer :- A Explanation :- Let, the present ages of your uncle and you be 3x and x years respectively

15 years later, ratio between your uncle and your age = 2 :1

Now, (3x + 15)/(x + 15) = 2/1 => 3x + 15 = 2x + 30 => x = 15

Thus, Sum of the ages of your uncle and your  = ( 3x + x ) = 4 × 15 = 60 years Here the correct answer is A .

11. Question The present age of Kamal is 5 times the age of Shiva. After 10 yr, Kamal will be 3 times as old as Shiva. What are the present ages of Kamal and Shiva?

A) 45 years, 9 years B) 55 years, 11 years C) 65 years, 13 years D) 40 years, 8 years E) 50 years, 10 years

Answer :- E Explanation:- Let the present age of Shivam = x years

Then, present age of Kamal = 5x years

After 10 years, the ratio of ages of Kamal and Shiva = 3 : 1

(5x + 10)/(x + 10) = 3/1 => 5x + 10 = 3x + 30 => 2x = 20 => x = 10

Present age of Shiva = 10 years Present age of Kamal = 10×5 = 50 years

12. Question 5 yr ago, the age of Sanjay was 4 times the age of Vikram and after 10 yr, Sanjay will be twice as old as Vikram. Find the present ages of Sanjay and Vikram.

A) 35 years, 12.5 years

B) 33 years, 11 years

C) 45 years, 16 years

D) 50 years, 18.5 years

Explanation:-

Let the present ages of Sanjay and Vikram be x years and y years, respectively.

According to the question, 5 years ago,

(x – 5)/( y – 5) = 4/1 => x – 5 = 4y  – 20 => x = 4y – 15      —————–(1)

10 years later, ( x + 10)/(y + 10) = 2 : 1 => x + 10 = 2y + 20 => x = 2y + 10      —————-(2)

Equating both the equations (1) and (2) ,we get,

4y – 15 = 2y + 10 => 2y = 25 => y = 12.5

Putting the value of y in equation (2),we get,

x = 2 × 12.5 + 10 => x = 35

Thus,the present age of Sanjay = 35 years And the present age of Vikram = 12.5 years

13.Question Four years ago, Sharma’s age was 3/4 times that of Raman. Four years hence, Sharma’s age will be 5/6 times that of Raman. What is the present age of Sharma?

(a) 16 yrs (b) 20 yr  (c) 15 yr  (d) 24 yr (e) 8 yr

Answer :-A Explanation :-

4 years ago,  let, Raman’s age = x years And Sharma’s age = 3x/4 years

Now, Present age of Raman = x + 4 years And present age of Sharma =( 3x/4 + 4 ) years

5/6 ( x + 4 + 4) = (3x/4 + 4 + 4) => 5/6 ( x + 8) = ( 3x/4 + 8) => 4(5x + 40) = 6( 3x + 32) => 20x + 160 = 18x + 192 => 2x = 32 => x = 16

Thus the present age of Sharma  =( 3/4 × 16 + 4) years = 16 years Here the correct answer is A.

14. Question The present ages of the two women are 36 and 50 yr, respectively. After p years, the ratio of their ages will be 3 : 4, then find the value of p.

(e) none of these

Answer :-C Explanation :-

According to the question, (36 + p)/( 50 + p) = 3/4 => 144 + 4p = 150 + 3p => p = 6 Thus, the value of p = 6.

15. Question The present ages of Rohit and Urvashi are 36 and 48 yr, respectively. What was the ratio between the ages of Urvashi and Rohit respectively 8 yr ago?

(e) None of the above

Answer :- D Explanation :-

Ratio = (48 – 8) : (36 – 8)          = 40 : 28          = 10 : 7 Here the correct answer is D.

16. Question The average age of two teachers is 35 yr. The average age of the two teachers and a student is 27 yr. What is the age of the student?

(c) 10.5 yr

Answer :- A

Explanation :-

Total age of two teachers = 35 x 2 = 70 yr

and total age of two teachers and students together

= 27 x 3 = 81 yr

: Age of the student = 81-70 = 11 yr

17.Question The average of the present ages of Saikat and Shivani is 36 yr. If Saikat is 8 yr older than Shivani, what is the Shivani’s present age?

Answer :- C

Total age of Saikat and Shivani

= 36 × 2 = 72 yr

Let age of Shivani = x yr

Then, age of Saikat = (x + 8) yr

Hence, Shivani’s present age = 32 years  

18. Question

The ratio between the present ages of Irfaan and Kapil is 3 : 8. After 8 years, Irfaan’s age will be 20 yr. What was Kapil’s age 5 years hence?

Answer :-A Explanation:- Let present ages of Irfaan and Kapil are

3x years and 8x years respectively.

3x + 8 = 20

=> 3x = 12

Kapil’s present age = 8x = 8×4 = 32 years

Hence, Kapil’s age 5 years hence = 32+5 = 37 years

19. Question The ratio between the present ages of Tanmay and Vivek is 3:7, respectively. After 4 yr, Vivek’s age will be 39 yr. What was Tanmay’s age 4 yr later?

Answer :- C Explanation :-

Let Tanmay and Vivek’s ages are 3x yr and 7x yr, respectively. 7x + 4 = 39 ⇒ x = 5 =3×5-4=11 yr Hence, Tanmay’s age 4 yr later = 3×5 + 4  = 19 years.

20. Question

The average age of a group of five girls is 24. If the present age of the youngest girl is 8 year, what was the average age of the group at the time of the birth of the youngest girl?

Total age of the group of five girls

= 24 x 5 = 120

Total age of four girls at the time of

birth of youngest girl=( 120 – 8×5) years

= 120 – 40 = 80 years

Hence, required average age = 80/4 = 20 yr

Hence the correct answer is E.

21. Question 30. The present age of Rahul’s mother is four times Rahul’s present age. Five years ago, Rahul’s mother was seven times as old as Rahul. What is the present age of Rahul’s mother?

Let the present age of Rahul = x.

Then, present age of Rahul’s mother= 4x

Now, 5 yr ago,

Rahul’s mother’s age = 7 x Rahul’s age

4x -5= 7(x – 5)

=> 4x – 5 = 7x – 35

=>3x = 30

=>x = 10

.. Rahul’s present age = x= 10 years

.. Rahul’s mother’s present age= 4x  = 4 x 10 = 40 yr

22. Question The age of Prakash is three times Arvind’s age. After 7 yr, Prakash will be twice Arvind’s age, then how many times will Prakash’s age be in another 14 yr time with respect to Arvind’s age then?

Let Arvind’s age = x yr

Then, Prakash’s age = 3x yr

3x + 7 = 2(x + 7)

3x + 7 = 2x + 14

:: Age of Arvind after 14 yr = 7+ 14 = 21 yr

Prakash’s present age = 21 yr

Hence, Prakash’s age is one time of Arvind’s age.

23. Question

10 years ago, the ratio of ages of a man and a woman was 13:17. After 17 years from now, the ratio of their ages will be 10:11 . What is the present age of the woman?

Let the ages of man and woman 10 yr before were 13x yr and 17x yr, respectively.

Then, present age of man = 13x + 10

and present age of woman = 17x + 10

 (13x + 10 + 17)/(17x + 10 + 17) = 10/11

=> (13x + 27)/(17x + 27) = 10/11

=> 143x + 297 = 170x + 270

=> 27x = 27

=> x = 1

Hence ,the present age of the woman  = (17×1 + 10) = 27 years

24.Question Kailash age is twice of Shekher’s age. 8 yr hence, the respective ratio between Kailash’s and Shekhar’s ages will be 22:13.What is Kailash’s present age?

(b) 18 yr (d) 30 yr (a) 26 yr (c) 42 yr (e) None of the above

Explanation:- Let Shekhar’s present age = x yr Then, Kailash’s present age = 2x yr According to the question, (2x + 8)/(x + 8 ) = 22/13 26x + 104 = 22x + 176 4x = 72 x = 18 Hence, Kailash’s age = 2 x 18 = 36 yr

25.Question Before 7 years, the ages of P and Q were in the ratio 4:5 and 7 yr hence, they will be in the ratio 5: 6. Find the age of Q in present.

Let, 7 yr ago, ages of P and Q were 4x years and 5x years, respectively.

Then, present age of P = 4x + 7  and present age of Q = 5x + 7

Now, according to the question,

(4x + 7+ 7)/(5x +7+7) =5/6

24x + 84 = 25x + 70

Hence, Q’s present age =(5x + 7)

= 5×14 + 7 years

= 77 years.

Hence the correct answer is D.

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20-Somethings Are in Trouble

They’re more depressed, more anxious, and lonelier than any other age group in America—but their distress has gone widely unnoticed.

Produced by ElevenLabs and News Over Audio (NOA) using AI narration.

What if I told you that one age group is more depressed, more anxious, and lonelier than any other in America?

You might assume I’m talking about teens. Mood disorders, self-harm, and suicide have become more common among adolescents in recent years ; article after article reports that social media is toxic for teen girls especially, eroding their self-esteem and leaving them disconnected. Or you might think of older adults, often depicted in popular culture and news commentary as isolated and unhappy, their health declining and their friends dropping away.

So perhaps you’d be surprised to hear the results of a Harvard Graduate School of Education survey on mental health in America: Young adults are the ones most in crisis. Even Richard Weissbourd, who led the study in 2022, was taken aback. His team found that 36 percent of participants ages 18 to 25 reported experiencing anxiety and 29 percent reported experiencing depression—about double the proportion of 14-to-17-year-olds on each measure. More than half of young adults were worried about money, felt that the pressure to achieve hurt their mental health, and believed that their lives lacked meaning or purpose. Teenagers and senior citizens are actually the two populations with the lowest levels of anxiety and depression, Weissbourd’s research has found.

Other studies of young adults have similarly alarming findings. According to the CDC , in 2020, depression was most prevalent among 18-to-24-year-olds (and least prevalent among those 65 or older). A 2023 Gallup poll found that loneliness peaked at ages 18 to 29. And, according to one meta-analysis spanning four decades, more and more young adults reported loneliness each year. When Weissbourd repeated his survey last year, young-adult anxiety and depression had also risen, to 54 and 42 percent, respectively. Still, the struggles of young adults have gone widely unnoticed. When Weissbourd got his data, “it was really upsetting,” he told me. “What is going on here? And why aren’t we talking about it more?”

The phase between adolescence and adulthood has long been daunting: You’re expected to figure out who you are, to create a life for yourself. That might sound exciting, as if all the doors are wide open, but much of the time it’s stressful—and modern challenges are making it harder. Young adults are more vulnerable than ever, but much of American society doesn’t see them that way.

One thing that gets Jennifer Tanner fired up is the myth that young adulthood is a carefree time. Many people see it as a perfect juncture, when you’re old enough to have agency but young enough to be free of big responsibilities. Commonly, though, it’s the inverse: You have new obligations but not the wisdom, support, or funds to handle them. Tanner is a developmental researcher studying “emerging adulthood,” typically defined as the years from age 18 to 29, and she thinks that many more established adults wish they could go back to that period and do things differently; in hindsight, it might seem like a golden age of possibility. “Everybody who’s 40 is like, I wish I was 18 .” Meanwhile, young adults are “like, The world’s on my shoulders and I have no resources ,” she told me. “We’re gaslighting the hell out of them all the time.”

Of course, being a teen isn’t easy either. Depression and anxiety are increasing among adolescents. But in high school, you’re more likely to have people keeping an eye on you, who’ll notice if you’re upset at home or if you don’t show up to school. Adults know that they should protect you, and they have some power to do it, Weissbourd said. After you graduate from high school or college, though, you might not have anyone watching over you. The friends you had in school may scatter to different places, and you may not be near your family. If you’re not regularly showing up to a workplace, either, you could largely disappear from the public eye. And if life is taking a toll, mental-health resources can be hard to come by, Tanner told me, because psychologists tend to specialize either in childhood and adolescence or adult services, which generally skew older.

Read: The real reason young adults seem slow to ‘grow up’

As soon as you become independent, you’re expected to find housing, land a satisfying job, and connect with a community. But achieving those hallmarks of adulthood is getting harder . College tuition has skyrocketed, and many young people are saddled with student loans. With or without such debt, finding a place to live can feel impossible, given the current dearth of affordable housing. In 2022, a full half of renters spent more than 30 percent of their income on rent and utilities—a precarious situation when you haven’t yet built up savings. Under rising financial stress, finding fulfilling work can come second to paying the bills, Weissbourd explained. But that might mean missing out on a career that gives you a sense of self-worth and meaning. Jillian Stile, a clinical psychologist who works with young adults, told me that a lot of her clients are “feeling like a failure.”

On top of that, the social worlds that young people once occupied are crumbling. In the recent past, young adults were more likely to marry and have kids than they are today. They might have befriended other parents or co-workers, or both. Commonly, they’d belong to a religious congregation. Now they’re marrying and starting families later, if at all. Those with white-collar jobs are more likely to work remotely or to have colleagues who do, making it hard to find friends or mentors through work, Pamela Aronson, a sociologist at the University of Michigan at Dearborn, told me. Religious-participation rates have plunged. Americans in general are spending more time alone , and they have fewer public places to hang out and talk with strangers. For young adults who haven’t yet established social routines, the decline of in-person gatherings can be especially brutal. “Until you build those new systems around yourself that you contribute to, and they contribute back to your health and well-being,” Tanner told me, “you’re on shaky ground.”

Read: The new age of endless parenting

Sources of companionship inevitably shift. Today, for example, more young people are getting support (emotional and financial) from parents ; 45 percent of 18-to-29-year-olds live with their folks. But that can be isolating if you don’t also have friends nearby. Family bonds, no matter how wonderful, aren’t substitutes for a group of peers going through this sometimes-scary life phase at the same time.

Without a sense of belonging, the world can seem bleak. In Weissbourd’s study, 45 percent of young adults said they had a “sense that things are falling apart,” 42 percent said gun violence in schools was weighing on them, 34 percent said the same of climate change, and 30 percent reported worrying about political leaders being incompetent or corrupt. These issues don’t affect only young adults, but they might feel particularly grim if you can’t imagine what your life will look like in a decade. When it comes to “anxiety and depression,” Weissbourd told me, “it’s not only about your past—it’s about how you imagine your future.” And young adults? “They’re not hopeful.”

A rocky start to adulthood could cast a shadow over the rest of someone’s life. Aronson reminded me that, on average, Millennials have “less wealth than their predecessors at the same age—because their incomes were lower, because they started their jobs during a recession.” Gen Z spends a greater portion of its money on essentials than Millennials did at their age. That doesn’t bode well for Gen Z’s future finances. And there are other concerns: Maybe, if you can’t afford to pursue a rewarding job when you’re young, you’ll work your way up in a career you don’t care about—and end up feeling stuck. Perhaps if you don’t make genuine friends in young adulthood— commonly a time when people form long-lasting bonds—you’ll be lonelier in middle age. And if you lean exclusively on your parents, what will you do when they die?

Leaving individual young adults responsible for overcoming societal obstacles clearly isn’t working. “I don’t think we’re going to therapize or medicate our way out of this problem,” Weissbourd, a therapist himself, told me. He wants to see more “social infrastructure”: Libraries might arrange classes, volunteer opportunities, or crafting sessions that would be open to people of all ages but that could allow isolated young people to feel part of something. Doctors might ask young-adult patients about loneliness and offer resources to connect them with other people. Colleges could assign students an adviser for all four years and offer courses to guide students through the big questions about their place in the world. (Weissbourd teaches one at Harvard called “Becoming a Good Person and Leading a Good Life.”) Aronson suggested that workplaces should hold mentoring programs for young employees. And of course, student-loan-debt forgiveness, government support for higher education, affordable housing, and more extensive mental-health-care coverage wouldn’t hurt.

First, older adults need to acknowledge this crisis. Seeing young people as worthy of empathy means understanding today’s challenges, but it might also involve recalling one’s own youth as it really was—and finding compassion for one’s past self. While older adults may have regrets, they probably did their best with the perspective and resources they had. And they could stand to remind the young adults in their lives: Even flawed choices can lead to a life that, however imperfect, encompasses real moments of joy, accomplishment, and self-knowledge. If our culture romanticized that growth a little more and the golden glow of youth a little less, young adults might feel less alone in their distress. They might even look forward to finding out what’s next.

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