"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:

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- 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:

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

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

3W + 34 = 11W + 10

34 = 8W + 10

24 = 8W

3 = W

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.

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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

2N = 12

N = 6

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.

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- 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 now: H

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:

½ of age 2 years from now:

of age 3 years ago:

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:

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.

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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

7*m* = 77

*m* = 11

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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