"Age" Word Problems

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:

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 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:

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 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:

Solving, I get:

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?

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.

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

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.

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.

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:

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".

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:

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:

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.

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

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:

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

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.