One of the things I love about "age" problems is that they require students to not just convert sentences to equations, but also to identify different time frames in which parts of the problem may occur. It is helpful for students to recognize that for any given person in a problem, there may be multiple "unknowns." We might not assign a separate variable for each unknown, but it's helpful to write out each one. For example, in the following statement, we have one person, but two unknowns:

Unkowns
John's age NOW: J
John's age TEN YEARS AGO: J - 10

Now we use this to write an equation:

J = 2(J - 10). From here, students can solve the simple equation

Here is another example with one person, but with three unknowns:

Unknowns
Georgia's age NOW: G
Georgia's age 30 YEARS AGO: G - 30
Georgia's age IN TEN YEARS: G + 10

G = 10 + 3(G - 30)
G = 10 + 3G - 90
2G = 80
G = 40

Therefore, in ten years, she will be 40 + 10 = 50 years old.

The following problem involves two people:

Unknowns
Mark's age NOW: M
Mark's age IN EIGHT YEARS: M + 8
David's age NOW: D
David's age  TWENTY YEARS AGO: D - 20

The equations, then, are M = 12 + D - 20 and D = M + 8

Combining like terms in the first equation makes it evident that these are actually different forms of the same equation, which results in an indeterminate system.

Have students take a close look at the last statement. Are both of those ages eighteen years in the future? Or just one? Since a time frame is not specified for Evan's age, we assume it to be the same as the time frame for Ollie's age - 18 years in the future.

Unknowns
Ollie's age NOW: L
Ollie's age IN TWO YEARS: L 
Ollie's age IN TEN YEARS: L + 18
Evan's age NOW: E
Evan's age TEN YEARS AGO: E - 10
Evan's age IN EIGHTEEN YEARS: E + 18

From this, we can write our equations:

L + 2 = 2(E + (E - 10)) or L = 4E - 22
L + 18 = 2(E + 18) or L = 2E + 18

This gives Ollie's age as 58 years.

Students should note that, intstead of asking for either Joan's age or Betty's age, we're being asked for the sum of their ages, or B + J.

Unknowns
Joan's age NOW: J
Betty's age NOW: B
Betty's age LAST YEAR: B - 1
Betty's age in TWENTY FOUR YEARS: B + 24

So our equations are J = (B - 1) + 5 = B + 4
B + 24 = 2J

Solving these equations gives B = 16; J = 20. Thus, the answer is 36.

Much more interesting problems can be obtained by including time frames which are unknown. For example, in the sentence, "John's age is twice what it was when he was as old as Bob is now," not only do we not know Bob's age and John's age, we also don't know how long ago John was as old as Bob is now. This introduces another unknown into the problem. This a good topic to explore in a more advanced lesson.

NOTE: The final problem in the slide show above is intended as a JOKE. Students who are learning to solve the problems in this lesson will not be solving "The Infamous James Problem" (as my students call it).