Lesson Tips for solving age-related word problems

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It is not possible to provide a "cookbook" which will explain a specific sequence of steps that will solve every age-related problem. But here are some general tips:<ol><li>Use as few variables as possible. One variable is the best.</li><li>Most age-related problems mention one or more people and one or more points in time (now, in the past, in the future). Write an expression for each possible combination of persons and points in time. For example, if the problem mentions persons A and B and their ages now and 3 years ago then you will write 4 expressions:<ul><li>A's age now</li><li>B's age now</li><li>A's age 3 years ago</li><li>B's age 3 years ago</li></ul>4 people and 3 points in time would mean 4*3 = 12 different expressions.</li><li>As in most word problems it is generally advantageous to make your variable represent the smallest value. This allows you to use addition and/or multiplication to express the other values. In the example above, if A is the younger person, then A's age 3 years ago would be the smallest number. So make "x" represent A's age 3 years ago. And A's age now would be x+3. (B's ages now and 3 years ago would be expressed in terms of x according to the information given in the problem.)</li><li>Using the relationships described in the problem write as many equations as you have variables. 1 variable -> 1 equation, 2 variables -> 2 equations, etc.</li><li>Solve the equation (or system of equations)</li><li>Answer the question! "x" may not be the answer to the question. But you can use "x" and the expressions you wrote in the second step above to answer the question.</li></ol>
Here's an example problem:
Four years ago Heather was four times as old as Tom. Today she is twice as old. How old is Heather now?
Expressions. With two people and two points in time I need 4 expressions. Since Tom is younger, I will make "x" represent his age 4 years ago:
x = Tom's age 4 years ago
4*x = Heathers age 4 years ago
x+4 = Tom's age now
4*x+4 = Heather's age now
Notice how there is only addition and multiplication. This is because I chose "x" to be the smallest number. Although it may not always be possible to to this, it is advantageous to make "x" the smallest number most of the time.
Equations. 1 variable = 1 equation.
The relationship described in the problem which has not been used is: "Today she is twice as old". Another way to word this is: "Heather's age now is twice Tom's age now". Translating this into an equation we get:
4*x+4 = 2*(x+4)
Look at the expressions, the equation and the sentences "Today she is twice as old" or "Heather's age now is twice Tom's age now" to see how the sentences translate into the equation.
Now we solve. Simplify first:
4x + 4 = 2x + 8
Subtract 2x from each side:
2x + 4 = 8
Subtract 4 from each side:
2x = 4
Divide both sides by 2:
x = 2
Answer the question! "x" as you can see from the expressions list above, stand for Tom's age 4 years ago. The question is about Heather's age now. The expressions list tells us that Heather's age now is 4*x+4. So we use the value we found for x and this expression to answer the question:
Heather's age now = 4*x+4 = 4*(2)+4 = 8 + 4 = 12.