Answer to old chestnut: age puzzles:

The solution to all puzzles of this general type is to rewrite them in the form of algebraic equations. In the common form you will have two (or more) equations in two variables (the current ages of everybody involved). There are two important steps: first interpret the language of the problem mathematically, then solve the resulting equations.

William is now 1/4 the age of Xavier.
This is equivalent to the equation

W = (1/4)X,

where W represents William's current age, and X represents Xavier's current age.

In five years, William will be 1/3 the age of Xavier.
This tells us that William's age in 5 years (or W+5) is 1/3 of Xavier's age in five years (or X+5), so we get the equation

W+5 = (1/3)(X+5), or

W+5 = X/3 + 5/3.

Now, solve this system of equations by first solving one equation for one variable. (That is, convert it to a mathematically identical equation in which one variable stands along on one side of the equation, and that variable does not appear on the other side of the equation.)

In this case, the first equation is already solved for W (but if we needed to solve the second equation for W, we would subtract 5 from both sides). In any case, you'll end up with something like W = aX + b where a and b are some numbers. (If more than two people are involved, you might have extra variables multiplied by other factors, like W = aX + bY + cZ + d but the same general principle applies.

Now we will use this equation to eliminate W from the other equation(s). Do this by replacing W with aX + b wherever it appears.

In the example problem, we get

X/4 + 5 = X/3 + 5/3.

This step reduces the number of equations and the number of variables by one. Repeat it, choosing one of the remaining equations and one of the remaining variables, solving for that variable, and using the resulting equation to eliminate that variable from the remaining equations. Eventually, you'll have one equation in one variable, which you can solve to find an answer for that variable.

In this case, we solve the one equation we have left for X:

X/4 - X/3 = 5/3 - 5.

(3/12 - 4/12)X = 5/3 - 15/3

(-1/12)X = - 10/3

X = (10/3)/(1/12) = 10*12/3 = 40.

To find the values of the other variable(s), plug this answer into the equations in which you solved for the other variables. Here, we'd have:

W = (1/4) (40) = 10.

And this is the answer we want, William is 10 years old.

Check your work when you are done with such a problem, in two different ways.

First, check that the resulting ages have the mathematical description given in the problem. If William is 10 and Xavier is 40, then the first statement holds. In five years, William will be 15 and Xavier will be 45, so the second statement holds as well.

Second, make sure the answer makes logical sense. These problems are generally constructed so that the answers make sense; ages should not be negative or beyond the range of human life spans (unless the problem explicitly specifies aliens or the future). Also, there shouldn't be any impossible relationships; nobody can be older than his father. Usually they are constructed so that the ages are all whole numbers.

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