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A problem with one variable: How old is Al?

Many single-variable algebra word problems have to do with the relations between different people's ages. For example:
Al's father is 45. He is 15 years older than twice Al's age. How old is Al?
We can begin by assigning a variable to what we're asked to find. Here this is Al's age, so let Al's age = x.

We also know from the information given in the problem that 45 is 15 more than twice Al's age. How can we translate this from words into mathematical symbols? What is twice Al's age?

Well, Al's age is x, so twice Al's age is 2x, and 15 more than twice Al's age is 15 + 2x. That equals 45, right? Now we have an equation in terms of one variable that we can solve for x: 45 = 15 + 2x.

    original statement of the problem:  45 = 15 + 2x
    subtract 15 from each side: 30 = 2x
    divide both sides by 2: 15 = x
Since x is Al's age and x = 15, this means that Al is 15 years old.

It's always a good idea to check our answer:

    twice Al's age is 2 x 15:  30
    15 more than 30 is 15 + 30:  45
This should be the age of Al's father, and it is.

Solving a problem using one or two variables: How old is Karen?

We can solve this problem using either one or two variables:
Karen is twice as old as Lori. Three years from now, the sum of their ages will be 42. How old is Karen?

One-variable solution:

We'll let Lori's age be x. We can set up a chart:

             now   in 3 years
     Karen   2x    2x + 3
     Lori    x     x + 3
The sum of their ages in 3 years will be 42, so we have:
     (2x + 3) + (x + 3) = 42
                 3x + 6 = 42
                     3x = 36
                      x = 12
If Lori is 12, Karen is 24; in three years they will be 15 and 27, and the sum of their ages will be 42.

Two-variable solution:

If we want to use two variables to express the given information, we will need two equations to solve for these variables. Here's an example:

Start by assigning variables. We want to find Karen's age, so let's call that K. But we need a variable for Lori's age too, so we will call her age L.

We know that Karen is twice as old as Lori. Another way of saying this is that Karen's age is 2 times Lori's age. This gives us our first equation: K = 2L.

We also know that:

  1. in three years the sum of Karen's and Lori's ages will be 42;
  2. in three years, Karen's age will be 3 more than it is now, or K + 3;
  3. the same is true of Lori's age: in three years; it will be L + 3.

Since the sum of the girls' ages in three years is 42, we have our second equation: K + 3 + L + 3 = 42.

    simplify by adding the numbers:  K + L + 6 = 42
    subtract 6 from each side:  K + L = 36

Now we have two equations in two variables:

  1. K = 2L
  2. K + L = 36

Since Equation 1 provides an expression for K in terms of L that needs no simplification, we can plug the value for K in Equation 1 into the value for K in Equation 2: 2L + L = 36.

    add like terms:  3L = 36
    divide both sides by 3:  L = 12
Now we know that Lori is 12 years old, which makes Karen's age easy to find. All we need to do is plug L = 12 into either Equation 1 or Equation 2 and solve for K:
    Equation 1    Equation 2   
    K = 2L    K + L = 36
    K = 2 x 12    K + 12 = 36
    K = 24  K = 24 
As we can see, Karen is 24 years old. It doesn't matter which equation we use, since the the value for Karen's age must be the same in both cases.

Again, it's always a good idea to check our answer.

  1. Karen is supposed to be twice as old as Lori. Karen is 24; Lori is 12. Is 24 twice 12? Yes.

  2. In three years, the sum of Karen's and Lori's ages should be 42. In 3 years, Karen will be 24 + 3 = 27 years old. In 3 years, Lori will be 12 + 3 = 15 years old. Is the sum of 27 and 15 equal to 42? Yes.

We can see that we have found the correct answer.

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