Basic Word Problems

A word problem in algebra is the equivalent of a story problem in math.
When you solved story problems in your math class you had to decide what
information you had and what you needed to find out. Then you decided what
operation to use. Addition was used to find a totals and subtraction was
used to find changes in values.

The approach to solve problems with algebra is usually quite different.
Word problems are solved by separating information from the
problems into two equal groups, one for each side of an equation.
Examine this problem.

Sara has 15 apples and 12 oranges. How many pieces of fruit does she have?

We know that the sum of 15 and 12 is equal to the the total amount of fruit.
As explained in the Basics of the Equation
lesson, an unknown number or value is represented by a letter.
The total number of pieces of fruit is unknown, so we will represent that
amount with x. When the value that a particular variable will represent
is determined, it is defined by writing a statement like,

Let x = Total Amount of Fruit

Once again, the sum of 15 apples and 12 oranges is equal to the total
amount of fruit. This can be used to translate the problem into an equation,
like the following:

15 + 12 = x

The next step is to solve this equation.

Basics of Word Problems

Now solve the equation which was created in the last step.

Let x = Total Pieces of Fruit

Initial Equation 15 + 12 = x
After combining like terms 27 = x

The answer is then rewritten as a sentence.

There are 27 Total Pieces of Fruit.

By using simple arithmetic, this problem probably could have been solved faster without setting
up an algebra equation.
But, knowing how to use an equation for this problem builds awareness of
concepts which are useful, and sometimes critical to solving much harder
problems. One such problem will be presented in the next example.

Examine this word problem.

Two consecutive numbers have a sum of 91. What are the numbers?

Take notice, this problem has two numbers which are unknown, unlike the previous
one which only had one unknown value. In order for this problem to be solved
using basic algebra methods, we must set up an equation that has only one variable
(such as x).

Consecutive Integer Word Problems

The problem is shown again below for reference.

Two consecutive numbers have a sum of 91. What are the numbers?

To begin solving this problem, define the variable. You do not know what the first consecutive number is, so you can call
it x.

Let x = The First Consecutive Number

Since the numbers are consecutive, meaning one number comes right after the other, the second number must be one more than
the first. So, x + 1 equals the second number.

Let x + 1 = The Second Consecutive Number

The problem says that the sum of the two numbers is 91. This can be shown in the equation like the following:

x + (x + 1) = 91

The equation which you just wrote can be solved as follows:

Initial Equation x + (x + 1) = 91
After combining like terms 2x + 1 = 91
After subtracting 1 from each side 2x = 90
After dividing each side by 2 x = 45

Consecutive Integer Word Problems

In the previous problem, we determined that

x = 45

The answer to the equation (shown above) must now be used to find the answer
to the word problem. Go back to the top of the paper you used to solve this
problem. It should contain the following work:

Let x = The First Consecutive Number
Let x + 1 = The Second Consecutive Number

Since you now know that x equals 45, and that the First Consecutive Number
equals x, you can show this in the work like we did below.

Let x = The First Consecutive Number = 45
Let x + 1 = The Second Consecutive Number

Since x equals 45 and the Second Consecutive Number equals x + 1 you can simply add 1 to 45, to find that the Second Consecutive
Number is 46. It should be shown like the work below.

Let x = The First Consecutive Number = 45
Let x + 1 = The Second Consecutive Number = 46

This problem is now completed. If you did all of the work correctly, it should appear as ours does below.

Sample Problem Work

Let x = The First Consecutive Number = 45
Let x + 1 = The Second Consecutive Number = 46
x + (x + 1) = 91
2x + 1 = 91
2x = 90
x = 45

Keep going to learn about what to do when you encounter more than two consecutive numbers, negative consecutives,
and even or odd consecutives.

Variations of Consecutive Integer World Problems

More than 2 consecutive integers

Sometimes you will encounter a problem which has more than two consecutive numbers, such as the one below.

When added, four consecutive numbers have a sum of 18. What are the numbers?

You can solve this much like the previous problem. The difference is that
you will have to define four numbers (instead of two), like we did below.
Note: each consecutive number is found by adding 1 to the previous number.

Let x = The First Consecutive Number
Let x + 1 = The Second Consecutive Number
Let x + 2 = The Third Consecutive Number
Let x + 3 = The Fourth Consecutive Number

Your equation will look like this.

x + (x + 1) + (x + 2) + (x + 3) = 18

Negative consecutive integers

To solve problems which involve negative consecutive numbers, it is
important that you ignore the negative sign, and that you do not do
anything differently.

The sum of two consecutive integers is -9. What are the integers?

Keep the variable x positive, as shown, so that the answer does not come out wrong.

Let x = The First Consecutive Number
Let x + 1 = The Second Consecutive Number

Thus the equation will have the form

x + (x + 1) = -9

Even or Odd Consecutive Numbers

The only difference between ordinary consecutive numbers and even or odd
consecutive numbers is the space between each number. The next consecutive
number after 16 can be found by adding 1. The next consecutive even
number can be found by adding 2. Similarly, the next consecutive odd
number after 7 is 7 + 2, or 9.

Examine this problem

Two consecutive even numbers have a sum of 30. What are the numbers?

Since each even number is 2 away from the next, it is logical that you should define each number like the following

Let x = The First Consecutive Even Number
Let x + 2 = The Second Consecutive Even Number

Leading to the equation

x + (x + 2) = 30

Word Problem Basics Resources

Basic Word Problems Worksheet
20 consecutive integer word problems.

Next Lesson:
Basic Rules

An introduction to basic algebraic properties. These properties apply to all real numbers and include the cumulative property, the commutative property, and more.


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