Algebra - Word Problems Lesson

WORD PROBLEMS

They fall into categories. Below are examples. The only difficulty will be translating verbal language into algebraic language.

Example 1. ax ± b = c. All problems like the following one lead to an equation in that simple form.

Jane spent $42 for shoes. This was $14 less than twice what she spent for a blouse. How much was the blouse?

Solution. Every word problem has an unknown number. In this problem, it is the price of the blouse. Always let x represent the unknown number. That is, let x answer the question.

Let x, then, be how much she spent for the blouse. The problem states that "This" -- that is, $42 -- was $14 less than two times x.

Here is the equation:

2x − 14 = 42.

2x = 42 + 14

= 56.

x = 56
2

x = 28.

The blouse cost $28.

Example 2. There are b boys in the class. This is three more than four times the number of girls. How many girls are in the class?

Solution. Again, let x represent the unknown number that we are asked to find: Let x be the number of girls.(Although b is not known, it is not what we are asked to find.)

The problem states that "This" -- b -- is three more than four times x:

4x + 3 = b.

Therefore,

4x = b − 3

x = b − 3 / 4.



The solution in this example is not a number, because it will depend on the value of b. This is a type of "literal" equation, which is very common in algebra.



Example 3. The sum of two consecutive numbers is 37. What are they?

Solution. Two consecutive numbers are like 8 and 9, or 51 and 52.

So, let x be the first number. Then the next number is x + 1.

The problem states that their sum is 37:

x + (x + 1) = 37

2x = 37 − 1

x = 36 / 2

x = 18.

The two numbers are 18 and 19.

Example 4. Divide $80 among three people so that the second will have twice as much as the first, and the third will have $5 less than the second.

Solution. Let x be how much the first person gets.

Then the second gets twice as much, 2x.

And the third gets $5 less than that, 2x − 5.

Their sum is $80:

5x = 80 + 5

x = 85 / 5

x = 17.

This is how much the first person gets. Therefore the second gets

2x = 34.

And the third gets

2x − 5 = 29.

The sum of 17, 34, and 29 is in fact 80.

Problems

Problem 1. Julie has $50, which is eight dollars more than twice what John has. How much has John?

First, what will you let x represent?

The unknown number -- which is how much that John has.

What is the equation?

2x + 8 = 50.

Here is the solution: _____________

Problem 2. Carlotta spent $35 at the market. This was seven dollars less than three times what she spent at the bookstore; how much did she spend there?

Here is the equation.

3x − 7 = 35

Here is the solution: ________________



Problem 3. There are b black marbles. This is four more than twice the number of red marbles. How many red marbles are there?

Here is the equation.

2x + 4 = b

Here is the solution: __________________

Problem 4. Janet spent $100 on books. This was k dollars less than five times what she spent on lunch. How much did she spend on lunch?

Here is the equation.

5x- k = 100

Here is the solution: __________________

Problem 5. The whole is equal to the sum of the parts.

The sum of two numbers is 99, and one of them is 17 more than the other. What are the two numbers?

Here is the equation.

Here is the solution: _____________________

Problem 6. A class of 50 students is divided into two groups; one group has eight less than the other; how many are in each group?

Here is the equation.

Here is the solution: ___________________

Problem 7. The sum of two numbers is 72, and one of them is five times the other; what are the two numbers?
Here is the equation.

x + 5x = 72.

Here is the solution: _____________________

Problem 8. The sum of three consecutive numbers is 87; what are they?

Here is the equation.

Here is the solution: _____________________

Problem 9. A group of 266 persons consists of men, women, and children. There are four times as many men as children, and twice as many women as children. How many of each are there?

(What will you let x equal -- the number of men, women, or children?)

Let x = The number of children. Then

4x = The number of men. And

2x = The number of women.

Here is the equation:

x + 4x + 2x = 266

Here is the solution: ___________________

Problem 10. Divide $79 among three people so that the second will have three times more than the first, and the third will have two dollars more than the second.

Here is the equation.

Here is the solution: ______________________

Problem 11. Divide $15.20 among three people so that the second will have one dollar more than the first, and the third will have $2.70 more than the second.

Here is the equation.

Here is the solution:_______________________