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:_______________________

Algebra - Factoring Trinomials

FACTORING IS THE REVERSE of multiplying. Skill in factoring, then, depends upon skill in multiplying: As for a quadratic trinomial --

2x² + 9x − 5

-- it will be factored as a product of binomials:

(? ?)(? ?)

Now, how will 2x² be produced? There is only one way: 2x• x :

(2x ?)(x ?)

And how will 5 be produced? Again, there is only one way: 1• 5. But does the 5 go with 2x or with x ?

(2x 5)(x 1) or (2x 1)(x 5) ?

Notice: We have not yet placed any signs!

How shall we decide between these two possibilities? It is the combination that will correctly give the middle term, 9x :

2x² + 9x − 5.

Consider the first possibility:

(2x 5)(x 1)

Is it possible to produce 9x by combining the outers and the inners: 2x• 1 with 5x ?

No, it is not. Therefore, we must eliminate that possibility and consider the other:

(2x 1)(x 5)

Can we produce 9x by combining 10x with x ?
Yes -- if we choose +5 and −1:

(2x − 1)(x + 5)
(2x − 1)(x + 5) = 2x² + 9x − 5

Skill in factoring depends on skill in multiplying -- particularly in picking out the middle term

Problem 1 Place the correct signs to give the middle term.

a) 2x² + 7x − 15 = S:(2x 3)(x 5)

b) 2x² − 7x − 15 = S:(2x 3)(x 5)

c) 2x² − x − 15 = S:(2x 5)(x 3)

d) 2x² − 13x + 15 = S:(2x 3)(x 5)

Note: When the constant term is negative, as in parts a), b), c), then the signs in each factor will be different. But when that term is positive, as in part d), the signs will be the same. Usually, however, that happens by itself.

Nevertheless, can you correctly factor the following?

2x² − 5x + 3 = S:(2x − 3)(x − 1)

Problem 2 Factor these trinomials.

a) 3x² + 8x + 5 = S: ___________________

b) 3x² + 16x + 5 = S: _________________

c) 2x² + 9x + 7 = S:___________________

d) 2x² + 15x + 7 = S: __________________

e) 5x² + 8x + 3 = S:____________________

f) 5x² + 16x + 3 = S: ___________________

Problem 3. Factor these trinomials. The first one is done for you.

a) 2x² − 7x + 5 = S:(2x − 5)(x − 1)

b) 2x² − 11x + 5 = S: ______________________

c) 3x² + x − 10 = S:_____________________

d) 2x² − x − 3 = S:_______________________

e) 5x² − 13x + 6 = S:_____________________

f) 5x² − 17x + 6 = S:_______________________

g) 2x² + 5x − 3 = S:_______________________

h) 2x² − 5x − 3 = S:______________________

i) 2x² + x − 3 = S:________________________

j) 2x² − 13x + 21 = S:_________________________

k) 5x² − 7x − 6 = S:__________________________

i) 5x² − 22x + 21 = S:__________________________

Algebra - FACTORING A NUMBER

TO FACTOR A NUMBER or an expression, means to write it as a product of factors.

Example 1 Factor 30

SOLUTION: 30 = 2• 15 = 2• 3• 5

If we begin 30 = 5• 6, we still obtain -- apart from the order-- 30 = 5• 2• 3.

SOLUTION: 50 = 2• 25 = 2• 5• 5

Factoring, then, is the reverse of multiplying. When we multiply, we write
2(a + b) = 2a + 2b.

But if we switch sides and write 2a + 2b = 2(a + b),

then we have factored 2a + 2b as the product 2(a + b).

In the sum 2a + 2b, 2 is a common factor of each term. It is a factor of 2a, and it is a factor of 2b. This Lesson is concerned exclusively with recognizing common factors.

Problem 2 Factor 3x − 3y.

SOLUTION: 3x − 3y = 3(x − y)

Problem 3 Rewrite each of the following as the product of 2x and another factor. The first one is done for you.

a) 8x = SOLUTION: 2x• 4

b) 6ax = SOLUTION: ___________

c) 2x² = SOLUTION: ___________

d) 2x to the 3rd power = SOLUTION: ___________

e) 4x to the 10th power = SOLUTION: ___________

f) 6x to the 5th power = SOLUTION: ____________

g) 2ax to the 6th power = SOLUTION: _____________

h) 2x = SOLUTION: _______________

SOLUTION: 5 is a common factor of each term. Display it on the left of the parentheses:

10a − 15b + 5 = 5(2a − 3b + 1)

If we multiply the right-hand side, we will get the left-hand side. In that way, the student can always check factoring.

Also, the sum on the left has three terms. Therefore, the sum in parentheses must also have three terms -- and it should have no common factors.

Problem 4 Factor each sum. Pick out the common factor. Check your answer. The first one is done for you.

a) 4x + 6y = SOLUTION: 2(2x + 3y)

b) 6x − 6 = SOLUTION: _______________

c) 8x + 12y − 16z = SOLUTION: __________________

d) 12x + 3 = SOLUTION: ___________________

e) 18x − 30 = SOLUTION: ____________________

f) 2x + ax = SOLUTION: ______________________

g) x² + 4x = SOLUTION: ___________________

h) 8x² − 4x = SOLUTION: _____________________

Algebra - Negative Numbers & Rule of Signs

The Algebraic Definition of the Negative of a Number

Finally, the way we define a negative number in algebra is as follows. −5, for example, is that number which when added to 5 itself, results in 0.
5 + (−5) = 0.

That is, to each number a there corresponds one and only one number −a called its negative. And when we add it to a, we get 0.

a + (−a) = −a + a = 0

Problem 1 What number is the negative of xyz? Why?

SOLUTION: −xyz , because xyz + (−xyz) = 0.

Problem 2 What number is the negative of −q? Why?

SOLUTION: ___________________

Problem 3 If s + t = 0, then what is the relationship between t and s?

SOLUTION: _________________________

Problem 4 If you had to prove that b − a is the negative of a − b,
how would you do it?

SOLUTION: Show that a − b + b − a = 0 .

To prove a fact about anything, whether in mathematics, logic, or the law, we have simply to show that it satisfies the definition of that fact.

An Explanation of the Rule of Signs

To decide how negative numbers should behave, we are not able to copy arithmetic. Rather, we have to respect the either-or, yes-or-no nature of logic.

For example, the introduction of the word not into a statement changes its truth value. If the statement was true, "not" makes it false, and vice-versa. If the statement Today is Monday is true, then Today is not Monday is false. But if we write Today is not not Monday, then that changes its truth value again -- that statement is true

Now in algebra we do not have true or false, but we do have the logical equivalent: positive or negative. Thus if the value of x is positive, then the value of −x must be negative, and vice-versa.

And so since we call the positive or negative value of a number its sign, then we can state the following principle:

A minus sign changes the sign of a number.


Geometrically, a minus sign reflects a number symmetrically about 0 with the number line to illustrate it visually.

We saw that with −(−3) = 3.
(As for 0, it is best to say that it has both signs: −0 = +0 = 0). If we now apply this principle to multiplication:

A negative factor changes the sign of a product.


Thus if ab is positive, then (−a)b cannot also be positive. It must be negative -- it must be the negative of ab.

(−a)b = −ab.

That is, "Unlike signs produce a negative number."

And upon introducing another negative factor, the sign changes back:
(−a)(−b) = ab.

"Like signs produce a positive number."

This same logical principle will apply to division and fractions. Hence we have the Rule of Signs.

To prove that (−a)b is the negative of ab in what some call a rigorous manner, we would have to apply the definition of the negative of a number given above. We would have to prove:

ab + (−a)b = 0.

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