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# Essential Mathematics for Economics and Business / Edition 3

Essential Mathematics for Economics and Business / Edition 3 available in Paperback

## Paperback - Rent for

**Temporarily Out of Stock Online**

## Overview

In recent years increasing numbers of economics and business students have experienced difficulty with the mathematical nature of their courses.

## Product Details

ISBN-13: | 9780470018569 |
---|---|

Publisher: | Wiley |

Publication date: | 07/15/2008 |

Edition description: | Older Edition |

Pages: | 682 |

Product dimensions: | 7.40(w) x 9.60(h) x 1.20(d) |

## Read an Excerpt

#### Essential Mathematics for Economics and Business

**By Teresa Bradley Paul Patton**

** John Wiley & Sons **

**ISBN: 0-470-84466-3**

#### Chapter One

**Mathematical Preliminaries**

At the end of this chapter you should be able to:

Perform basic arithmetic operations and simplify algebraic expressions

Perform basic arithmetic operations with fractions

Solve equations in one unknown, including equations involving fractions

Understand the meaning of no solution and infinitely many solutions

Currency conversions

Solve simple inequalities

Calculate percentages

In addition, you will be introduced to the calculator and a spreadsheet.

**Some mathematical preliminaries **

** Brackets** in mathematics are used for grouping and clarity. Brackets may also be used to indicate multiplication. Brackets are used in functions to declare the independent variable (see later). **Powers:** positive whole numbers such as [2.sup.3], which means 2 x 2 x 2 = 8:

[(anything).sup.3] = (anything) x (anything) x (anything)

[(*x*).sup.3] = *x* x *x* x *x*

[(*x* + 4).sup.5] = (*x* + 4)(*x* + 4)(*x* + 4) (*x* + 4) (*x* + 4)

**Note**

Brackets: (*A*)(*B*) or *A* x *B* or *AB* all indicate *A* multiplied by *B*.

**Variables and letters:** When we don't know the value of a quantity, we give that quantity a symbol, such as *x*. We may then make general statements about the unknownquantity, *x*, for example 'For the next 15 weeks, if I save £*x* per week I shall have £4000 to spend on a holiday'. This statement may be expressed as a mathematical equation:

15 x weekly savings = 4000 15 x *x* = 4000 or 15*x* = 4000

Now that the statement has been reduced to a mathematical equation, we may solve the equation for the unknown, *x*:

15*x* = 4000 15*x*/15 = 4000/15 divide both sides of the equation by 15 *x* = 266.67

*Square roots:* the square root of a number is the reverse of squaring:

[(2).sup.2] = 4 [right arrow] [square root of (4)] = 2

[(2.5).sup.2] = 6:25 [right arrow] [square root of (6.25)] = 2:5

**1.1 Arithmetic Operations**

**Addition and subtraction **

** Adding**: If all the signs are the same, simply add all the terms and give the answer with the common overall sign.

**Subtracting:** When subtracting any two numbers or two similar terms, give the answer with the sign of the largest number or term.

If terms are identical, for example all *x*-terms, all *xy*-terms, all [*x*.sup.2]-terms, then they may be added or subtracted as shown in the following examples:

**Add/subtract with numbers, mostly** **Add/subtract with variable terms**

5 + 8 + 3 = 16 similarity [right arrow] 5*x* + 8*x* + 3*x* = 16*x*

5 + 8 + 3 + *y* = 16 + *y* similarity [right arrow] (i) 5*x* + 8*x* + 3*x* + *y* = 16*x + y*

The *y*-term is different, so it cannot be (ii) 5*xy* + 8*xy* + 3*xy* + added to the others *y* = 16*xy* + *y*

The *y*-term is different, so it cannot be added to the others

7 - 10 = -3 similarity [right arrow] (i) 7*x* - 10*x* = -3*x*

(ii) 7[*x*.sup.2] - 10[*x*.sup.2] = -3[*x*.sup.2]

7 - 10 - 10*x* = -3 - 10*x* similarity [right arrow] 7[*x*.sup.2] - 10[*x*.sup.2] - 10*x* = -3[*x*.sup.2] - 10*x*

The *x*-term is different, so it cannot be The *x*-term is different, so it cannot be subtracted from the others subtracted from the others

**Worked example 1.1 Addition and subtraction**

For each of the following, illustrate the rules for addition and subtraction:

(a) 2 + 3 + 2·5 = (2 + 3 + 2.5) = 7.5

(b) 2*x* + 3*x* + 2.5*x* = (2 + 3 + 2.5)*x* = 7.5*x*

(c) -3*xy* - 2.2*xy* - 6*xy* = (-3 -2.2 -6)*xy* = -11.2 *xy*

(d) 8*x* + 6*xy* -12*x* + 6 + 2*xy* = 8*x* 12*x* + 6*xy* + 2*xy*)6 = -4*x* + 8*xy* + 6

(e) 3[*x*.sup.2] + 4*x* + 7 - 2[*x*.sup.2] - 8*x* + 2 = 3[*x*.sup.2] - 2[*x*.sup.2] + 4*x* - 8[*x* + 7 + 2 = [*x*.sup.2] - 4*x* + 9

Multiplying and dividing

Multiplying or dividing two quantities with like signs gives an answer with a positive sign. Multiplying or dividing two quantities with different signs gives an answer with a negative sign.

**Worked example 1.2 Multiplication and division**

Each of the following examples illustrate the rules for multiplication.

(a) 5 x 7 = 35 (b) -5 x -7 = 35 (c) 5 x -7 = -35 (d) -5 x 7 = -35 (e) 7/5 = 1.4 (f) (-7)/(-5) = 1.4 (g) (-7)/5= -1.4 (h) 7/(-5) = -1.4 (i) 5(7) = 35 (j) (-5)(-7) = 35 (k) (-5)*y* = -5*y* (l) (-*x*)(-*y*) = *xy* (m) 2(*x* + 2) = 2*x* + 4 (n) (*x* + 4)(*x* + 2) = *x*(*x* + 2) + 4(*x* + 2) = [*x*.sup.2] + 2*x* + 4*x* + 8 = [*x*.sup.2] + 6*x* + 8

(o) [(*x* + *y*).sup.2]

= (*x* + *y*)(*x* + *y*) = *x*(*x* + *y*) + *y*(*x* + *y*) = *xx* + *xy* + *yx* + *yy* = [*x*.sup.2] + 2*xy* + [*y*.sup.2]

* **Remember**

It is very useful to remember that a minus sign is a -1, so -5 is the same as -1 x 5

* **Remember**

0 x (any real number) = 0 0 / (any real number) = 0 **But you cannot divide by 0**

multiply each term inside the bracket by the term outside the bracket multiply the second bracket by *x*, then multiply the second bracket by (+4) and add,

multiply each bracket by the term outside it add or subtract similar terms, such as 2*x* + 4*x* = 6*x*

multiply the second bracket by *x* and then by *y* add the similar terms: *xy* + *yx* = 2*xy*

The following identities are important:

1. (*x* + *y*).sup.2] = [*x*.sup.2] + 2*xy* + [*y*.sup.2] 2. (*x* - *y*).sup.2] = [x.sup.2] - 2*xy* + [*y*.sup.2] 3. (*x* + *y*) = (*x* - *y*) = [*x*.sup.2] -[*y*.sup.2]

* **Remember:** Brackets are used for grouping terms together in maths for:

(i) Clarity (ii) Indicating the order in which a series of operations should be carried out

**1.2 Fractions **

** Terminology:**

fraction = numerator/denominator

3/7

3 is called the numerator

7 is called the denominator

**1.2.1 Add/subtract fractions: method**

The method for adding or subtracting fractions is:

**Step 1:** Take a common denominator, that is, a number or term which is divisible by the denominator of each fraction to be added or subtracted. A safe bet is to use the product of all the individual denominators as the common denominator.

**Step 2:** For each fraction, divide each denominator into the common denominator, then multiply the answer by the numerator.

**Step 3:** Simplify your answer if possible.

**Worked example 1.3 Add and subtract fractions**

Each of the following illustrates the rules for addition and subtraction of fractions.

**Numerical example**

1/7 + 2/3 - 4/5

Step 1: The common denominator is (7)(3)(5)

Step 2: 1/7 + 2/3 - 4/5

=1(3)(5)+2(7)(5)-4(7)(3)/(7)(3)(5)

Step 3: = 15 + 70 - 84/105 =1/105

1/7 + 2/3

Step 1: The common denominator is (7)(3)

Step 2: 1/7 + 2/3 = 1(3) + 2(7)/(7)(3)

Step 3: = 3 + 14/21 = 17/21

**Same example, but with variables**

*x*/7 + 2*x*/3-4*x*/5

= *x*(3)(5) + 2*x*(7)(5)-4*x*(7)(3) (7)(3)(5)

= 15*x* + 70*x* - 84*x*/105

= *x*/105

1/*x* + 4 + 2/*x* = 1(*x*) + 2(*x* + 4)/(*x* + 4) (*x*)

= *x* + 2*x* + 8/[*x*.sup.2]+ 4*x*

= 3*x* + 8/[*x*.sup.2]+ 4*x*

**1.2.2 Multiplying fractions**

In multiplication, write out the fractions, multiply the numbers across the top lines and multiply the numbers across the bottom lines.

*Note:* Write whole numbers as fractions by putting them over 1.

*Terminology:* RHS means right-hand side and LHS means left-hand side.

**Worked example 1.4 **

** Multiplying fractions**

(a) (2/3) (5/7) = (2)(5)/(3)(7) = 10/21

(b) (-2/3) (7/5) = (-2)(7)/(3)(5) = -14/15

(c) 3 x 2/5 (3/1)(2/5)=(3)(2)/(1)(5) = 6/5 = 1 1/5

The same rules apply for fractions involving variables, *x, y*, etc.

(d) (3/*x*) (*x* + 3)/(*x* - 5) = 3(*x* + 3)/*x*(*x* - 5) = 3*x* + 9/[*x*.sup.2]- 5*x*]

**1.2.3 Dividing fractions **

** General rule:**

Dividing by a fraction is the same as multiplying by the fraction inverted

**Worked example 1.5 **

** Division with fractions**

The following examples illustrate how division with fractions operates.

(a) (2/3)/(5/11) = (2/3)(11/5) = 22/15

(b) 5/(3/4) = 5 x 4/3 = 5/1 x 4/3 = 20/3 = 6 2/3

(c) (7/3)/8 = (7/3)/8/1 = 7/3 x 1/8 = 7/24

(d) 2*x*/*x* + *y*/3*x*/2(*x* - *y*) = 2*x*/*x* + *y* 2(*x* - *y*)/ 3*x*

= 4*x*(*x - y*)/3*x*(*x + y*) = 4(*x - y*)/3(*x + y*)

**Note:** The same rules apply to all fractions, whether the fractions consist of numbers or variables.

*Progress Exercises 1.1 Revision on Basics*

Show, step by step, how the expression on the left-hand side simplifies to that on the right.

**1.** 2*x* + 3*x* + 5(2*x* - 3) = 15(*x* - 1)

**2.** 4[*x*.sup.2] + 7*x* + 2*x* (4*x* - 5) = 3x(4*x* - 1)

**3.** 2*x* (*y* + 2) - 2*y*(*x* + 2) = 4(*x* - *y*)

**4.** (*x* + 2)(*x* - 4) - 2(*x* - 4) = *x*(*x* - 4)

**5.** (*x* + 2)(*y* - 2) + (*x* - 3)(*y* + 2)

= 2*xy* - *y* - 10

**6.** [(*x* + 2).sup.2] + [(*x* - 2).sup.2] = 2[(*x*.sup.2] + 4)

**7.** [(*x* + 2).sup.2] - [(*x* - 2).sup.2] = 8*x*

**8.** (*x* + 2)2 - *x*(*x* + 2) = 2(*x* + 2)

**9.** 1/3 + 3/5 + 5/7 = 1 68/105

**10.** *x*/2 - *x*/3 = *x*/6

**11.** (2/3)/(1/5) = 10/3

**12.** (2/7)/3 = 2/21

**13.** 2(2/*x* - *x*/2) = 4-[*x*.sup.2]/*x* = 4/*x* - *x*

**14.** -12/*P* (3*P*/2 + *P*/2) = -24

**15.** (3/*x*)/*x* + 3 = 3/*x*(*x* +3)

**16.** (5*Q*/*P* + 2)/(1/(*P* + 2)) = 5*Q*

**1.3 Solving Equations**

The solution of an equation is simply the value or values of the unknown(s) for which the left-hand side (LHS) of the equation is equal to the right-hand side (RHS).

For example, the equation, *x* + 4 = 10, has the solution *x* = 6. We say *x* = 6 'satisfies' the equation. We say this equation has a **unique solution**.

Not all equations have solutions. In fact, equations may have no solutions at all or may have infinitely many solutions. Each of these situations is demonstrated in the following examples.

**Case 1: Unique solutions** An example of this is given above: x + 4 = 10 etc.

**Case 2: Infinitely many solutions** The equation, *x* + *y* = 10 has solutions (*x* = 5, *y* = 5); (*x* = 4, *y* = 6); (*x* = 3, *y* = 7), etc. In fact, this equation has infinitely many solutions or pairs of values (x, y) which satisfy the formula, *x* + *y* = 10.

**Case 3: No solution** The equation, 0(*x*) = 5 has no solution. There is simply no value of *x* which can be multiplied by 0 to give 5.

* Methods for solving equations

Solving equations can involve a variety of techniques, many of which will be covered later.

*(Continues...)*

Excerpted fromEssential Mathematics for Economics and BusinessbyTeresa Bradley Paul PattonExcerpted by permission.

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

## First Chapter

#### Essential Mathematics for Economics and Business

**By Teresa Bradley Paul Patton**

** John Wiley & Sons **

**ISBN: 0-470-84466-3**

#### Chapter One

**Mathematical Preliminaries**

At the end of this chapter you should be able to:

Perform basic arithmetic operations and simplify algebraic expressions

Perform basic arithmetic operations with fractions

Solve equations in one unknown, including equations involving fractions

Understand the meaning of no solution and infinitely many solutions

Currency conversions

Solve simple inequalities

Calculate percentages

In addition, you will be introduced to the calculator and a spreadsheet.

**Some mathematical preliminaries **

** Brackets** in mathematics are used for grouping and clarity. Brackets may also be used to indicate multiplication. Brackets are used in functions to declare the independent variable (see later). **Powers:** positive whole numbers such as [2.sup.3], which means 2 x 2 x 2 = 8:

[(anything).sup.3] = (anything) x (anything) x (anything)

[(*x*).sup.3] = *x* x *x* x *x*

[(*x* + 4).sup.5] = (*x* + 4)(*x* + 4)(*x* + 4) (*x* + 4) (*x* + 4)

**Note**

Brackets: (*A*)(*B*) or *A* x *B* or *AB* all indicate *A* multiplied by *B*.

**Variables and letters:** When we don't know the value of a quantity, we give that quantity a symbol, such as *x*. We may then make general statements about the unknownquantity, *x*, for example 'For the next 15 weeks, if I save £*x* per week I shall have £4000 to spend on a holiday'. This statement may be expressed as a mathematical equation:

15 x weekly savings = 4000 15 x *x* = 4000 or 15*x* = 4000

Now that the statement has been reduced to a mathematical equation, we may solve the equation for the unknown, *x*:

15*x* = 4000 15*x*/15 = 4000/15 divide both sides of the equation by 15 *x* = 266.67

*Square roots:* the square root of a number is the reverse of squaring:

[(2).sup.2] = 4 [right arrow] [square root of (4)] = 2

[(2.5).sup.2] = 6:25 [right arrow] [square root of (6.25)] = 2:5

**1.1 Arithmetic Operations**

**Addition and subtraction **

** Adding**: If all the signs are the same, simply add all the terms and give the answer with the common overall sign.

**Subtracting:** When subtracting any two numbers or two similar terms, give the answer with the sign of the largest number or term.

If terms are identical, for example all *x*-terms, all *xy*-terms, all [*x*.sup.2]-terms, then they may be added or subtracted as shown in the following examples:

**Add/subtract with numbers, mostly** **Add/subtract with variable terms**

5 + 8 + 3 = 16 similarity [right arrow] 5*x* + 8*x* + 3*x* = 16*x*

5 + 8 + 3 + *y* = 16 + *y* similarity [right arrow] (i) 5*x* + 8*x* + 3*x* + *y* = 16*x + y*

The *y*-term is different, so it cannot be (ii) 5*xy* + 8*xy* + 3*xy* + added to the others *y* = 16*xy* + *y*

The *y*-term is different, so it cannot be added to the others

7 - 10 = -3 similarity [right arrow] (i) 7*x* - 10*x* = -3*x*

(ii) 7[*x*.sup.2] - 10[*x*.sup.2] = -3[*x*.sup.2]

7 - 10 - 10*x* = -3 - 10*x* similarity [right arrow] 7[*x*.sup.2] - 10[*x*.sup.2] - 10*x* = -3[*x*.sup.2] - 10*x*

The *x*-term is different, so it cannot be The *x*-term is different, so it cannot be subtracted from the others subtracted from the others

**Worked example 1.1 Addition and subtraction**

For each of the following, illustrate the rules for addition and subtraction:

(a) 2 + 3 + 2·5 = (2 + 3 + 2.5) = 7.5

(b) 2*x* + 3*x* + 2.5*x* = (2 + 3 + 2.5)*x* = 7.5*x*

(c) -3*xy* - 2.2*xy* - 6*xy* = (-3 -2.2 -6)*xy* = -11.2 *xy*

(d) 8*x* + 6*xy* -12*x* + 6 + 2*xy* = 8*x* 12*x* + 6*xy* + 2*xy*)6 = -4*x* + 8*xy* + 6

(e) 3[*x*.sup.2] + 4*x* + 7 - 2[*x*.sup.2] - 8*x* + 2 = 3[*x*.sup.2] - 2[*x*.sup.2] + 4*x* - 8[*x* + 7 + 2 = [*x*.sup.2] - 4*x* + 9

Multiplying and dividing

Multiplying or dividing two quantities with like signs gives an answer with a positive sign. Multiplying or dividing two quantities with different signs gives an answer with a negative sign.

**Worked example 1.2 Multiplication and division**

Each of the following examples illustrate the rules for multiplication.

(a) 5 x 7 = 35 (b) -5 x -7 = 35 (c) 5 x -7 = -35 (d) -5 x 7 = -35 (e) 7/5 = 1.4 (f) (-7)/(-5) = 1.4 (g) (-7)/5= -1.4 (h) 7/(-5) = -1.4 (i) 5(7) = 35 (j) (-5)(-7) = 35 (k) (-5)*y* = -5*y* (l) (-*x*)(-*y*) = *xy* (m) 2(*x* + 2) = 2*x* + 4 (n) (*x* + 4)(*x* + 2) = *x*(*x* + 2) + 4(*x* + 2) = [*x*.sup.2] + 2*x* + 4*x* + 8 = [*x*.sup.2] + 6*x* + 8

(o) [(*x* + *y*).sup.2]

= (*x* + *y*)(*x* + *y*) = *x*(*x* + *y*) + *y*(*x* + *y*) = *xx* + *xy* + *yx* + *yy* = [*x*.sup.2] + 2*xy* + [*y*.sup.2]

* **Remember**

It is very useful to remember that a minus sign is a -1, so -5 is the same as -1 x 5

* **Remember**

0 x (any real number) = 0 0 / (any real number) = 0 **But you cannot divide by 0**

multiply each term inside the bracket by the term outside the bracket multiply the second bracket by *x*, then multiply the second bracket by (+4) and add,

multiply each bracket by the term outside it add or subtract similar terms, such as 2*x* + 4*x* = 6*x*

multiply the second bracket by *x* and then by *y* add the similar terms: *xy* + *yx* = 2*xy*

The following identities are important:

1. (*x* + *y*).sup.2] = [*x*.sup.2] + 2*xy* + [*y*.sup.2] 2. (*x* - *y*).sup.2] = [x.sup.2] - 2*xy* + [*y*.sup.2] 3. (*x* + *y*) = (*x* - *y*) = [*x*.sup.2] -[*y*.sup.2]

* **Remember:** Brackets are used for grouping terms together in maths for:

(i) Clarity (ii) Indicating the order in which a series of operations should be carried out

**1.2 Fractions **

** Terminology:**

fraction = numerator/denominator

3/7

3 is called the numerator

7 is called the denominator

**1.2.1 Add/subtract fractions: method**

The method for adding or subtracting fractions is:

**Step 1:** Take a common denominator, that is, a number or term which is divisible by the denominator of each fraction to be added or subtracted. A safe bet is to use the product of all the individual denominators as the common denominator.

**Step 2:** For each fraction, divide each denominator into the common denominator, then multiply the answer by the numerator.

**Step 3:** Simplify your answer if possible.

**Worked example 1.3 Add and subtract fractions**

Each of the following illustrates the rules for addition and subtraction of fractions.

**Numerical example**

1/7 + 2/3 - 4/5

Step 1: The common denominator is (7)(3)(5)

Step 2: 1/7 + 2/3 - 4/5

=1(3)(5)+2(7)(5)-4(7)(3)/(7)(3)(5)

Step 3: = 15 + 70 - 84/105 =1/105

1/7 + 2/3

Step 1: The common denominator is (7)(3)

Step 2: 1/7 + 2/3 = 1(3) + 2(7)/(7)(3)

Step 3: = 3 + 14/21 = 17/21

**Same example, but with variables**

*x*/7 + 2*x*/3-4*x*/5

= *x*(3)(5) + 2*x*(7)(5)-4*x*(7)(3) (7)(3)(5)

= 15*x* + 70*x* - 84*x*/105

= *x*/105

1/*x* + 4 + 2/*x* = 1(*x*) + 2(*x* + 4)/(*x* + 4) (*x*)

= *x* + 2*x* + 8/[*x*.sup.2]+ 4*x*

= 3*x* + 8/[*x*.sup.2]+ 4*x*

**1.2.2 Multiplying fractions**

In multiplication, write out the fractions, multiply the numbers across the top lines and multiply the numbers across the bottom lines.

*Note:* Write whole numbers as fractions by putting them over 1.

*Terminology:* RHS means right-hand side and LHS means left-hand side.

**Worked example 1.4 **

** Multiplying fractions**

(a) (2/3) (5/7) = (2)(5)/(3)(7) = 10/21

(b) (-2/3) (7/5) = (-2)(7)/(3)(5) = -14/15

(c) 3 x 2/5 (3/1)(2/5)=(3)(2)/(1)(5) = 6/5 = 1 1/5

The same rules apply for fractions involving variables, *x, y*, etc.

(d) (3/*x*) (*x* + 3)/(*x* - 5) = 3(*x* + 3)/*x*(*x* - 5) = 3*x* + 9/[*x*.sup.2]- 5*x*]

**1.2.3 Dividing fractions **

** General rule:**

Dividing by a fraction is the same as multiplying by the fraction inverted

**Worked example 1.5 **

** Division with fractions**

The following examples illustrate how division with fractions operates.

(a) (2/3)/(5/11) = (2/3)(11/5) = 22/15

(b) 5/(3/4) = 5 x 4/3 = 5/1 x 4/3 = 20/3 = 6 2/3

(c) (7/3)/8 = (7/3)/8/1 = 7/3 x 1/8 = 7/24

(d) 2*x*/*x* + *y*/3*x*/2(*x* - *y*) = 2*x*/*x* + *y* 2(*x* - *y*)/ 3*x*

= 4*x*(*x - y*)/3*x*(*x + y*) = 4(*x - y*)/3(*x + y*)

**Note:** The same rules apply to all fractions, whether the fractions consist of numbers or variables.

*Progress Exercises 1.1 Revision on Basics*

Show, step by step, how the expression on the left-hand side simplifies to that on the right.

**1.** 2*x* + 3*x* + 5(2*x* - 3) = 15(*x* - 1)

**2.** 4[*x*.sup.2] + 7*x* + 2*x* (4*x* - 5) = 3x(4*x* - 1)

**3.** 2*x* (*y* + 2) - 2*y*(*x* + 2) = 4(*x* - *y*)

**4.** (*x* + 2)(*x* - 4) - 2(*x* - 4) = *x*(*x* - 4)

**5.** (*x* + 2)(*y* - 2) + (*x* - 3)(*y* + 2)

= 2*xy* - *y* - 10

**6.** [(*x* + 2).sup.2] + [(*x* - 2).sup.2] = 2[(*x*.sup.2] + 4)

**7.** [(*x* + 2).sup.2] - [(*x* - 2).sup.2] = 8*x*

**8.** (*x* + 2)2 - *x*(*x* + 2) = 2(*x* + 2)

**9.** 1/3 + 3/5 + 5/7 = 1 68/105

**10.** *x*/2 - *x*/3 = *x*/6

**11.** (2/3)/(1/5) = 10/3

**12.** (2/7)/3 = 2/21

**13.** 2(2/*x* - *x*/2) = 4-[*x*.sup.2]/*x* = 4/*x* - *x*

**14.** -12/*P* (3*P*/2 + *P*/2) = -24

**15.** (3/*x*)/*x* + 3 = 3/*x*(*x* +3)

**16.** (5*Q*/*P* + 2)/(1/(*P* + 2)) = 5*Q*

**1.3 Solving Equations**

The solution of an equation is simply the value or values of the unknown(s) for which the left-hand side (LHS) of the equation is equal to the right-hand side (RHS).

For example, the equation, *x* + 4 = 10, has the solution *x* = 6. We say *x* = 6 'satisfies' the equation. We say this equation has a **unique solution**.

Not all equations have solutions. In fact, equations may have no solutions at all or may have infinitely many solutions. Each of these situations is demonstrated in the following examples.

**Case 1: Unique solutions** An example of this is given above: x + 4 = 10 etc.

**Case 2: Infinitely many solutions** The equation, *x* + *y* = 10 has solutions (*x* = 5, *y* = 5); (*x* = 4, *y* = 6); (*x* = 3, *y* = 7), etc. In fact, this equation has infinitely many solutions or pairs of values (x, y) which satisfy the formula, *x* + *y* = 10.

**Case 3: No solution** The equation, 0(*x*) = 5 has no solution. There is simply no value of *x* which can be multiplied by 0 to give 5.

* Methods for solving equations

Solving equations can involve a variety of techniques, many of which will be covered later.

*(Continues...)*

Excerpted fromEssential Mathematics for Economics and BusinessbyTeresa Bradley Paul PattonExcerpted by permission.

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## Table of Contents

Perform Basic Arithmetic Operations & Simplify Algebraic Expressions.

Perform Basic Arithmetic Operations with Fractions.

Solve Equations in One Unknown, including Equations involving Fractions.

Solve Quadratic Equations by Formula.

Calculate Percentages.

Use the calculator.

Use Spreadsheets.

The Straight Line and Applications.

The Straight Line.

Mathematical Modelling.

Applications: Demand, Supply, Cost, Revenue, Consumption & Savings.

Elasticity of Demand & Supply.

Income and Cost Constraints.

Simultaneous Equations.

Solving simultaneous equations.

Equibilirium in various markets.

Consumer & producer surplus.

National Income model and IS-LM model.

Non-linear Functions and Applications.

Quadratics and and Cubic Functions.

Exponential Functions.

Logarithmetic Functions.

Hyperbolic Functions.

Financial Mathematics.

Arithmetic and Geometric Sequences and Series.

Simple Interest, Compound Interest and Annual Percentage Rates (APRs).

Depreciation.

Net Present Value and Internal Rate of Return.

The Relationship between the Price of a Bond and the Interest Rate.

Exchange Rates.

Introduction to Differention and Applications.

Introduction to Differention.

Applications of Differention, Marginal Functions.

Optimisation of Functions of One Variable.

Economic Applications of Maximum/Minimum.

Curvature and Applications.

Further rules for Differention and Applications to Economic Models.

Elasticity, Total Revenue and Marginal Revenue.

Production Functions and Applications.

Functions of Several Variables.

Partial Differention.

Applications of PartialDifferention.

Unconstrained Optimisation and Applications.

Constrained Optimisation, Lagrange Multipliers.

Applications of Constrained Optimisation.

Integration and Applications.

Integration as the Reverse of Differention.

The Power Rules for Integration.

Integration of Linear Functions Raised to Powers.

The Definite Integral and the Area Under a Curve.

Consumer and Producer Surplus.

The Integration of Natural Exponentials.

Integration by substituition.

Integration by parts.

First-Order Differential Equations and Applications.

Differential Equations for Limited and Unlimited Growth.

Linear Algebra and Applications.

Linear Programming for Functions of Two Variables.

Matrices: Arithmetic Operations: Applications.

Determinants: 2x2 and 3x3 determinants: Cramer's Rule: Applications.

Difference Equations and Applications.

Difference equations and differential equations.

Solution of First Order Difference equations.

Lagged Income, cobweb, Harrod growth models and other applications.

Solution of second order Difference equations and applications.

## Reading Group Guide

Mathematical Preliminaries.

Perform Basic Arithmetic Operations & Simplify Algebraic Expressions.

Perform Basic Arithmetic Operations with Fractions.

Solve Equations in One Unknown, including Equations involving Fractions.

Solve Quadratic Equations by Formula.

Calculate Percentages.

Use the calculator.

Use Spreadsheets.

The Straight Line and Applications.

The Straight Line.

Mathematical Modelling.

Applications: Demand, Supply, Cost, Revenue, Consumption & Savings.

Elasticity of Demand & Supply.

Income and Cost Constraints.

Simultaneous Equations.

Solving simultaneous equations.

Equibilirium in various markets.

Consumer & producer surplus.

National Income model and IS-LM model.

Non-linear Functions and Applications.

Quadratics and and Cubic Functions.

Exponential Functions.

Logarithmetic Functions.

Hyperbolic Functions.

Financial Mathematics.

Arithmetic and Geometric Sequences and Series.

Simple Interest, Compound Interest and Annual Percentage Rates (APRs).

Depreciation.

Net Present Value and Internal Rate of Return.

The Relationship between the Price of a Bond and the Interest Rate.

Exchange Rates.

Introduction to Differention and Applications.

Introduction to Differention.

Applications of Differention, Marginal Functions.

Optimisation of Functions of One Variable.

Economic Applications of Maximum/Minimum.

Curvature and Applications.

Further rules for Differention and Applications to Economic Models.

Elasticity, Total Revenue and Marginal Revenue.

Production Functions and Applications.

Functions of Several Variables.

Partial Differention.

Applications of PartialDifferention.

Unconstrained Optimisation and Applications.

Constrained Optimisation, Lagrange Multipliers.

Applications of Constrained Optimisation.

Integration and Applications.

Integration as the Reverse of Differention.

The Power Rules for Integration.

Integration of Linear Functions Raised to Powers.

The Definite Integral and the Area Under a Curve.

Consumer and Producer Surplus.

The Integration of Natural Exponentials.

Integration by substituition.

Integration by parts.

First-Order Differential Equations and Applications.

Differential Equations for Limited and Unlimited Growth.

Linear Algebra and Applications.

Linear Programming for Functions of Two Variables.

Matrices: Arithmetic Operations: Applications.

Determinants: 2x2 and 3x3 determinants: Cramer's Rule: Applications.

Difference Equations and Applications.

Difference equations and differential equations.

Solution of First Order Difference equations.

Lagged Income, cobweb, Harrod growth models and other applications.

Solution of second order Difference equations and applications.

## Interviews

Mathematical Preliminaries.

Perform Basic Arithmetic Operations & Simplify Algebraic Expressions.

Perform Basic Arithmetic Operations with Fractions.

Solve Equations in One Unknown, including Equations involving Fractions.

Solve Quadratic Equations by Formula.

Calculate Percentages.

Use the calculator.

Use Spreadsheets.

The Straight Line and Applications.

The Straight Line.

Mathematical Modelling.

Applications: Demand, Supply, Cost, Revenue, Consumption & Savings.

Elasticity of Demand & Supply.

Income and Cost Constraints.

Simultaneous Equations.

Solving simultaneous equations.

Equibilirium in various markets.

Consumer & producer surplus.

National Income model and IS-LM model.

Non-linear Functions and Applications.

Quadratics and and Cubic Functions.

Exponential Functions.

Logarithmetic Functions.

Hyperbolic Functions.

Financial Mathematics.

Arithmetic and Geometric Sequences and Series.

Simple Interest, Compound Interest and Annual Percentage Rates (APRs).

Depreciation.

Net Present Value and Internal Rate of Return.

The Relationship between the Price of a Bond and the Interest Rate.

Exchange Rates.

Introduction to Differention and Applications.

Introduction to Differention.

Applications of Differention, Marginal Functions.

Optimisation of Functions of One Variable.

Economic Applications of Maximum/Minimum.

Curvature and Applications.

Further rules for Differention and Applications to Economic Models.

Elasticity, Total Revenue and Marginal Revenue.

Production Functions and Applications.

Functions of Several Variables.

Partial Differention.

Applications of PartialDifferention.

Unconstrained Optimisation and Applications.

Constrained Optimisation, Lagrange Multipliers.

Applications of Constrained Optimisation.

Integration and Applications.

Integration as the Reverse of Differention.

The Power Rules for Integration.

Integration of Linear Functions Raised to Powers.

The Definite Integral and the Area Under a Curve.

Consumer and Producer Surplus.

The Integration of Natural Exponentials.

Integration by substituition.

Integration by parts.

First-Order Differential Equations and Applications.

Differential Equations for Limited and Unlimited Growth.

Linear Algebra and Applications.

Linear Programming for Functions of Two Variables.

Matrices: Arithmetic Operations: Applications.

Determinants: 2x2 and 3x3 determinants: Cramer's Rule: Applications.

Difference Equations and Applications.

Difference equations and differential equations.

Solution of First Order Difference equations.

Lagged Income, cobweb, Harrod growth models and other applications.

Solution of second order Difference equations and applications.

## Recipe

Perform Basic Arithmetic Operations & Simplify Algebraic Expressions.

Perform Basic Arithmetic Operations with Fractions.

Solve Equations in One Unknown, including Equations involving Fractions.

Solve Quadratic Equations by Formula.

Calculate Percentages.

Use the calculator.

Use Spreadsheets.

The Straight Line and Applications.

The Straight Line.

Mathematical Modelling.

Applications: Demand, Supply, Cost, Revenue, Consumption & Savings.

Elasticity of Demand & Supply.

Income and Cost Constraints.

Simultaneous Equations.

Solving simultaneous equations.

Equibilirium in various markets.

Consumer & producer surplus.

National Income model and IS-LM model.

Non-linear Functions and Applications.

Quadratics and and Cubic Functions.

Exponential Functions.

Logarithmetic Functions.

Hyperbolic Functions.

Financial Mathematics.

Arithmetic and Geometric Sequences and Series.

Simple Interest, Compound Interest and Annual Percentage Rates (APRs).

Depreciation.

Net Present Value and Internal Rate of Return.

The Relationship between the Price of a Bond and the Interest Rate.

Exchange Rates.

Introduction to Differention and Applications.

Introduction to Differention.

Applications of Differention, Marginal Functions.

Optimisation of Functions of One Variable.

Economic Applications of Maximum/Minimum.

Curvature and Applications.

Further rules for Differention and Applications to Economic Models.

Elasticity, Total Revenue and Marginal Revenue.

Production Functions and Applications.

Functions of Several Variables.

Partial Differention.

Applications of PartialDifferention.

Unconstrained Optimisation and Applications.

Constrained Optimisation, Lagrange Multipliers.

Applications of Constrained Optimisation.

Integration and Applications.

Integration as the Reverse of Differention.

The Power Rules for Integration.

Integration of Linear Functions Raised to Powers.

The Definite Integral and the Area Under a Curve.

Consumer and Producer Surplus.

The Integration of Natural Exponentials.

Integration by substituition.

Integration by parts.

First-Order Differential Equations and Applications.

Differential Equations for Limited and Unlimited Growth.

Linear Algebra and Applications.

Linear Programming for Functions of Two Variables.

Matrices: Arithmetic Operations: Applications.

Determinants: 2x2 and 3x3 determinants: Cramer's Rule: Applications.

Difference Equations and Applications.

Difference equations and differential equations.

Solution of First Order Difference equations.

Lagged Income, cobweb, Harrod growth models and other applications.

Solution of second order Difference equations and applications.