Review of Basic Mathematics

QNT 130 Lecture Notes

Review of Basic Mathematics:

In this chapter, we present a review of basic mathematical concepts that the students have studied in their high school mathematics subjects or courses given in the first year of college. A sound review in mathematics is extremely useful in understanding applications in business. We cover topics ranging from order of arithmetic operations to functions and graphs.

2.1 Order of Arithmetic Operations:

In simplifying arithmetic expression it is conventional to follow the order given below:

a) Parenthesis in the sequence ( ), {}, [],

b) Exponents

c) Multiplication or division

d) Addition or subtraction

Example 1:

[{(12-4)}+3]

=[]

______

Example 2:

Solve

Steps:

Example 3:

Simplify

2.2 Rules of Exponents

Exponent is the power to which a number or variable is to be raised.

Example:

In general times

Some Basic Rules of Exponents

Example:

10)

Example:

11)

Example:

Other Examples:

2.4 Fractions:

A fraction is a ratio of the form where y is different from zero and x and y are both whole numbers. X is known as numerator and y is the denominator. Ratios are examples of such fractions.

Addition or subtraction of factors:

i. If the denominator is common for the fractions, add the numerators and divide by the common denominator to obtain the results.

Examples:

ii. If the denominator is not common, then a procedure is followed to obtain common denominator. First, note that multiplying the numerator and denominator by the same non-zero number does not change the value of the fraction. Second, usual procedure of obtaining the common denominator is to multiply the numerator and the denominator by the ratio of the least common multiple (l.c.m) and the denominator of each fraction.

Example:

In this case, 12 is the l.c.m.

then

Therefore,

Example:

L.C.M =36

36 then,

Therefore,

Multiplication and Division involving fractions:

The product of any two fractions is obtained by the ratio of the product of the two numerators and the product of two denominators.

Example:

When dividing a fraction by another fraction, take the reciprocal of the second fraction and multiply it by the first fraction, following the previous procedure.

Example:

Mixed Fraction:

The sum of whole number and fraction less than one.

Example:

Problem: Express as ratio.

Decimals: A number that involves three parts is usually known as a decimal number. The three parts are

i. Whole number

ii. Decimal point and

iii. Numbers whose denominators are 10, or some powers of 10.

Example:

13.127

Here 13 is the whole number.

. is the decimal point.

The number after decimal are respectively 1 in “tenths” digit, 2 in “hundredths” digit and 7 in “thousands” digit.

Thus,

13.127 =

Operations with decimals:

Addition or subtraction: These operations are carried out after aligning the numbers in their appropriate positions.

Example: 164.21+121.3719

=164.21

Problem: Simplify 90.03-107.4271

Multiplication:

First, obtain the product of two numbers ignoring the decimal points. Count the “total” number of positions after the decimal points in both the numbers. Place the decimal point in the product number after the “total number of positions counting from right to left.

Example:

2.12

Step 1.

Step 2. Total Number of positions

Step 3. 0.01272

Problem: Simplify 6.37

Division: To divide a decimal by another, such as 62.744 or ,

First move the decimal point in the divisor to the right until the divisor becomes an integer, then move the decimal point in the dividend the same number of places;

This procedure determines the correct position of the decimal point in the quotient (as shown). The division can then proceed as follows:

Conversion from a given decimal to an equivalent fraction is straightforward. Since each place value is a power of ten, every decimal can be converted easily to an divided by a power of ten. For example,

The last example can be reduced to lowest terms by dividing the numerator and denominator by 4, which is their greatest common factor. Thus,

(In lowest terms)

Any fraction can be converted to an equivalent decimal. Since the fraction means , we can divide the numerator of a fraction by its denominator to convert the fraction to a decimal. For example, to convert to a decimal, divide 3 by 8 as follows.

Changing Fractions to Decimals: To change a fraction to a decimal, simply do what the operation says. In other words, means 13 divided by 20. So do just that (insert decimal points and zeros accordingly):

Changing Decimals to Fractions: To change a decimal to a fraction,

1. Move the decimal point two places to the right

2. Put that number over 100

3. Reduce if necessary.

Read it:

Write it:

Reduce it:

Finding Percent of a Number: To determine percent of a number, change the percent to a fraction or decimal (whichever is easier for you) and multiply. Remember, the word of means multiply.

What is 20% of 80?

What is 12% of 50?

What is % of 18?

Changing from Decimals to Percents: To change decimals to percents,

1. Move the decimal point two places to the right.

2. Insert a percent sign.

0.75=75% 0.05=5%

Changing from Fractions to Percents: To change a fraction to a percent,

1. Multiply by 100

2. Insert a percent sign.

Changing from Percents to Fractions: To change percents top fractions,

1. Divide the percent by 100

2. Eliminate the percent sign

3. Reduce if necessary

60%=

Percentages less than 1:

Other Applications of Percent: Turn the question word -for – word into an equation. For what substitute the letter x; for is substitute an equal sign; for of substitute a multiplication sign. Change percents to decimals or fractions, whichever you find easier. Then solve the equation.

18 is what percent of 90?

10 is 50% of what number?

What is 15% of 60?

0.15(60)=9

Finding Percentage Increase or Percentage Decrease: To find the percentage change (increase or decrease), use this formula:

Change
= Percentage Change
Starting point

What is the percentage decrease of a $500 item on sale for $400?

Change: 500-400=100

Change
=
Starting point
= decrease

What is the percentage increase of Jon’s salary if it went from $150 a week to $200 a week?

Change: 200-150=50

Change
=
Starting point
increase

References to Handout 1

1. Mathematics for Business and Economics by Robert H. Nicholson, McGraw-Hill, 1986

2. Introductory Algebra by Alan Wise, Harcourt Brace Jovanovich, 1986

Week 1. Review of Basic Mathematics

Problem Set 1

1) Evaluate the following:

(a)

(b) 0

(c)

2) Simplify the following:

(a)

(b)

3) Factor the following.

4) Simplify the following:

b) An investor bought 125 shares of stock at a price of $ a share.

i) What was the cost of the stock?

ii) If this represents of the total amount of investment, find the total amount of investment.

c) Complete the following blanks with correct answers

Decimal / Fraction / Percent
---- / / ----
0.1875 / ---- / ----
---- / ---- / 72

d) If a technology stock price has increased from $146 per share to $267 during 1998, find the percent increase in share price.

QNT 130 Lecture Notes

Week 2

y= f (x) is defined as a function of x if for one or more values of x, there corresponds one and only one value of y. The expression “f(x)” does not mean ‘ f times x ’. Instead, f (x) is another symbol for the variable y. Here f represents a particular rule by which x is converted into y.

In y= f (x), the set of values of x is called the domain of the function and the set corresponding to the y values is known as the range of the function. x, y are respectively known as independent and dependent variables in the context of the relation y =f (x).

y= f(x)

Example 1: y= x2 is a function.

y= x2

Example 2: y2=x is not a function.

If we solve for y, y=

Therefore, for a given value x, there are two values for y.

y2=x

y=±Öx

y2=16

y=±Ö16=±4

Example 3. x2+y2=k2 is not a function.(circle)

y2 =k2-x2 , y=

ÞFor a given x, we have two values of y.

Graphically, find if the following are functions are not.

Y Y

Y 1 Y = X2

Y = mX + c

X X

Yes Yes

Y Y

Y=X3

X2 + Y2 = 16

X (0,0) X

NO NO

Y= f(X)

YES

Note: Vertical Line Test: For a given value of x, find y by drawing a vertical line at x. If you find more than one value of y, then it is not a function.

WORKING WITH THE FUNCTION NOTATION

Evaluating functions given by a formula can involve algebraic simplification, as the following example shows. Similarly, solving for the input, or independent variable, involves solving an equation algebraically.

EVALUATING A FUNCTION

A formula like f(x)= is a rule that tells us what the function f does with its input value. In the formula, the letter x is placeholder for the input value. Thus, to evaluate f(x), we replace each occurrence of x in the formula with the value of the input.

Example1: Let g(x)= . Evaluate the following expressions. Some of your answers will contain a, a constant.

(a) g(3) (b) g(-1) (c) g(a)

(d) g(a-2) (e) g(a)-2 (f) g(a)-g(2)

Solution (a) To evaluate g(3), replace every x in the formula with 3.

g(3)=

(b) To evaluate g(-1), replace every x in the formula with (-1).

g(-1)=

g(a)=

(d) To evaluate g(a-2), replace every x in the formula with (a-2).

g(a-2)=

(e) To evaluate g(a)-2, first evaluate g(a) (as we did in part (c)), then subtract 2:

g(a)-2=

(f) To evaluate g(a)-g(2), subtract g(2) from g(a):

g(2)=

From part c, g(a)=. Thus,

g(a)-g(2)=

Graphing Functions

1. a. Point method: Graph 2x+3y=6

Solution: Solve for y.

Therefore,

x /
3 / 0
2 /
1 /
0 / 2
-1 /
-2 /
-3 / 4

b. Intercept method

In the function, 2x+3y=6,

If x=0, 3y=6, and so, y==2 (y-intercept)

If y=0, 2x=6, and so, y==3 (x-intercept)

The straight line is obtained by joining the two points (0,2) and (3,0).

If x=1, y=4/3 (one extra point)

Example: Graph the function 4x+5y-20=0

2. Graph Absolute value of x.

x / y
3 / 3
2 / 2
1 / 1
0 / 0
-1 / 1
-2 / 2
-3 / 3

3.Graph y=x2

x / y
3 / 9
2 / 4
1 / 1
0 / 0
-1 / 1
-2 / 4
-3 / 9

4. Graph the function y=2x ( the exponential function)

x / y
3 / 8
2 / 4
1 / 2
0 / 1
-1 /
-2 /
-3 /

5. Graph the following exponential growth function

where R= exponential growth rate

t=time period

P= the value of the function at time t.

Po = Value of P at t=0.

You are given

Po=8, 28, R=0.02

t / P
0 / 8
3 / 8.5
6 / 9.0
8 / 9.39

Example:

In 1985 the population of the United States was 234 million and the exponential growth rate was 0.8% per year. What will be the population in the year 2000?

Solution:

At t=0 (1985), the population was Po=234 million. Here R=0.008, therefore, the exponential equation is 234 e0.008t.

In the year 2000, we have t=15

Example: The cost of a first class postage stamp was 3 ¢ in 1932 and the exponential growth rate was 3.8% per year. What will be the cost of a first class stamp in year 2000.

Linear Functions and Applications in Business and Economics

Definition: A function with a constant rate of change, is known as a linear function.

In y=b0+ b1 x , b1 is called the slope of the linear function and it measures the change in the value of the dependent variable y as a result of one unit change in the value of the independent variable x. bo is known as the y-intercept and measures the value of y when x=0.

Note the following:

A horizontal line has zero slope.

A vertical line has no slope or its slope is undefined.

A line rising from left to right has positive slope.

A line falling from left to right has a negative slope.

Examples: Q=50-2P+0.6Y is called a demand function where Q=quantity demanded of a product, P is the price in $ of the product and Y is the family income in thousands of dollars.

i) Draw the demand curve(graphical relationship between Q and P for fixed Y) for Y=50 at price levels of P=0,5,10,15 and 20.

ii) Repeat the above for Y=100 and for the same price levels on the same graph.

iii) Comment on the demand curve in ii) in relation to the demand curve in i)

Solution

P / Y / Q
0 / 50 / 80
5 / 50 / 70
10 / 50 / 60
15 / 50 / 50
20 / 50 / 40

(II)

20 (I)