Applied Math Basics

Applied Math Basics

This book covers basic arithmetic concepts including whole numbers, decimals, fractions, and English and metric measurements. Instructional emphasis is on application to typical problems one would encounter in the workplace. Simple and yet effective algorithms (methods) are used to solve them. Furthermore you should notice that all digits here involve the base 10 numbering system.

Basic Math Terminology

The Decimal System

The decimal system, sometimes referred to as base 10, contains a total of ten identifiers called digits. The decimal system is widely used because humans have ten fingers on which to count. For now, we will disregard other number systems for the sake of simplicity with the understanding that the decimal system is not unique.

The ten digits of the decimal system, arranged from lowest to greatest, are:

• 0 (zero)
• 1 (one)
• 2 (two)
• 3 (three)
• 4 (four)
• 5 (five)
• 6 (six)
• 7 (seven)
• 8 (eight)
• 9 (nine)

The decimal system uses positional notation to represent numbers larger than 9. This means that a digit's position in relation to other digits affects its meaning. Digits in the furthest right position represent the number of ones being counted, while digits in the second position from the right represent the number of tens. Digits in the third position from the right represent the number of hundreds, and digits in the fourth position from the right represent the number of thousands. This pattern can continue forever; for more information, see orders of magnitude.

For example, the number 535,254 means 5 hundreds of thousands, 3 tens of thousands, 5 thousands, 2 hundreds, 5 tens, and 4 ones. We would say this number as "five hundred thirty-five thousand two hundred fifty-four".

The Basic Sets of Numerals

Counting numbers are the numbers we use every day to count things. Mathematicians sometimes refer to this set of numbers as the Natural Numbers.

$1,2,3,4...$

Whole numbers include all the counting numbers and zero.

$0,1,2,3,4...$

Basic Whole Number Operations

Rounding Whole Numbers

Rounding is the process of finding the closest number to a specific value. You round a number up or down based on the last digit you are interested in.

$\genfrac{}{}{0ex}{}{down}{\overbrace{0,1,2,3,4,}}\genfrac{}{}{0ex}{}{up}{\overbrace{5,6,7,8,9}}$

For example, rounding the number 245 to the nearest tens place would round up to 250, while the number 324 rounded to the nearest tens place would be rounded down to 320.

Following the same logic, one could round to the nearest whole number. For example, 1.5 (pronounced as "one point five" or "one and a half") would be rounded up to 2, and 2.1 would be rounded down to 2.

Adding Whole Numbers

First, arrange the numbers in columns. For example, take 134+937

  134
+937
  ___

Add the first column (starting on the right)

  134
+937
     1
     1
Note the 10's digit put under the next column. Now add the next column and the number underneath
  134
+937
    71
      1
Finish it off with the other columns
  134
+937
1071
  1  1 So the answer to 134+937 is 1071

Subtracting Whole numbers

To subtract numbers think of a basket of oranges. If you have ten oranges in a basket and you remove eight oranges you are left with two oranges. For example:

 10
- 8
_____
  2  

If you have ten oranges in a basket and you remove all ten then you will no longer have any oranges so you are left with zero oranges. For example:

 10
-10
____
  0

to subtract large numbers use this method:

1. Arange the number that is being subtracted on top of how much its being subtracted by(ex. 2594-1673)

 2 5 9 4
-1 6 7 3
________

2. Subtract each column starting from the right and going to the left

 2 5 9 4
-1 6 7 3
________
     2 1

3. If you encounter a number that can't be subtracted without becoming negative,"borrow" subtract(if possible) 1 from the next digit over and add 10 to the digit that can't be subtracted(if not possible continue to borrow from the next digit).

 1 15
 X X 9 4
-1 6 7 3
________
   9 2 1

4. continue until done

note: that 921+1673=2594.

Multiplying Whole Numbers

Single number times Single number producing a Single number

Take the first number as $n_1$. Take the second number as $n_2$. Repeatedly add $n_1$ for $n_2$ times. (add $n_1$ to itself $(n_2-1)$ times).

$n_1=3\times 10^0$

$n_2=2\times 10^0$

$(n_1)\times (n_2)=2(3\times 10^0)=6\times 10^0=6$

(break this down back to "basic" level, instead of all this confusing "n1 x n2" junk please.)

Dividing Whole Numbers

Dividing whole numbers is the process of representing fractions as decimal numbers (eg. 0, 2.2, 4.55).

$4÷2=2$

In the above example, 2 goes into 4 twice; therefore, the answer (or the quotient) would be 2.(for more information about fractions, refer to Introduction to Fractions)

Divisions are often represented as fractions. For example,

$68÷43=\frac{68}{43}$

Factoring Whole Numbers

Factoring is the process of determining what prime numbers (numbers that cannot be divided by any number but 1 and itself; 2,3, and 5 are prime numbers) when multiplied will give a specific number. This process of factoring is very important in reducing fractions, which is covered in the Fractions chapter of this book. For example:

$4=2\times 2$

Or a more complicated example:

$180=2\times 2\times 3\times 3\times 5$

Introduction to Fractions

When you divide something into two or more parts, you have what is known as a common fraction. A common fraction is usually written as two numbers; a top number and a bottom number. Common fractions can also be expressed in words. The number that is on top is called the numerator, and the number on the bottom is called the denominator (the prefix 'de-' is latin for reverse).

$\frac{numerator}{denominator}$

These two numbers are always separated by a line, which is known as a fraction bar. This way of representing fractions is called display representation. Common fractions are, more often than not, simply known as fractions in everyday speech.

The numerator in any given fraction tells you how many parts of something you have on hand. For example, if you were to slice a delicious pepperoni pizza for your birthday party into six equal pieces, and you took two slices of pizza for yourself, you would have $\frac{2}{6}$ (pronounced two-sixths) of that pizza. Another way to look at it is by thinking in terms of equal parts; when that pizza was cut into six equal parts, each part was exactly $\frac{1}{6}$ (one-sixth) of the whole pizza.

The denominator tells you how many parts are in a whole, in this case your pizza. Your pizza was cut into six equal parts, and therefore the entire pizza consists of six equal slices. So when you took two slices for yourself, only four slices of pizza remain, or $\frac{4}{6}$ (four-sixths).

Also keep in mind that denominators can never be zero. It makes no sense to have something divided into zero parts. If the numerator is zero, however, than the value of the fraction is zero. For instance, the fraction $\frac{0}{6}$ is equal to zero, because you have nothing of the six slices.

Another way of representing fractions is by using a diagonal line between the numerator and the denominator.

$1/2$

In this case, the separator between the numerator and the denominator is called a slash, a solidus or a virgule. This method of representing fractions is called in-line representation, meaning that the fraction is lined up with the rest of the text. You will often see in-line representations in texts where the author does not have any way to use display representation.

Proper Fractions

The fraction in the pizza analogy we just used is known as a proper fraction. In a proper fraction, the numerator (top number) is always smaller than the denominator (bottom number). Thus, the value of a proper fraction is always less than one. Proper fractions are generally the kind you will encounter most often in mathematics.

Improper Fractions

When the numerator of a fraction is greater than, or equal to the denominator, you have an improper fraction. For example, the fractions $\frac{5}{3},\frac{2}{1}$ and $\frac{6}{6}$ are all considered improper fractions. Improper fractions always have a value of one whole or more. So with $\frac{6}{6}$, the numerator says you have 6 pieces, but 6 is also the number of the whole, so the value of this fraction is one whole. It is as if no one took a slice of pizza after you cut it.

In the case of $\frac{5}{3}$, one whole of something is divided into three equal pieces, but on hand you have five pieces (you had two pizzas, each divided into three slices, and you ate one slice). This means you have two pieces extra, or two pieces greater than one whole. This concept may seem rather confusing and strange at first, but as you become better in math you will eventually put two and two together to the get the whole picture (okay, bad pun).

Mixed Fractions

When a whole number is written next to a fraction, such as $2\frac{1}{3}$ (two and one-third) you are seeing what is called a mixed fraction. A mixed fraction is understood as being the sum, or total, of both the whole number and fraction. The number two in $2\frac{1}{3}$ stands for two wholes - you also have a third more of something, which is the $\frac{1}{3}$.

Simplifying Fractions

Sometimes in mathematics you will need to rewrite a fraction in smaller numbers, while also keeping the value of the fraction the same. This is known as simplifying, or reducing to lowest terms. It should be mentioned that a fraction which is not reduced is not intrinsically incorrect, but it may be confusing for others reviewing your work. There are two ways to simplify fractions, and both will be useful anytime you work with fractions, so it is recommended you learn both methods.

To reiterate, reducing fractions is essentially replacing your original fraction with another one of equal value, called an equivalent fraction. Below are a few examples of equivalent fractions.

$\frac{4}{8}=\frac{1}{2},\frac{8}{12}=\frac{2}{3},\frac{6}{10}=\frac{3}{5}$

When the fraction $\frac{4}{8}$ is reduced to lowest terms, it then becomes $\frac{1}{2}$, because four pieces out of a total of eight is exactly one-half of all available pieces. A fraction is also in its lowest terms when both the numerator and denominator cannot be divided evenly by any number other than one.

Division Method

To reduce a fraction to lowest terms, you must divide the numerator and denominator by the largest whole number that divides evenly into both. For example, to reduce the fraction $\frac{3}{9}$ to lowest terms, divide the numerator (3) and denominator (9) by three.

$\frac{3÷3}{9÷3}=\frac{1}{3}$

If the largest whole number is not obvious, and many times it is not, divide the numerator and denominator by any number (except one) that divides evenly into each, and then repeat the process until the fraction is in lowest terms.

For clarity, below are a few examples of reducing fractions using this method.

Example

Reduce $\frac{12}{20}$ to lowest terms.

Solution

In this problem, the largest whole number is difficult to see, so we first divide the numerator and denominator by two, as shown:

$\frac{12÷2}{20÷2}=\frac{6}{10}$

Next divide by two again:

$\frac{6÷2}{10÷2}=\frac{3}{5}$
There are no whole numbers left which can divide evenly into $\frac{3}{5}$, so the problem is finished.

Answer

Hence $\frac{12}{20}$ reduced to lowest terms is $\frac{3}{5}$

Example

Reduce $\frac{18}{24}$ to lowest terms.

Solution

Divide the numerator and denominator by six, as shown:

$\frac{18÷6}{24÷6}=\frac{3}{4}$

Answer

Hence $\frac{18}{24}$ reduced to lowest terms is $\frac{3}{4}$.

Example

Reduce $\frac{112}{126}$ to lowest terms.

Solution

In this problem, the largest whole number is not immediately apparent, so we first divide the numerator and denominator by two, as shown:

$\frac{112÷2}{126÷2}=\frac{56}{63}$

Next divide by seven:

$\frac{56÷7}{63÷7}=\frac{8}{9}$

Answer

Hence $\frac{112}{126}$ reduced to lowest terms is $\frac{8}{9}$.

Greatest Common Factor Method

The second method of simplifying fractions involves finding the greatest common factor between the numerator and the denominator. We do this by breaking up both the numerator and the denominator into their prime factors (greatest common factor = 2x2 = 4):

$\frac{12}{16}=\frac{2\times 2\times 3}{2\times 2\times 2\times 2}=\frac{2/\times 2/\times 3}{2/\times 2/\times 2\times 2}=\frac{3}{4}$

It is implied that any part is multiplied by one. If we divide 2s out of the factorized fraction, we are left with one 2 in the denominator.

$\frac{4}{8}=\frac{1\times 2/\times 2/}{2\times 2/\times 2/}=\frac{1}{2}$

It is best to practice these skills of reducing fractions until you feel confident enough to do them on your own. Remember, practice makes perfect.

Raising Fractions to Higher Terms

To raise a fraction to higher terms is to rewrite it in larger numbers while keeping the fraction equivalent to the original in value. This is, for all intents and purposes, the exact opposite of simplifying a fraction.

Example

Convert $\frac{2}{5}$ to a fraction with denominator of 15.

$\frac{2}{5}=\frac{?}{15}$

Step 1

Ask yourself “what number multiplied by 5 equals 15?” To find the answer simply divide 5 into 15.

$15÷5=3$

Step 2

After dividing the two denominators, you must take the answer and multiply it by the numerator you already have. So in this case, we multiply 2 by 3 to find the missing numerator.

$2\times 3=6$

Answer

$\frac{2}{5}=\frac{6}{15}$

Changing Improper Fractions to Mixed Numbers

Often times you will encounter fractions in their improper form. While this may be useful in some instances, it is usually best to convert the fraction into simplest form, or mixed fraction.

To convert an improper fraction into a mixed fraction, divide the denominator into the numerator.

Example

Change $\frac{13}{2}$ into a mixed fraction.

Solution

Divide 13 by 2, use long division to obtain quotient and a remainder. $13÷2=6\text{r}1$

Answer

To form the proper fraction part of the answer, we use the divisor (2) as the denominator, and the remainder (r1) as the numerator. Finally, we take the answer to the division problem, in this case 6, and use that as the whole number.

Hence $\frac{13}{2}$ in mixed fraction form is $6\frac{1}{2}$

Adding Fractions

Adding Fractions With The Same Denominator

In order to add fractions with the same denominator, you only need to add the numerators while keeping the original denominator for the sum.

$\frac{1}{5}+\frac{3}{5}=\frac{1+3}{5}=\frac{4}{5}$

Adding fractions with the same denominator is the rule but it begs the question why? Why can’t (or shouldn't) I add both numerators and denominators?

$\frac{1}{4}+\frac{1}{4}=\frac{1+1}{4+4}=\frac{2}{8}$

To make sense of this try taking a 12 inch ruler and drawing a 3 inch horizontal line (1/4 of a foot) and then on the end add another 3 inch line (1/4 of a foot). What is the total length of the line? It should be 6 inches (1/2 a foot) and not 2/8 of a foot (3 inches). In essence it seems we can only add like items and like items are terms that have the same denominator and we add them up by adding up numerators.

Adding Fractions With Different Denominators

When adding fractions that do not have the same denominator, you must make the denominators of all the terms the same. We do this by finding the least common multiple of the two denominators.

Least common multiple of 4 and 5 is 20; therefore, make the denominators 20:

$\frac{1}{4}+\frac{2}{5}=\frac{1\times 5}{4\times 5}+\frac{2\times 4}{5\times 4}=\frac{5}{20}+\frac{8}{20}$

Now that the common denominators are the same, perform the usual addition:

$\frac{5+8}{20}=\frac{13}{20}$

Subtracting Fractions

Subtracting Fractions With The Same Denominator

To subtract fractions sharing a denominator, take their numerators and subtract them in order of appearance. If the numerator's difference is zero, the whole difference will be zero, regardless of the denominator.

$\frac{6}{13}-\frac{2}{13}=\frac{6-2}{13}=\frac{4}{13}$

Subtracting Fractions With Different Denominators

To subtract one fraction from another, you must again find the least common multiple of the two denominators.

Least common multiple of 4 and 6 is 12; therefore, make the denominator 12:

$\frac{3}{4}-\frac{1}{6}=\frac{3\times 3}{4\times 3}-\frac{1\times 2}{6\times 2}=\frac{9}{12}-\frac{2}{12}$

Now that the denominator is same, perform the usual subtraction.

$\frac{9-2}{12}=\frac{7}{12}$

Multiplying Fractions

Multiplying fractions is very easy. Simply multiply the numerators of the fractions to find the numerator of the answer. Then multiply the denominators of the fractions to find the denominator of the answer. In other words, it can be said “top times top equals top”, and “bottom times bottom equals bottom.” This rule is used to multiply both proper and improper fractions, and can be used to find the answer to more than two fractions in any given problem.

Example

Multiply $\frac{1}{2}\times \frac{3}{4}$

Step 1

Multiply the numerators to find the numerator of the answer. $1\times 3=3$

Step 2

Multiply the denominators to find the denominator of the answer. $2\times 4=8$

Answer

$\frac{1}{2}\times \frac{3}{4}=\frac{3}{8}$

Make sure to reduce your answer if possible.

Canceling

You can also use a technique called canceling. This shortcut is very similar to reducing a fraction; however, when you cancel you merely simplify your fractions before multiplying.

To cancel, you divide the numerator and denominator by the largest common whole number. For example, to cancel the problem $\frac{4}{9}\times \frac{3}{8}$ just divide the left numerator (4) and right denominator (8) by 4. Likewise, divide the right numerator (3) and the left denominator (9) by 3, as shown below.

$\frac{4}{9}\times \frac{3}{8}=\frac{1\times 1}{3\times 2}=\frac{1}{6}$

Whole and Mixed Numbers

When you need to multiply a fraction by a whole number, you must first convert the whole number into a fraction. This, fortunately, is not as difficult as it may sound; just put the whole number over the number one. Then proceed to multiply as you would with any two fractions. An example is given below.

$\frac{3}{4}\times 5=\frac{3}{4}\times \frac{5}{1}=\frac{15}{4}=3\frac{3}{4}$

If a problem contains one or more mixed numbers, you must first convert all mixed numbers into improper fractions, and multiply as before. Finally, convert any improper fraction back to a mixed number.

$1\frac{1}{2}\times 2\frac{1}{4}=\frac{3}{2}\times \frac{9}{4}=\frac{27}{8}=3\frac{3}{8}$

Dividing Fractions

To divide fractions, simply exchange the numerator and the denominator of the second term in the problem, then multiply the two fractions.

$\frac{1}{2}÷\frac{3}{4}$

Invert the second fraction:

$\frac{1}{2}\times \frac{4}{3}$

Multiply:

$\frac{1\times 4}{2\times 3}=\frac{4}{6}$

Always check to see if simplifying the resulting fraction can be done:

$\frac{4}{6}=\frac{2}{3}$

Decimals

Decimals are basically fractions expressed without a denominator, rather replaced by a power of ten, and then the decimal point is inserted into the numerator at a position corresponding to the power of ten of the denominator. It is usual to add a leading zero to the left of the decimal point when the number is less than one.

$\frac{2}{5}=\frac{2\times 2}{5\times 2}=\frac{4}{10^1}=0.4$

Adding Decimal Numbers

Add decimal numbers much the same way you would add integers. Line up decimal points, and then proceed to add each column and carry at the top. The decimal point in the answer should line up with all of the others. Here is an example:

 12.3
+24.2
 ----
 36.5

Subtracting Decimal Numbers

Subtract as you would integers, but remember to follow all the rules from addition of decimals.

 312.9
-111.4
 -----
 201.5

Converting Fractions to Decimal Numbers

To convert a fraction to a decimal number, divide the numerator by the denominator.

• $\frac{3}{4}=0.75$
• $\frac{10}{3}=3.333333333333333...$

Repeating and Terminating Decimals

A repeating decimal is a decimal that is infinite. For instance,3.33 ... Instead of writing many 3's, you can draw a line above the number that is repeating(3). When there are two numbers repeating, such as .232323, you have to draw a line above the 2 and the 3. A terminating decimal is a decimal that ends at one point and does not go on forever. ex. 1.25

Multiplying Decimal Numbers

Multiplying decimal numbers can be tricky at times, but most of the time, it is similar to multiplying any integers. Although there are easier methods of multiplying, this is one of the methods.

You can make both decimal numbers have same multiple of a power of ten.

$0.6\times 0.75=(60\times 10^{-2})\times (75\times 10^{-2})$

Then multiply the first terms together, and the second terms.

$(60\times 75)\times (10^{-2}\times 10^{-2})=4500\times (10^{-4})$

Then insert the decimal point into a corresponding power of ten.

$4500\times (10^{-4})=450\times (10^{-3})=45\times (10^{-2})=4.5\times (10^{-1})=.45\times (10^0)=.45\times (1)=.45$

Dividing Decimal Numbers

Dividing decimal numbers is similar to multiplying them.

Make both decimal numbers have same multiple of a power of ten.

$0.3/0.4=(3\times 10^{-1})/(4\times 10^{-1})$

Then divide the first terms together, and the second terms.

$(3\times 10^{-1})/(4\times 10^{-1})=(3/4)/10^0$

Then insert the decimal point into a corresponding power of ten.

$(3/4)/10^0=0.75/1=0.75$

Alternatively, you can make the numbers integers (if the decimal is finite) and perform a simple division.

$0.3/0.4=(0.3\times 10)/(0.4\times 10)=3/4=0.75$

Measurements

Converting English Measurements

Here are some conversions that you should know in the English system:

1 foot = 12 inches
1 yard = 3 feet
1 mile = 5280 feet
1 gallon = 16 cups
1 gallon = 4 quarts

The Metric System

Metric system is based on the SI (Système International d'Unités) units.

Length

Length is measured in metres (m). There are 100 centimetres (cm) in 1 metre, and 10 millimeters (mm) in each centimetre. Therefore, there are 1000 millimeters in each metre. There are 1000 metres in each kilometre (km).

Mass

Mass is measured in kilograms (kg). The kilogram was originally defined as the mass of one litre of pure water at a temperature of 4 degrees Celsius and standard atmospheric pressure.

Time

Time is measured in seconds (s). Second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium-133 atom at zero kelvins. There are 60 seconds in a minute, 3600 seconds in an hour (or 60 minutes), and 86,400 seconds in a day (or 24 hours, or 1,440 minutes).

Temperature

Temperature is measured in kelvins or degrees Celsius (°C) in SI; the latter is used more frequently in general applications. The degree Celsius is a unit of temperature named after the Swedish astronomer Anders Celsius (1701–1744), who first proposed a similiar system in 1742. The Celsius temperature scale was designed so that the freezing point of water is 0 degrees Celsius, and the boiling point is 100 degrees Celsius at standard atmospheric pressure.

Volume

Volume is measured cubic metre (m³). The volume of a solid object is a numerical value given to describe the three-dimensional concept of how much space it occupies. One-dimensional objects (such as lines) and two-dimensional objects (such as squares) are assigned zero volume in the three-dimensional space. Litre (L and l) is used commonly. 1000 L makes a cubic metre.

Metric Length Measurement Conversions

1 meter = 10 decimeters = 100 centimeters = 1000 millimeters
1 decimeter = 10 centimeters = 100 millimeters
1 centimeter = 10 millimeters
1 kilometer = 1000 meters = 10,000 decimeters = 100,000 centimeters = 1,000,000 millimeters
1 kilogram = 1000 grams = 1,000,000 milligrams
1 gram = 1000 milligrams

Converting English and Metric Measurements

To convert between one measurement unit and another, requires a conversion factor. For instance;

1 inch = 25.4 mm

and so 12 inches = 12 x 25.4 = 304.8mm

The same conversion factor can also be used to convert mm to inches by using division. For example;

400mm = 400/25.4 = 15.75 inches (to 2 decimal places)

Glossary

Conversion

Greatest Common Factor

Improper Fraction

Least Common Multiple

Litre - The basic unit of volume in the metric system

Multiplication

Meter - The basic unit of length in the metric system.

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