The Student Success Commons at York County Community College: MAT 107: Technical Mathematics

Chapter 4-1

A ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six. This can be written as 8:6, which is can be reduced to the equivalent the ratio 4:3.

In other words, a ratio says how much of one thing there is compared to another thing.

 

GEAR RATIO = (number of teeth on the driven gear)/(number of teeth on the driving gear)

Video links:

• Math Calculations & Conversions : Definition of Gearing Ratios by ehow on YouTube
• How Do Gear Ratios Work? by Concerning Reality on YouTube
• How to Calculate RPM | How to Calculate RPM or Diameter of Driven Pulley by 

COMPRESSION RATIO= (expanded volume)/(compressed volume)

Video links:

• Compression Ratio - Explained by Engineering Explained on YouTube

 

SKILL CHECK: practice simplifying fractions. Download practice sheet. 

 

A proportion is an equation in which two ratios are set equal to each other. You can solve for an unknown value in a proportion by using the cross-product rule (sometimes called the "butterfly method").

Video links:

• How to Solve Proportions Using Cross Multiplication | Solving Proportions | Math with Mr. J on YouTube
• Applying cross multiplication to solve a proportion
• Proportional Ratios with the Butterfly Method by EduK8 Collection on YouTube
• Ratios and Proportions by instructor Jill Smith

 

SKILL CHECK: practice solving proportions.

 

Chapter 4-2

Video links:

• Similar Triangles by MathHelp.com on YouTube
• Ratios and Proportions: gear teeth and RPM by YCCCSSC
• What Does Inversely Proportional Mean? by cognito on YouTube
  • The beginning of this video gives an excellent definition of inverse proportions

Note: The symbol used to denote the proportionality is ∝. For example, if we say, a is proportional to b, then it is represented as “a ∝ b” and if we say, a is inversely proportional to b, then it is denoted as 'a∝1/b'. You may or may not see this symbol used when working with proportions.

 

Chapter 4-3 through 4-5

A percentage is a ratio, fraction, or portion of a whole (which is represented as 100) usually denoted with the percent symbol: %. 

 

Video links:

• Intro to Percent by Khan Academy
• Percents by instructor Jill Smith (this comprehensive video covers several percent problem solving examples; visit The Math Success Program page for tons of resources and practice problems!)

 

(Click to enlarge or print)

SKILL CHECK: practice converting between decimals, fractions, and percents. 

Chapter 5

Accuracy refers to how close a given set of measurements (observations or readings) are to their true or accepted value.

For example, let's say you buy a 5-pound bag of oranges and weigh the bag when you get home. If the result is 5 pounds then your measurement is highly accurate. If the result is 23.2 pounds then the result has low accuracy.

 

Precision refers to how close multiple measurements of the same object are to each other.

If you weigh one orange five different times and the result is 4.7 oz every time, then your measurement is very precise. On the other hand, if you get five different measurements then the precision is lower. 

 

Significant digits, also known as significant figures or sig figs, are the digits in a number that are reliable and necessary for conveying a quantity. They are important for determining the accuracy and precision of measurements, especially in scientific or technical fields.

Chapter 6

Negative numbers are numbers that are less than zero and are represented by a minus sign (-) before them. On a number line, negative numbers are located to the left of zero and are the mirror image of the positive numbers on the right.

As the absolute value of a negative number increases, it gets farther away from zero and becomes smaller. For example, -2 is smaller than 1, and -7 is smaller than -2

Think: which is colder -7°F or -2°F?

-7°F is colder than -2°F, thus -7 is smaller -2.

Another way to think about negative numbers with money. Which bank account is worse off: Overdrawn $750.00 (-750) or overdrawn $2.35 (-2.35)? The account overdrawn by $750 would be much worse to have than an account overdrawn by a little over two dollars (and much harder to recover from). Thus, -750 is less than than -2.35.

 

COMMON SIGN KEYWORDS
POSITIVE	NEGATIVE
Profit Increase Add Above More than Surplus	Debt Loss Decrease Lower Below Defecit Subtract

Adding a negative value is equivalent to subtracting its absolute value, and subtracting a negative value is like adding its absolute value. Here are some examples:

• 4 + -3 is the same as 4 - 3 = 1.

• 7 + -10 is the same as 7 - 10 = -3.

• 23 + -64 is the same as 23 - 64 = -41

• -7 - - 7 is the same as -7 + 7 = 0

• 10 - - 7 = 10 + 7 = 17

Remember: math is the only time that "two wrongs make a right"! When adding or subtracting, two successive negative symbols turn into a plus sign. You'll see that this applies to multiplication and division as well.

Additional examples:

• You have $4.00. An item you wish to purchase will cost you $3.00. You will have $1.00 remaining.
• You have $7.00. An item you wish to purchase will cost you $10.00. You do not have enough so are left with a defecit of $3.00, or -$3.00

 

Multiplying two negative numbers will result in a positive. Multuplying two positive numbers will result in a positive. Multiplying a negative and a positive will result in a negative. 

Remember: when multiplying, if the signs are the same the result is positive. If the signs are different the result it negative.

Chapter 7

In algebra, letters are used to represent unknown numbers.These letters are called variables, and we usually use the rules of mathematics to solve for, or identify what the unknown number could be.

The lowercase letter "x" is often used in algebra which can cause confusion with the traditional multiplication symbol "×". They look a lot alike, don't they? To avoid confusion, we will use other methods to imply multiplication. The most common are:

A simple dot: 4 • x

Parentheses: 4(x)

No sign: 4x

All three of these examples represent the same expression: 4 is being multiplied by some unknown number.

SKILL CHECK: get into the habit of avoiding confusion with the multiplication symbol by practicing using one of the above methods to write out multiplication problems.

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 When evaluating mathematical expressions, be sure to follow the order of operations:

1) G: Groupings. These include parentheses, brackets, and braces.
2) E: Exponents. This includes roots and radicals.
3) M and D: Multiplication or Division. These both have the same "weight" so when you get to this step do whichever operation comes first when working from left to right.
4) A and S: Addition or Subtraction. These both have the same "weight" so when you get to this step do whichever operation comes first when working from left to right.

Get Everyone More Drinks And Snacks

According to an experiment run by Dr. Peter Price of the Classroom Professor website, about 75% of students get the wrong answer by not using the order of operations. In his experiment Dr. Price posted the following brainteaser:

7 - 1 × 0 + 3 ÷ 3 = ?

Out of the 6,000 respondents, only 26% arrived at the right answer. Try out this problem for yourself and see what you get.

. . . . . . . . .

Did you get 8? If so, congrats, you're using the order of operations! If you got a different answer check out Order of Operations 101, an article from the Calculator Site linked below for a complete explanation about GEMDAS (also known as PEMDAS).

 Order of Operations 101

 You can also visit Khan Academy if you'd like more review and practice with the order of operations. 

 

SKILL CHECK: practice simpliflying expressions by using the order of operations.

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PROPERTIES TO KNOW

PROPERTY	DESCRIPTION	EXAMPLE
Commutative Property of Addition	When two numbers are added, the sum is the same regardless of the order in which the numbers are added.	2 + 3 =  5 3 + 2 = 5
Associative Property of Addition	When three or more numbers are added, the sum is the same regardless of the way in which the numbers are grouped.	6 + (4 + 3) = 13 (6 + 4) + 3 = 13
Commutative Property of Multiplication	When two numbers are multiplied together, the product is the same regardless of the order in which the numbers are multiplied.	3 • 5 = 15 5 • 3 = 15
Associative Property of Multiplication	When three or more numbers are multiplied, the product is the same regardless of the way in which the numbers are grouped.	6 • (4 • 3) = 72 (6 • 4) • 3 = 72
Distributive Property	The sum of two numbers times a third number is equal to the sum of each addend times the third number.	5(7 + 2) = 45 5 • 7 + 5 • 2 = 45
Identity Property of Addition	Adding 0 to any number does not change the original value.	76 + 0 = 76
Identity Property of Multiplication	Multiplying any number by 1 does not change the original value.	19 • 1 = 19
Inverse Property of Addition	The sum of any number and its additive inverse is equal to 0.	5 + -5 = 0
Inverse Property of Multiplication	The product of any number and its reciprocal is equal to 1.	4(1/4) = 1
Zero Property of Multiplication	Any number multiplied by 0 is equal to 0.	123(0) = 0

SKILL CHECK: practice using these rules, as well as the order of operations, to solve equations.

 

TYPES OF NUMBERS

• Natural Numbers (N), (also called positive integers, counting numbers, or natural numbers); They are the numbers {1, 2, 3, 4, 5, …}
• Whole Numbers (W). This is the set of  natural numbers, plus zero, i.e., {0, 1, 2, 3, 4, 5, …}.
• Integers (Z). This is the set of all whole numbers plus all the negatives (or opposites) of the natural numbers, i.e., {… , ⁻2, ⁻1, 0, 1, 2, …}
• Rational numbers (Q). This is all the fractions where the top and bottom numbers are integers; e.g., 1/2, 3/4, 7/2, ⁻4/3, 4/1 [Note: The denominator cannot be 0, but the numerator can be].
• Real numbers (R), (also called measuring numbers or measurement numbers). This includes all numbers that can be written as a decimal. This includes fractions written in decimal form e.g., 0.5, 0.75 2.35, ⁻0.073, 0.3333, or 2.142857. It also includes all the irrational numbers such as π, √2 etc. Every real number corresponds to a point on the number line

 

EQUATION-SOLVING TIPS THAT MAY SAVE YOUR LIFE:

Be familiar with the rules.

Know the order of operations, basic mathematical properties, and how to use your calculator properly.

Read carefully.

Know what you know and know what you don't--especially when solving a word problem. Clarify for yourself what it is you have and what you're being asked to find. 

Write clearly.

Don't skimp on writing things down. Write down the problem simply, in your own words. Set aside space to list out the variables in your problem, what they stand for, and what values are known to be associated with them. Color code to help you keep track.

PRACTICE.

Practice, practice, practice. Practice so much that the word practice no longer feels like a real word. Repeat the same problem multiple times.

Show all your steps.

Avoid skipping steps and definitely avoid not writing things down. This goes along with writing clearly: write each step on a new line. Don't sacrifice clarity for the sake of saving paper.

Double check.

As you solve ensure you're following the correct steps and using the correct values. Your instinct might be to rush in order to get the pain over with as fast as possible but please resist this temptation. 

 

 

Video links

• Isolating variables and practice with the order of operations
• Solving for an unkonwn, then subsituting values (practical example word problem)
• Subsituting values, then solving for an unknown (practical example word problem)