Ratios, Rates, Fractions, Percentages
What do a baker and a professional athlete have in common? Whether it's in the kitchen or out on the field ratios, rates, fractions, and percentages are at work. Known as some of the most useful math concepts for real life, ratios, rates, fractions, and percentages are present in many of our days to day activities and are helping make our lives a lot easier.
Let’s start with rates and ratios. A ratio is a relationship between two quantities or numbers. For example, in cooking you might hear something like “2 cups of water is to 3 cups of oil”, that would be an example of a ratio and can be written in different ways like “2 is to 3” or “2:3”. Ratios can also be expressed as fractions.
A Rate is a special type of ratio that compares two different kinds of measures. In our previous example, both parts of the ratio are in cups, but if we were to compare the measure of distance (in kilometers) and time (in hours) like “ 2 kilometer is to 3 hours” we would then have a rate.
As mentioned earlier, ratios can also be expressed in the form of fractions. A fraction can simply be defined as a “part of a whole”. Using again the example of 2 cups of water and 3 cups of oil, our ratio of water to the whole mixture (oil and water combined) is a ratio of 2:5. Written as a fraction this would be 25. This means that two-fifths of our mixture is water.
Similar to a ratio and a fraction, percentages can also show you “a part of a whole”. A percentage is a value that tells you how much out of 100, with “Percent” meaning “per 100”. Using again the earlier mixture of water to oil ratio 2:5, when converted to a percentage it is equal to 40%. Meaning that 40 parts out of the 100 parts of the mixture are made up of water. Using what we call “proportions” we take the initial ratio of 2:5 and find a ratio equal to it. In this case, 40:100 is equal to 2:5.
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Statistics and Probability
What if someone told you that you could learn how to predict the future? You might think it's impossible, but mathematicians might argue otherwise! Although Mathematics cannot accurately tell you what your life will be like 3 years from now, statistics and probability might be able to predict the likelihood of a certain outcome for a certain event. Ever wonder how weather specialists can tell it what the weather will be like 2 days ahead of time? What about sports analysts who can give you a better idea of which sports team will win a tournament? These are just some examples of statistics and probability at work!
Statistics is a branch of mathematics that involves gathering information, summarizing it, and interpreting what it means. The numerical data we get from this process is called statistics. An example of statistics in action would be how weather specialists study can predict future weather patterns and old weather patterns. Probability is the study of ‘chance’ or the likelihood of an occurrence and is represented by the formula of Probability = EventOutcomes. An application of probability might be solving for the likelihood of you getting a ‘heads’ when flipping a coin. Using the formula for probability we can say that you have a ½ chance of getting ahead when flipping a coin.
It's important to remember that, even though statistics and probability are closely related, they aren’t the same thing. Statistics involves predicting the outcome by studying and understanding past events while probability deals with predicting future events using mathematical definitions. Statistics and probability go hand in hand in helping mathematicians, researchers, scientists, and analysts conduct all sorts of statistical studies by applying probability theories to draw conclusions.
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Negative Numbers: Addition, Subtraction, Multiplication, and Division
Negative numbers can be a little bit difficult for some to understand. Let’s say you have 10 apples in a basket and 10 of your friends take 1 apple each from you, how much would you have left? This would leave you with 0 apples and it would be impossible for you to have any less than that wouldn’t it? Imagining anything smaller than 0 is hard can be hard for some. But in real life, we can go less than 0 by using negative numbers. By definition, a negative number is any value smaller than 0 such as negative 1, negative 2, negative 3, and so on. To know a number is a negative value, a minus sign or (-) is placed right before the number. Just as how there is an infinite number of values after 0, there is also an infinite number of values before 0.
Real-life applications of negative numbers can be seen in many everyday things. One example would be temperature which can be shown in Fahrenheit or Celsius scales. A temperature of -1 degrees celsius would mean that the temperature is below freezing. Negative numbers are also used in banking and finance to show how money flows. A negative sign before an amount of money will mean that the money was either spent or lost.
Because negative numbers are after 0 on the number line, they have special rules when you add, subtract, multiply or divide with them.
Adding Negative Numbers
When adding negative numbers to something is it the same as subtracting a positive number from it. For example, to add the negative number “-1” to the number “5” is the same as subtracting one from five.
5 + (-1) = 5 - 1 = 4
Subtracting Negative Numbers
When subtracting negative numbers to something it is the same as adding a positive number to it. For example, to subtract the negative number “- 6” to the number “4” is the same as adding “4” and “6”.
4 - (-6) = 4 + 6 = 11
Multiplying Negative Numbers
When multiplying a negative number with another negative number, it produces a positive number. For example “- 2” multiplied by “-1” is the same as multiplying “2” by “1”.
(-2) (-1)= 2 1 = 2
When multiplying a negative number with a positive number, it produces a negative number. For example, “-3” multiplied by “2” is the same as multiplying “3” and “2” but the answer will always be negative.
(-3) 2 = -(32) = -6
Dividing Negative Numbers
When dividing a negative number with another negative number, it produces a positive number. For example “-12” divided by “-3” is the same as dividing “12” and “3” with the answer always being positive.
(-12) -(3) =12 3 = 4
When dividing a negative number by a positive number or vise versa, it produces a negative number. As an example, let's check out the examples below
(-6) 2 = -(6 2) = -3
6 (-2) = -(6 2) = -3
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Imagine a world wherein you couldn’t add, subtract, multiply or divide, what would it be like? Let’s say that you had 5 coins in your pocket and your friend gave you another coin, without using addition how would you describe what you now have? What if your friend took a coin away instead of giving you one, how do you describe how much you have in your pocket without using subtraction? Without arithmetic operations, a lot of things just wouldn’t make any sense.
Arithmetic is the branch of mathematics that focuses on the study of numbers and using different kinds of operations on them. You’ve probably heard of arithmetic operations before and know them better as addition (+), subtraction (-), multiplication (x), and division ()!
It is super important to learn arithmetic operations because it is the foundation of all advanced mathematics, sciences, art, music, and even everyday tasks. In addition we put values together, in subtraction we remove a value, in multiplication we add the same number to itself a certain number of times, and in division, we take large values and group them into smaller parts.
An important thing to remember when working with arithmetic operations is the importance of order. If you ever come across a problem that has more than one operation, always remember to follow the MDAS (or PEMDAS for those more complex problems) rule. The rule dictates that you should solve the equation in the order of Multiplication, Division, Addition, then subtraction.
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Linear Equations and Functions
All algebraic expressions can be graphed on what we call a plane. To better understand this, you can visualize a plane as a flat surface with a grid on it. Different algebraic expressions can be graphed and form different shapes and patterns. One of the most basic types of algebraic expressions is a Linear Equation, which is an equation that will form a straight line on a graph with an X and Y-axis. In general, a linear equation is an equation that can be written in the form y = ax + b where a and b are real numbers and x and y are variables.
Sometimes a linear equation is written as a function. A function is a special relationship where each input has a single output. Functions are usually expressed in a form of an equation, written with a special notation. We can think of a function as a machine that is programmed to apply certain rules that help define the input and output. Functions in the form of equations use the notation f(x). Sometimes a linear equation can be written as a function using f(x) instead of y. For example y = 3x + 5is the same as f(x) = 3x + 5. There are different kinds of Linear Functions, one example would be the Identity Function, where f(x) = x, which means that the output is identical to the input.
To get a more in-depth understanding of Linear Equations and Functions, check out the resources below!
Geometry (intro to Pythagorean Theorem)
Have you ever wondered how video game designers get virtual landscapes to look so realistic? You can thank Geometry for that! You can actually thank geometry for a lot of things like buildings, space travel, maps, and even medicine!
Geometry is a branch of mathematics that deals with lines, angles, space, figures, and shapes. It has a lot to do with defining, constructing, measuring, comparing, and proving figures and shapes. The most basic terms of geometry are points (a single location in a space), lines (the shortest distance between two points that have a length but no width), and angles (formed by two rays with a shared endpoint and is measured in degrees).
The most basic and one of the most popular geometric formulas is the Pythagorean Theorem. Pythagoras, a Greek mathematician and philosopher, was said to have made the discovery of triangles over 2,000 years ago. The theorem states that in a right triangle, the square of the long side, called the hypotenuse, is equal to the sum of squares of the other two sides. It is expressed in the formula a2+ b2 = c 2.
The Pythagorean Theorem is important in many real-life situations. For example, in architecture, it helps you calculate the length for certain parts such as roofings and support braces. It is also helpful in two-dimensional navigation. You can use the theorem to calculate the distance of your ship to a certain point and also how to better angle your course to get there faster!
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