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Continuously Compounding Interest Formula

Hi everyone and welcome to MathSux! In this post, we are going to go over the continuously compounding interest formula! This is a great topic as it relates to finance and real world money situations (sort of). We are going to breakdown everything step by step by understanding what the continuously compounding interest rate formula is, identify each component of the formula, and then apply it to an example. If you’re looking for more, don’t forget to check out the video and the practice questions at the end of this post. Happy calculating! 🙂

What is Continuous Compounding Interest Formula?

Let’s say we have $500 and we want to invest it.

What if it compounded interest once a year? 

Twice a year? 

Once a day, or 365 days a year?

What if we compounded interest every second of the day for a total of 86,400 seconds throughout the year!?

And what if we kept going, making the number of times compounded annually more and more often to occur every half second? This is what Continuous Compounding Interest is, and it tells us how much we earn on a principle (original amount) if the compound interest rate for the year were to be granted an infinite number of times.

The weird thing is that continuous compounding interest is technically impossible (I’ve yet to see a bank that offers an infinite number of compounding interest!).  Even though it is impossible, in math and finance, we look at continuous compounding interest for theoretical purposes, in other words, it’s for money nerds! Luckily, it comes with an easy-to-use formula, let’s take a look:

Continuously Compounding Interest Formula

Now, let’s see this formula in action with the following Example:

Step 1: First, let’s write out our formula and identify what each value represents based on the question.

Continuously Compounding Interest Formula

Step 2: Fill in our formula with the given values and solve.

Practice Questions:

1) Sally invested $1000 which was then continuously compounded by 4%. How much money will Sally have after 5 years?

2) Brad invested $1500 into an account continuously compounded by 5%. How much money will he have after 7 years?

3) Fran invests $2000 into an account that is continuously compounded by 1%. How much money will Fran earn by year 5?

Solutions:

1) $1,221.40

2) $2,128.60

3) $2,102.54

Want to make math suck just a little bit less? Subscribe to my Youtube channel for free math videos every week! 🙂

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Looking to review more topics in Algebra? Check our Algebra page here!

NumWorks Calculator Review

Greeting math friends and welcome to MathSux! In today’s post we are going to review and take a look at how to use the graphing calculator available by the French company, NumWorks.

In this NumWorks calculator review, first impressions are that this is a serious competitor for Texas Instruments and offers more features than a typical calculator with a focus on statistics, data analysis, and even computer programming! Check out the video below to see the un-boxing, full review, and how to use this calculator step by step. Happy calculating! 🙂

NumWorks Calculator Stand Out Features:

NumWorks Calculator Review

1) The Home Screen: Works and looks like apps on an iPhone. It is super easy to use, and includes apps such as the regular graphing calculator we’re all used to, as well as, Python, Statistics, Probability, Equation Solver, Sequences, and Regression.

2) The Equation Solver: Punch in any function and find it’s x-values and discriminant! Very cool!

3) Python: Yes, this calculator is programmable via Python! It also includes pre-made scripts that you can easily run. This is great for aspiring programmers and important for today’s economy.

4) Exam Mode: Teachers can make students put their calculators in exam mode and watch their students calculators light up in red to prove there’s no cheating funny business going on! Warning though, this will delete all of your data including the pre-made Python scripts. But you can always hit the reset button in the back to reset.

NumWorks Calculator Review

Did I mention math teacher’s can potentially get a free calculator from NumWorks? Check out the link here!

Has anyone else tried this graphing calculator from NumWorks? What were your first thoughts? Let me know in the comments and happy calculating!

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For more math resources, check out this post here and happy calculating! 🙂

Square Inscribed in a Circle Construction

Greetings math friends and welcome to MathSux! In this week’s post, we are going to take a step by step look on how a square inscribed in a circle construction works! We got videos, we got GIF’s, and we got a step by step written explanation below, the choice of learning this construction is up to you! Happy Calculating! 🙂

Square Inscribed in a Circle Construction

How to Construct a Square Inscribed in a Circle:

Step 1: Draw a circle using a compass.

Step 2: Using a ruler, draw a diameter across the length of the circle, going through its midpoint.

Step 3: Open up the compass across the circle. Then take the point of the compass to one end of the diameter and swing the compass above the circle, making a mark.

Step 4: Keeping that same length of the compass, go to the other side of the diameter and swing above the circle again making another mark until the two arcs intersect.

Step 5: Repeat steps 3 and 4, this time creating marks below the circle.

Step 6: Connect the point of intersection above and below the circle using a ruler. This creates a perpendicular bisector, cutting the diameter in half and forming 90º angles.

Step 7: Lastly, use a ruler to connect each corner point to one another creating a square.

Still got questions? No problem! Don’t hesitate to comment with any questions. Happy calculating! 🙂

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Looking for something similar to square inscribed in a circle construction? Check out this post here on how to construct and equilateral triangle here!

How to Calculate Z-Score?: Statistics

Hi there and welcome to MathSux! In this post, we are going to explore how to calculate z-score and the normal distribution. We’ll do this by examining the normal curve and learning how to find probability finding z-score and using the mean, standard deviation, and specific data points. Fore more info and more MathSix don’t forget to check out the video and practice questions below. Happy calculating!

What is a Normal Curve?

A normal curve is a bell shaped curve that shows the distribution of data evenly spread with respect to the mean. If you look at the normal curve below, the area under the curve shows all the possible probabilities of a certain data point occurring, notice the curve is higher towards the center mean, μ, and gets smaller as the distance from μ grows. The distance from μ is measured by the standard deviation, a unique unit of measurement that is specific to each group of data.

Mean: The mean always falls directly in the center of our normal curve. It is the average of our data, and always falls right in the middle.

Standard Deviation:  This value is used as a standard unit of measurement for the data, measuring the distance between each data point in relation to the mean throughout the entire data set. For a review on what standard deviation is and how to calculate it, check out this post here.

Now for our normal curve:

Notice half of the data is below the mean, μ, while the other half is above? The normal curve is symmetrical about the mean, μ!

How to Calculate Z-Score?

Z Score can tell us at what percentile a certain point in the data set falls in relation to the rest of the mean by using the standard deviation as a unit of measurement.  If this sounds confusing, it’s ok! Take a look at the following formula:

How to Calculate Z-Score?

We use the above formula in conjunction with  a z table which tells us the probability under the curve for a certain point.

Solution:

a) What percent of student scored below 500?

Step 1: First, let’s draw out our given information the mean=500, standard deviation=100, and the data point the question is asking for x=500 onto a normal curve. Notice that we want to find the value of the area under the curve shaded in pink.  This will tell us the percent of students that scored below 500.

How to Calculate Z-Score?

Step 2: We need to find the z-score by, using the data point given to us x=500, the mean=500, and the standard deviation, sigma=100.

Step 3: Yes, we have a zero! Now we need to take our z table and line up our chart. Notice that the chart finds the probability for everything at the beginning of the normal curve and on.  This is perfect for answering our question!

How to Calculate Z-Score?

Step 4: The table gives us our solution of .5000.  If we multiply .5000 times 100 it gives us the percent of students who scored below 500 at 50%.

b) What percent of student scored above 620?

Step 1: First, let’s draw out our given information the mean=500, standard deviation=100, and the data point the question is asking for x=620 onto a normal curve. Notice that we want to find the value of the area under the curve shaded in pink.  This will tell us the percent of students that scored above 620.

How to Calculate Z-Score?

Step 2: We need to find the z-score by, using the data point given to us x=620, the mean=500, and the standard deviation, sigma=100.

How to Calculate Z-Score?

Step 3: Yes, we got 1.2! Now we need to take our z table and line up our chart. Notice that the chart finds the probability for everything at the beginning of the normal curve and on.  This is means to find the percent we are looking for, we need to subtract our answer from one since we want the value of probability on the right side of the curve (the z-table only provides the left side).

How to Calculate Z-Score?

Step 4: The table gives us our solution of .8849.  If we subtract this value from 1 then multiply that value times 100 it gives us the percent of students who scored above 620.

How to Calculate Z-Score?

C) What is the highest score a student could receive if the students was in the 16.11th percentile?

Step 1: In this question we have to work backwards by first identifying, where on the z-score table is the number .1611 and then filling in our z score formula to find x, the missing data point (in this case test score).

Search the table for .1611:

How to Calculate Z-Score?

Notice that .1611 can be found on the z-table above with z-score -0.99.  This is what we’ll use to find the unknown data point!

Step 2: We need to find the unknown test score by, using the z score we just found z=-0.99, the mean=500, and the standard deviation, sigma=100.

How to Calculate Z-Score?

Step 3: Solve for x.

How to Calculate Z-Score?

Practice Questions:

The grades on a final English exam are normally distributed with a mean of 75 and a standard deviation of 10.

a) What percent of students scored below a 60?

b) What percent of students scored above an 89?

c) What is the highest possible grade that included in the 4.46th percentile?

d) What percent of students got at least a 77?

Solutions:

a) 6.68%

b) 8.08%

c) 58

d) 42.07%

Want to make math suck just a little bit less? Subscribe to my Youtube channel for free math videos every week! 🙂

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Simultaneous Equations: Algebra

Happy new year and welcome to Math Sux! In this post we are going to dive right into simultaneous equations and how to solve them three different ways! We will go over how to solve simultaneous equations using the (1) Substitution Method (2) Elimination Method and (3) Graphing Method. Each and every method leading us to the same exact answer! At the end of this post don’t forget to try the practice questions choosing the method that best works for you! Happy calculating! 🙂

What are Simultaneous Equations?

Simultaneous Equations are when two equations are graphed on a coordinate plane and they intersect at, at least one point.  The coordinate point of intersection for both equation is the answer we are trying to find when solving for simultaneous equations. There are three different methods for finding this answer:

We’re going to go over each method for solving simultaneous equations step by step with the example below:

Method #1: Substitution

The idea behind Substitution, is to solve for 1 variable first algebraically, and the plug this value back into the other equation solving for one variable.  Then solving for the remaining variable.  If this sounds confusing, don’t worry! We’re going to do this step by step:

Step 1: Let’s choose the first equation and move our terms around to solve for y.

Simultaneous Equations

Step 3: The equation is set up and ready to solve for x!

Simultaneous Equations

Step 4: All we need to do now, is plug x=3 into one of our original equations to solve for y.

Simultaneous Equations

Step 5: Now that we have solved for both x and y, we have officially found where these two simultaneous equations meet!

Simultaneous Equations

Method #2: Elimination

The main idea of Elimination is to add our two equations together to cancel out one of the variables, allowing us to solve for the remaining variable.  We do this by lining up both equations one on top of the other and adding them together.  If variables at first do not easily cancel out, we then multiply one of the equations by a number so it can. Check out how it’s done step by step below!

Step 1: First, let’s stack both equations one on top of the other to see if we can cancel anything out:

Simultaneous Equations

Step 2: Our goal is to get a 2 in front of y in the first equation, so we are going to multiply the entire first equation by 2.

Simultaneous Equations

Step 3: Now that we multiplied the entire first equation by 2, we can line up our two equations again, adding them together, this time canceling out the variable y to solve for x.

Simultaneous Equations

Step 4: Now, that we’ve found the value of variable x=3, we can plug this into one of our equations and solve for missing unknown variable y.

Simultaneous Equations

Step 5: Now that we have solved for both x and y, we have officially found where these two simultaneous equations meet!

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Method #3: Graphing

The main idea of Graphing is to graph each a equation on a coordinate plane and then see at what point they intersect.  This is the best method to visualize and check our answer!

Step 1: Before we start graphing let’s convert each equation into y=mx+b (equation of a line) form.

Equation 1:

Equation 2:

Step 2: Now, let’s graph each line, y=3x-4 and y=-x+8, to see at what coordinate point they intersect.

Simultaneous Equations

Need to review how to draw an equation of a line? Check out this post here! Notice we got the same exact answer using all three methods (1) Substitution (2) Elimination and (3) Graphing.

Ready to try the practice problems on your own?! Check them out below!

Practice Questions:

Solve the following simultaneous equations for x and y.

Solutions:

  1. (1, 3)
  2. (4,5)
  3. (-1, -6)
  4. (3, -3)

Want more MathSux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating!

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30 60 90 Special Triangles: Geometry

Hi everyone and welcome to MathSux! In this post we are going to break down 30 60 90 degree special triangles. What is it? Where did it come from? What are the ratios of it’s side lengths and how to do we use them? You will find all of the answers to these questions below. Also, don’t forget to check out the video below and practice questions at the end of this post. Happy calculating! 🙂

Want to make math suck just a little bit less? Subscribe to my Youtube channel for free math videos every week! 🙂

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What is a 30 60 90 Triangle and why is it “Special”?

The 30 60 90 triangle is special because it forms an equilateral triangle when a mirror image of itself is drawn, meaning all sides are equal!  This allows us to find the ratio between each side of the triangle by using the Pythagorean theorem. Check it out below!

30 60 90 Special Triangles

Now let’s draw a mirror image of our triangle.  Next, we can label the length of the new side opposite 30º “a,” and add this new mirror image length with the original we had to get, a+a=2a.

30 60 90 Special Triangles
30 60 90 Special Triangles

If we look at our original 30 60 90 triangle, we now have the following values for each side based on our equilateral triangle:

30 60 90 Special Triangles
30 60 90 Special Triangles

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two legs. This creates a ratio that applies to all 30 60 90 triangles!

30 60 90 triangle side lengths

How do I use this ratio?

30 60 90 triangle side lengths

Knowing the above ratio, allows us to find any length of any and every 30 60 90 triangle, when given the value of one of its sides.

Let’s try an Example:

30 60 90 triangle side lengths

-> First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

->Notice we are given the value of a, which equals 4, knowing this we can now fill in each length of our triangle based on the ratio of a 30 60 90 triangle.

30 60 90 triangle side lengths
30 60 90 triangle side lengths

Now let’s look at an Example where we are given the length of the hypotenuse and need to find the values of the other two missing sides.

30 60 90 triangle side lengths

->First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

-> Notice we are given the value of the hypotenuse, 2a=20. Knowing this we can find the value of a by dividing 20 by 2 to get a=10. Once we have the value of a=10, we can easily find the length of the last leg based on the 30 60 90 ratio:

30 60 90 triangle side lengths
30 60 90 triangle side lengths

Now for our last Example, when we are given the side length across from 60º and need to find the other two missing sides.

30 60 90 triangle side lengths

->First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

-> In this case, we need to use little algebra to find the value of a, using the ratio for 30 60 90 triangles.

30 60 90 triangle side lengths

Now that we have one piece of the puzzle, the value of a, let’s fill it in our triangle below:

Finally, let’s find the value of the length of the hypotenuse, which is equal to 2a.

Practice Questions:

Find the value of the missing sides of each 30 60 90 degree triangle.

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Looking to review 45 45 90 degree special triangles? Check out this post here!

NYE Ball Fun Facts: Volume & Combinations

Greetings math friends, and Happy New Year! In today’s post we’re going do something a little different and take a look at the math behind the very famous and very shiny New Year’s Eve Ball that drops down every year at midnight.  We’ll break down the shape, the volume, and the number of those dazzling Waterford crystals (and no this post isn’t sponsored) and look at some NYE Ball Fun Facts.

NYE Ball Fun Facts

Shape: Geodesic Sphere

Yes, apparently the shape of the New Year’s ball is officially called a “Geodesic Sphere.”  It is 12 feet in diameter and weighs 11,875 pounds.

Volume (Estimate): 288π ft3

If we wanted to estimate the volume of the New Year’s Ball we would could use the formula for volume of a sphere:

NYE Ball Fun Facts

Number of Waterford Crystals: 2,688

Talk about the ultimate shiny bauble! The NYE ball lights up the night with all 2,688 crystals in the shape of different sized triangles, each with heights of 5.75 inches or 4.75 inches.

NYE Ball Fun Facts

Number of Lights: 48 light emitting diodes (LED’s)

On each triangle, there are 48 LEDs: 12 red, 12 blue, 12 green, and 12 white, for a total of 32,256 LEDs on the entire NYE ball itself.

NYE Ball Fun Facts

Permutations and Combinations:

Permutations: With this many lights and colors, there are over a billion potential permutations of colors on the entire NYE ball.

Combinations: Let’s break down one triangle with 48 LED lights each with 12 red, 12 blue, 12 green, and 12 white LEDs. How many possible combinations of lights are possible if we were to choose 7 blue, 5 red, 10 green, and 1 white turned on all at the exact same time?

We end up with the combination formula below:

NYE Ball Fun Facts

That means that there are 496,793,088 possible ways that 7 blue lights, 5 red lights, 10 green lights, and 1 white light can be lit up on a triangle that is part of the entire NYE ball!

Interested in more NYE fun facts?  Check out the sources of this article here.

NYE Fact Sheet from: timessquarenyc.org

NYE Ball picture: Timesquareball.net

If you like finding the volume of the NYE ball maybe, you’ll want to find the volume of the Hudson Yards Vessal in NYC here.  Happy calculating and Happy New Year from MathSux!

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Trig Functions (Amp, Freq, Phase Shifts): Algebra 2/Trig.

Hi everyone, and welcome to MathSux! In this post we are going to break down how to graph trig functions by identifying its amplitude, frequency, period, and horizontal and vertical phase shifts. Fear not! Because we will breakdown what each of these mean and how to find them, then apply each of these changes step by step on our graph. And if you’re ready for more, check out the video and the practice problems below, happy calculating! 🙂

*For a review on how to derive basic Trig functions (y=sinx, y=cosx, and y=tanx), click here.

What are the Different Parts of a Trig Function?

Trig functions

Amplitude: The distance (or absolute value) between the x-axis and the highest point on the graph.

Frequency: This is the number of cycles that happen between 0 and 2π. (Α “cycle” in this case is the number of “s” cycles for the sine function).

Period: The x-value/length of one cycle. (Α “cycle” in this case is the number of “s” cycles for the sine function). This is found by looking at the graph and seeing where the first cycle ends, or, by using the formula:       

Horizontal Shift: When a trigonometric function is moved either left or right along the x-axis.

Vertical Shift: When a trigonometric function is moved either up or down along the y-axis.

Let’s try an Example, graphing a Trig Function step by step.

Step 1: First let’s label and identify all the different parts of our trig function.

Trig functions

Step 2: Now let’s transform our graph one step at a time.  First let’s start graphing y=cos(x) without any transformations.

Trig functions

Step 3: Let’s add our amplitude of 2, the distance to the x-axis.  To do this our highest and lowest points on the y-axis will now be moved to 2 and -2 respectively. 

Trig functions

Step 4: Next, we do a horizontal phase shift to the left by (π/2).  To do this, we look at where negative (π/2) is on our graph at (-π/2) and move our entire graph over to start at this new point, “shifting” it to (π/2).

Trig functions

Step 5: For our last transformation, we have a vertical phase shift up 1 unit.  All this means is that we are going to shift our entire graph up by 1 unit along the y-axis.

Trig functions

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for an in depth explanation. Happy calculating! 🙂

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Graphing Linear Inequalities: Algebra

graphing linear inequalities

Hi and welcome to MathSux! In this post, we are going to go over the rules for graphing linear inequalities on a coordinate plane when it comes to drawing lines, circles , and shading, then we are going to solve an example step by step. If you have any questions, check out the video below and try the practice questions at the end of this post! If you still have questions, don’t hesitate to comment below and happy calculating! 🙂

Graphing Linear Inequalities:

When graphing linear inequalities, we always want to treat the inequality as an equation of a line in  form y=mx+b….with a few exceptions:

Graphing Linear Inequalities
inequality shading above or below y-axis

Now that we know the rules, of graphing inequalities, let’s take a look at an Example!

graphing inequality example

Step 1: First, let’s identify what type of inequality we have here.  Since we are working with a > sign, we will need to use a dotted line and open circles when creating our graph.

graphing inequality example

Step 2: Now we are going to start graphing our linear inequality as a normal equation of a line, by identifying the slope and the y-intercept only this time keeping open circles in mind.  (For a review on how to graph regular equation of a line in y=mx+b form, click here)

graphing inequality example
graphing linear inequalities

Step 3: Now let’s connect our dots, by using a dotted line to represent our greater than sign.

graphing linear inequalities

Step 4: Now it is time for us to shade our graph, since this is an inequality, we need to show all of our potential solutions with shading.  Since we have a greater than sign, , we will be shading above the y-axis.  Notice all the positive y-values above are included to the left of our line.  This is where we will shade.

graphing linear inequalities

Step 5: Check!  Now we need to check our work.  To do that, we can choose any point within our shaded region, if the coordinate point we chose hold true when plugged into our inequality then we are correct!

Let’s take the point (-3,2) plugging it into our inequality where x=-3 and y=2.

graphing linear inequalities

Practice Questions:

graphing linear inequalities

Solutions:

graphing linear inequalities
graphing linear inequalities

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above for an in depth explanation. Happy calculating! 🙂

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Central Angles Theorems: Geometry

Hi everyone, and welcome to MathSux! In this post, we are going to go over the Central Angles Theorems of circles. We’ll go over the theorems associated with central angles and then solve a quick example. Make sure to test your understanding of central angles and arcs with the practice questions at the end of this post. And, if you want more, don’t forget to check out the video below, happy calculating!

Central Angles and Arcs:

Central angles and arcs form when two radii are drawn from the center point of a circle.  When these two radii come together they form a central angle. A central angle is equal to the length of the arc. When it comes to measuring the central angle, the central angle is always equal to arc length and vice versa:

Central Angles = Arc Length

central angles theorems

Central Angle Theorems:

There are a two central angle theorems to know, check them out below!

Central Angle Theorem #1:

central angles theorems

Central Angle Theorem #2:

central angles theorems

Let’s look at how to apply these rules with an Example:

central angles theorems

Let’s do this one step at a time.

central angles theorems
central angles theorems
central angles theorems
central angles theorems
central angles theorems
central angles theorems

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

*Also, if you want to check out Intersecting Secants click this link here!

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