Weekly Math Problem! #40

Molality. You weren't expecting that were you? Yeah...me either. 🤷🏿‍♀️😆 I wanted to post a calculus problem this week but, a conversation I had with one of my god-daughters (a college freshman) this evening, prompted me to post something different. I'm always interested in where mathematics shows up in the world and in other disciplines. So, this week we're putting our math skills to use in chemistry by doing a molality calculation. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to do substitution
        ✔️     How to do unit conversions
        ✔️     How to do metric conversions

You will also need the following chemistry skills:

        ✔️     How to use the periodic table
        ✔️     How to find molar mass
        ✔️     How to find moles of a compound

We'll definitely be looking up some things this week. WMP! #40 says:

Happy solving!

Check back on Friday, March 5th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Now, I've done Chemistry in college but it's been quite a few years since the last time I've done it. Nonetheless, the problem type is still familiar to me. Hopefully some of the resources I linked above were helpful. So...let's go! 💃🏿

Step 1. Calculate Molar Mass of the Solute

This is where you need to use the periodic table. You can't calculate the molality without knowing how many moles of the solute you have and you can't know how many moles of the solute you have without knowing the molar mass of the solute. Locate the elements of the solute (the thing being dissolved in water) in the periodic table to find what the mass of each is. Then proceed with the calculation as follows:

Step 2. Calculate Moles of Solute 

Now that we know the molar mass of NaCl, we can compute how many moles of the solute are present in 14.3 g of the substance. 

Step 3. Convert the Solvent to kg 

For whatever reason, the unit for the solvent (the things being dissolved into) is kg. Since our solvent is presented in g, we need to make the adjustment.

Step 4. Calculate the Molality of the Solution 

Here's the final step! 😄 Use the resulting information from steps 2 and 3 to calculate the molality.

Hmmm...not too bad. Now you know I just had to check my work--a way to ease the anxiety stemming from not having done this in years and to make sure I followed the steps properly. I used WolframAlpha to check my final answer: 

 

▪️ How did solving the problem this week go?

▪️ How did you feel about chemistry, in general?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next, WMP! #41.

Cheers!

The Younge Lady

Weekly Math Problem! #39

Contingency Table Probabilities. A contingency table is a nice way to summarize frequency related data for categories that have some of the data in common. You've probably seen one before. In the past I have helped some students prepare for an SAT exam and, without fail, there was at least one contingency table on every practice exam. You'll find that when a contingency table is presented, probabilities are asked for...especially conditional probabilities. That's what we're working on this week.

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to complete a contingency table with given information
        ✔️     How to use a contingency table to find probabilities of interest

Definitely have a calculator on-hand, in case you're interested in converting the probabilities to decimal format. (The question image is linked to the original source. I told y'all that I search the internet for all sorts of materials and things. 😁) WMP! #39 says the following:

Happy solving!

Check back on Friday, February 26th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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🏊🏿‍♀️ I'm jumping straight in. 

a. Let's organize the data in a contingency table! This part of the question has us placing the given information in the table so that we can see which cells need to filled-in. We're also finding the row and column totals. Totals are super important, as that information will be needed for the remining questions. The following are my partially filled table, calculations, and completed table:

Now that we have a completed table, let's move on the rest of the questions. Note: Make sure that 👏🏿 every 👏🏿 row and 👏🏿 column 👏🏿 total 👏🏿 check 👏🏿 out. If not, than any probability you pull from the table will be off or just incorrect.

b. The key word in this question, as small as it is, is the conjunction "and". "And" denotes an intersection, which is represented by '∩'. What does that mean for us? That means in order to find the probability that a randomly selected reader is female and [simultaneously] prefers Snapchat, we need to zero in on the cell that is in the Female row and Snapchat column; it's the cell where those two categories meet, or intersect. The cell contains 81. The reader being chosen at random is being chosen out of the 578 readers that responded to the survey. The probability is found by dividing the number of people in the event by the total number of people who participated in the survey. Here is the probability with proper notation:

c. This question threw us a little curve ball. Were you able to recognize that this probability is a conditional probability? 🤷🏿‍♀️ Conditional probability is represented by '|', which separates the category or event of interest and the condition. In effect, the condition changes the total to be considered. Instead of the total being all participants, it will be the total of the specified condition. What's the condition? In our case, the condition is "of the respondents who prefer Snapchat". This phrase is telling you to forget about everyone and just focus on those who prefer Snapchat--199 readers. How many of those 199 Snapchat folks are Female? 81. Just like the previous question, we'll divide but, this time, we're dividing the total number of Snapchat Females by the total number of people who prefer Snapchat. Here is the probability with proper notation:

d. Here's the second curve ball. What happens when you switch the categories in a conditional probability question? Let's find out because, that's what happened here. "Suppose a female reader is selected at random" is indicating to us that we're choosing from the Females only--362 of them. So, out of all 362 Females, how many of them prefer Snapchat? 81. Here is the probability with proper notation:

And we're done. YES! I actually like contingency tables. My favorite part about the data being organized in a contingency (or two-way) table is that I didn't need any formulas to answer the questions. 😁 The more I work with them, the more confident I become answering them and explaining them to other people. 

▪️ Were you able to find the probabilities?

▪️ How did you feel about PROBABILITY, in general?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

And now we move on to WMP! #40.

Cheers!

The Younge Lady

Weekly Math Problem! #38

Linear Regression. I scanned my current problems list and said, "Hmm...🤔... I don't see STATISTICS on here." I thought it would be fun to do a little linear regression so, here we are. I think STATISTICS is cool, however, there are many instances when performing the calculations can make you feel a little dizzy. 😵 This is why we're grateful for software. 🙃 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to find the regression line using the Least Squares method
        ✔️     How to use the regression line
        ✔️     How to interpret the slope regression line

You'll have to lookup the formulas and so will I. Perform the calculations by hand and use software to check. I know the numbers a little big but, we'll be okay. (You may want to do the calculations in stages, if doing it in one sitting is too much.) Alright, here goes WMP! #38:

Happy solving!

Check back on Friday, February 19th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Hopefully, you're not mad at me for these calculations. 🥴 Somehow, I found this to be a little, key phrase "a little", therapeutic. Yeah, I know, I know, I'm weird...but who isn't? 🤷🏿‍♀️ I will acknowledge that mistakes can easily be made, especially if you're not paying attention. I had to check my work...a must. I entered the data in Google Sheets to do so. Let's get started shall we...

a. To find the equation of the regression line, certain calculations are needed. We'll start with squaring and multiplying the data points, as needed, organizing them in chart. That information will, then, be used to calculate the intercept and the slope of the regression line. Finally, the equation of the regression line can be written.

   

b. To estimate the blood pressure, we'll use the equation of the regression line from part a. How? Substitute the age 50 in place of x, then perform the indicated operations. The answer you get will be the estimated blood pressure, assuming no computation errors were made.

c. In our situation, the residual is the difference between the blood pressure of a 42-year old as observed in the data and the estimated blood pressure of a 42-year old as computed from the regression line equation.

d. To interpret the slope, keep in mind the context of the problem and think about how the blood pressure number changes, when the age of a person increases by one year.

I hope you found this helpful. I sure did! It was super helpful for me, actually. Part d. was my favorite part of the problem because, it required to me explain the meaning of the slope. Interpreting/explaining numbers reminds me that it isn't always about what the number is. Just as important is what the number means. This is how reports get written. This helps with decision making in certain sectors. 

But before we go, I just have one more thing to share...a scatter plot of the data from the problem with the fitted regression line.

*This plot was created using Google Sheets.*

▪️ Did you find this problem hard, annoying, or something else?

▪️ How did you feel about STATISTICS, in general?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

L👀ks like WMP! #39 is quickly approaching.

Cheers!

The Younge Lady

Weekly Math Problem! #37

Completing the Square. One thing that can be super annoying--more than solving a problem--is being told how to solve a problem. 😒 Wait a minute. What happened to my mathematical freedom to choose? 😩 It seems like this is done when the instructor has something up his/her sleeve. You may even feel like you are being tricked on purpose. (Sometimes you are.) However, some instructors do that to see if you've payed attention, while some want to see if you have the ability to transfer prior skills to current topics. Even though many instructors have your best learning interests at heart, it is still annoying. 🙄

This week's WMP! is a very good example of the above. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to solve a quadratic equation
        ✔️     Completing the square method

WMP! #37 says:

Happy solving!

Check back on Friday, February 12th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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Jumping right in, the first thing that needs to be done is to get the quadratic equation in a form that accommodates the completing the square method to commence--a leading coefficient of 1 and the resulting constant isolated:

Now that the quadratic equation is in the proper form, we can go ahead and employ the completing the square method. (Right now it's incomplete. 🤪) This is where some students get tripped up. Notice that the b-term is not an integer but a fraction. It can be annoying to complete the square when the b-term is a fraction, it can be done. Multiply the fraction by 1/2, then square the result.

So, 4/25 is the number that will allow us to complete the square. Here is how to complete the rest of problem, which involves using the square root method to finish solving for x:

 

Remember, there are two solutions that will satisfy the original quadratic equation we were asked to solve. Lastly, we'll separate the plus-minus and round each solution to the nearest ten-thousandths.

Of course there is a graph. 😁

**This plot was created using Geogebra's Graphing Gaclulator.

▪️ How did you find this week's problem?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

WMP! #38 here we come!

Cheers!

The Younge Lady