GMAT

Permutations and Combinations Part II: Formulas

amanda chen — Fri, 19 Mar 2010 15:35:47 +0000

Once you’ve figured out whether to use permutation or combination, there is actually very little work to be done if you know the formulas for permutation and combination or if you know where the function is hidden on your calculator.

As I mentioned before, permutation is used when the order matters and combination when you just want to choose, but not order, the items. Let’s dive straight into the formulas then go through a few examples.

The number of ways of permuting r objects out of n objects is given by

n P r = n!/(n-r)! where n! = n * (n-1) * (n-2) * … * (2) * (1)

The number of ways of combining r objects out of n objects is given by

n C r = n!/r!(n-r)!

Note that 0! is defined as 1.

Example 1

How many ways are there of arranging 4 letters out of the following F R I E N D?

FRIEND has 6 different letters, and since the order of letters matter, I know I have to use the permutation formula. Applying the formula directly, the answer is given by

6 P 4 = 6!/(6-4)! -= 6!/2! =360

You can double-check this by using the fundamental counting principle that we covered in Part I. Drawing a tree diagram will also work, but it might get a little messy as the tree gets bigger and bigger.

Example 2

If there are 3 entrees and 5 desserts, how many ways are there of choosing 1 entrée and 2 desserts? Note, that the question does not say anything about the order in which the entrees and desserts are eaten, so we know to use the combination principle. Because we are choosing entrees and desserts separately, we have to apply the combination formula twice.

First, to choose 1 entrée out of 3, we apply the formula 3 C 1 = 3!/1!(3-1)! = 3!/1!2! = 3

Next, to choose 2 desserts out of 5, we apply the formula 5 C 2 = 5!/2!(5-2)! = 5!/2!3! = 10

Then because for each entrée, there are 10 possible 2-dessert combinations and there are 3 ways of choosing 1 entrée, to get the total number of possibilities, we take 3*10 = 30.

Example 3

What if the question throws you a curveball and asks you to permute something that has a repeated item. For example, how many ways are there of arranging the letters A G H A S T?

The ‘A’ is repeated twice so first we pretend that the letters are distinct and find the number of possibilities. Then divide that value by 2! because ‘A’ is repeated twice.

AGHAST has 6 letters, and if we permute all the letters, we get 6!

Because A is repeated, we divide 6! by 2! to get the answer 360

Supposing I asked you to permute only 4 out of 6 of the letters from AGHAST, then you would do 6 P 4 as per normal, and divide that answer by 2! since A is repeated. The answer should be 180.

Check out Grockit for more GMAT permutations and combinations practice!

GMAT Solving Linear Equations with 1 Unknown

amanda chen — Wed, 17 Mar 2010 14:38:07 +0000

A linear equation is any equation where the highest power of the unknown, which I shall call x, is 1. To illustrate more clearly with a few examples:

x+1 = 4; 10x = 3; x = 18 – 4x are three examples of linear equations

x² + 2 = 2x and x³ = 8 are not linear equations because there are x’s that are raised to a higher power than 1.

A linear equation with 1 variable is the simplest type to solve. There is 1 equation and 1 unknown, which means that the unknown can always be determined. To solve such an equation, you need to rearrange the equation to have like terms on either side of the equal sign. Put another way, you are trying to isolate x (or whatever the variable is called) on one side of the equation.

For example, if 2x = 234, to isolate x, we have to divide the entire equation by 2. Doing this, we get x = 117.

If there are x’s and numbers on either side of the equal sign, we add and subtract values to isolate x on one side. Suppose 2x – 17 = 18 – 3x

The first thing we could do is to add 17 to both sides to get: 2x – 17 + 17 = 18 – 3x + 17

This reduces to 2x = 35 – 3x

Now, we need to have all the x’s on one side so we add 3x to both sides to get: 2x + 3x = 35 – 3x + 3x

This reduces to 5x = 35

Dividing by 5 on both sides, we get x = 7

What I just went through was a fairly simple algebraic equation. The questions on the GMAT will look more complicated but you are essentially doing the same thing: manipulating both sides of the equation in the same way to isolate x. Let’s try a practice problem from Grockit.

To tackle this question, we multiply both sides by 2+3/x to get 3 = 2+3/x. . (This is also known as cross multiplying where in general if a/b-c/d,

then ad = bc

To simplify 3 = 2+3/x,

we multiply the entire equation by x to get 3x = 2x + 3. This leaves you with a much simpler equation that you already know how to solve.

What’s a little trickier than manipulating algebraic equations is translating a word problem into an algebraic equation. Here’s another practice problem:

Jack and his brother are sharing a monster piece of licorice that is 28 inches long. Since Jack is older, his share is 8 inches longer than his brother’s. How long, in inches, is Jack’s brother’s piece?

The way to solve this problem is to let something be x. Here’s what happens if we let Jack’s piece be x inches.

Jack’s piece = x inches

Jack’s brother’s piece = x – 8 inches

Total length of licorice = Jack’s piece + Jack’s brother’s piece = 28 = x + (x-8)

This means that x = 18 inches. But remember that the question wants the length of Jack’s brother’s piece, which we have defined as x – 8. So the correct answer is 10 inches.

Here’s what happens if we let Jack’s brother’s piece be x inches.

Jack’s brother’s piece = x inches

Jack’s piece = x + 8 inches

Total length of licorice = 28 = x + (x+8) and we determine that x = 10. In this case, since we have already defined Jack’s brother’s piece to be x, there is no further step we need to take.

In general, here are a few things to keep in mind.

if there is only one unknown, you only need one equation to determine the value of the unknown
in dealing with algebraic equations, remember that anything you do to one side (be it adding, subtracting, multiplying or dividing) you need to do to this other side too.
in dealing with word problems, define something to be x and see if you can define other things in terms of x only. (For example, in the question about licorice, you would not want to let Jack’s piece be x inches and his brother’s be y inches) Don’t introduce unnecessary variables if it can be expressed in terms of an existing variable.

MBA for Career Switching Part 4 (Final)

chase cairncross — Mon, 15 Mar 2010 16:40:49 +0000

The summer internship is a big part in switching. Not only will you be able to decide if this is in fact an industry you would like to be in, it is a long-term job interview. If you like the job and you do well it in, your second year of business school could be a breeze if they make you an offer. If you find out that you love it but they don’t love you, then at least you are part of the way home and now have good experience. If you find out not only do they not want you, but you don’t want them, then you have a lot of work in front of you.

Lets deal with the latter two, as the former is pretty easy. If you love the field, but they do not offer you a job, you will find yourself in a large camp of fellow business school students. A lot of firms simply do not have the budget to make an offer at that time, so the key is to keep in touch with the firm and specifically a person who liked you there. Try to meet up with them throughout the second year to see if things have changed on the hiring front. Make sure you are LinkedIn with them and have them write a recommendation for you when you are still fresh in their memory.

The next step is to get back to the informational interview grind. Get back in touch with your solid contacts, let them know about your internship and get back into their minds. Secondly, don’t fret too much as there will be a lot of companies coming on campus and not getting an offer is not a scarlet letter that should cause problems for you.

If you realized during your internship that you do not really want to go into that field, you are both lucky and disadvantaged. On one hand you got paid for three months of work to save you from getting into a lifetime of something that does not interest you, but on the other hand you now are back at close to square one with less time on your hands. You need to really focus on developing a new list of alumni contacts to speak to and possibly push for an academic internship.

Either way stay involved in the community, because you never know what type of opportunities will arise by just speaking to classmates, professors or people in the field.

Finding your job will be a job and you need to treat it as such. Do not get bogged down looking to get all A’s and graduate owing Sallie Mae and without a job. Make sure you continue to prioritize your time in a way that allows you to take advantage of all of the help you will have during this time, because it will recede after you graduate.

To sum it all up, business school is basically about getting you a job and the resources and time are set up that way. Do yourself a favor and use these resources fully and find that job that just cost you a pretty penny.

See my other articles in this series:

Permutations and Combinations Part I: Counting

amanda chen — Fri, 12 Mar 2010 17:01:14 +0000

When dealing with permutations and combinations, you are essentially trying to find the number of different outcomes given a set of items and a number of restrictions. The difference between permutation and combination merely depends on whether the order matters. Let me illustrate with an example. Suppose you have three food items, apples, bananas and carrots. If you were asked to only pick two of them, how many possibilities are there? If you got 3, that’s the right answer. You have apples & bananas, apples & carrots and bananas and carrots. This is combination. If I told you however, that the order in which you eat the food matters, then you have more possibilities, because instead of just apples & bananas, you have to consider bananas & apples too. The latter is permutation.

Listing

What I did earlier, when I listed out the 3 choices is called, not surprisingly, listing. But there is a method to listing to ensure that you don’t leave any possibilities out. Everyone tends to have his or her own method, let me share mine. If I have four items – a, b, c and d – and I’m supposed to choose 2:

Step 1: I take the first item a and combine it with b. Then I go down the list and combine it with c and d to get a total of three possibilities: ab, ac, ad.

Step 2: I take the second item b and combine it with the rest of the list. I don’t combine it with a because I already did that in step 1. So I get: bc, bd

Step 3: Keep doing the same thing for the next thing on the list, which is c. For c, there’s only one thing behind it on the list – d. So there’s only 1 possibility here: cd

Step 4: Keep going until you get to the second last item. In this case, we already reached it in step 3. Tally up your choices: ab, ac, ad, bc, bd, cd

Tree Diagram

If you noticed, listing is a good method for figuring out combinations. The Tree Diagram is a graphical method that helps you with permutation. Using the previous example of 4 items, here’s the tree diagram that illustrates the number of ways of permuting items.

If you look at the last line of the tree diagram and count the number of boxes, you will see that there are 12 possible ways to permute 2 items out of 4. Each item can be combined with 3 other things. Another way of seeing this is that you have two spaces you need to fill ___ and ___. Looking at the first space, you have 4 possible items you could place there. Looking at the second space, you have 3 possible items you could place there, since you have already put 1 item in the first space. So there are 4 x 3 = 12 ordered possibilities.

Fundamental Counting Principle

The spaces-and-number-of-options-to-fill-that-space method of thinking is essentially the fundamental counting principle. The principle states that

if task A can be done in m ways and after Task A is complete, Task B can be done in n ways, then there are m*n ways of completing Task A then Task B.

Let’s see if you understand the principle with a quick example. How many different ways are there are arranging these letters: T A N G O

Did you get 120?

Imagine five spaces ___ ___ ___ ___ ___

You have 5 possible letters to put in the first space, and when that’s done, you have 4 possible letters to put in the second space, 3 letters for the third space and so on. So the answer is 5*4*3*2*1 or 5!

If you’re comfortable using these methods to solve permutation and combination problems, stay tuned for the second installment that uses direct formulas to calculate the number of possibilities. In the meantime, check out Grockit for GMAT quantitative practice.

Interest and Compound Interest Problems

martin sobolewski — Wed, 10 Mar 2010 16:26:19 +0000

Interest and Compound Interest

There are two types of interest problems on the GMAT, and they include simple interest and compound interest. Simple interest is the most basic and is a function of P, the principle amount of money invested, the interest rate earned on the principle, i, and the amount of time the money is invested, t (this is usually stated in periods, such as years or months). The resulting equation is: I = iPt

In basic terms, the above equation tells us the amount of interest that would be earned on a principle amount invested (P), for a given time (t) at a given interest rate (i).

Example:

If you invested $1,000 (P = your principle) for one year (t = one year) at 6% simple interest (i = given interest rate), you would get $60 in interest at the end of the year and would have a total of $1,060.

For compound interest, you would earn slightly more.

Let’s look at similar type problem, though this one involves compound interest.

Mr. Riley deposits $500 into an account that pays 10% interest, compounded semiannually. How much money will be in Mr. Riley’s account at the end of one year?

For compound interest, first you need to divide the interest rate by how many compound periods there are. So for in the above question, because we are compounding semiannually, we need to divide 10% by 2 (because of 2 compounding periods), and if we were compounding quarterly, we would need to divide 10% by 4.

In the above question, Mr. Riley deposited $500 into his account at a rate of 10% compounded semiannually and the bank will divide his interest into two equal parts. They will pay 5% interest (10%/2) at the end of six months, and then will pay another 5% at the end of the year. Compound interest can essentially be translated into “interest paid on interest”, meaning that after one period, you are paid interest on the interest that was paid in prior periods, hence the phrase “compounding”.

So at the end of the six months, Mr. Riley has $525 because the bank paid $25 in interest ($500*5%) into his account. For the second half of the year, Mr. Riley is then paid 5% on the $525 balance that was in his account at the end of the first six months. This interest is equal to $525*5% = $26.25. Therefore, at the end of the year, Mr. Riley has $551.25, which is equal to his balance of $500, plus the $25 interest paid at the end of 6 months, plus $26.25 paid at the end of the year. Mr. Riley earns $1.25 more with this compound interest than he would have been paid if he were paid only 10% simple interest (would have been only $550). The lesson? Compound interest always pays more!

Let’s look at another similar type of problem that involves interest.

Money invested at x%, compounded annually, triples in value in approximately every 112/x years. If $2500 is invested at a rate of 8%, compounded annually, what will be its approximate worth in 28 years?

A. $3,750

B. $5,600

C. $8,100

D. $15,000

E. $22,500

At first glance, this one seems pretty tricky because you are given x% as the interest rate and it asks you about compounding and it might seem difficult where to find a starting point for this. For this one, it might be a bit easier to think about this without the use of compound interest, which might unnecessarily confuse you. Here, we are given x% as 8%, so all we need to do is take 112/8 = 14. Thus, we know that the money triples in value every 14 years. Further, we know that the money will triple exactly twice in 28 years, once in 14 years and one more time at the 28^th year. So first we need to multiply the original $2500 invested by 3 to get the balance at the end of year 14 (because it triples), to get $7,500 (or $2,500*3). Now, we know that this balance of $7,500 will triple again, so the final balance at the end of the next 14 year period will be $22,500 (or $7,500*3). The correct answer choice is E.

Overall, the three types of interest problems you will most likely encounter come test day will be simple interest, compound interest, and word problems involving the mention of interest, but that can be solved without the application of interest or compound interest methods. The key to deciphering between compound interest and simple interest is to see how many periods the interest is paid….interest paid in one period is simple interest and interest “paid on interest” in multiple periods is compound interest. Finally, remember that some questions can be solved intuitively.

Check out Grockit for more GMAT quantitative practice!

Grockit Live Online Courses Are Here

brian buser — Tue, 09 Mar 2010 18:40:39 +0000

We’re excited to announce that Grockit now offers personalized courses to students preparing for the SAT, ACT, GRE, and GMAT. Grockit courses offer a set schedule of live lessons with the best instructors in the country. Students pick the instructor and schedule that works best for them. Students and instructors are connected online and also through audio conferencing.

Each course lesson is a combination of slide based lecture and group practice questions. The curriculum adapts to each students’ strengths and weaknesses as they progress through the course material. Students also get 24/7 access to Grockit Group Study where they can work with or compete against students all over the world who preparing for the same exam. We’re so confident about the effectiveness of our courses that we’re offering a money back guarantee on score improvement.

Please select an exam to see how that Grockit courses compare to others in the industry:

SAT
ACT
GRE
GMAT

We look forward to your feedback!

MBA for Career Switching, Part III

chase cairncross — Mon, 08 Mar 2010 16:27:02 +0000

The internship. After a couple months of developing a solid network of industry alums, recruiting season will kick off. If you are going for the big three banking, consulting or CPG this will be a very structured process and take place in January to February generally.

The internship is very important for a couple reasons. First of all it will be nice to make some money again after taking almost a year off, and secondly it will be your first chance to establish yourself in your new industry of choice. At this point classes need to go on the back burner, but you need to be honest with your study group about this and make up for it later when they are going through the same process.

Once again you need to be well prepared for these interviews. You need to do plenty of background research on the company, as well as be well versed in the details of your resume. Be prepared to answer questions on the size, profit margin and revenue on past companies. If you do not know it, have a reason in hand as to why you cannot provide that answer. If you are going into banking for example, you need to have clear logic as to why, other than the huge paychecks, you want to spend 20 plus hours a day working in excel. I do not have the answer for you, so you need to come up with that one on your own.

If you are trying to do something that is outside of the common business school internships (my congrats first of all), this is when all of those contacts you developed will really come in handy. Hopefully at this point you have cultivated at least a handful of good contacts from the many informational interviews you have had. These good contacts are the people you should be corresponding with periodically to let them know you are ramping up efforts to find an internship and would appreciate if they would keep their eyes open for an opportunity. If the career you want is really hard to get into and you have limited relevant experience, you might also want to take a more aggressive step of trying to set up an academic internship. These internships are easier to sell to companies and a great way to get your foot in the door, plus your resume does not need to state “unpaid” on it.

In the mean time continue to go to industry event nights, which should be hosted by the clubs you are in. At this point you should be starting to narrow down your career path, which should enable you to drop some of the clubs and focus your efforts on the others.

If your career path is really a difficult industry to get into you might need to continue your academic internship through the summer and hope that they take some pity on you and pay you for your efforts.

See my other articles in this series:

How is the GMAT Scored?

vivian kerr — Thu, 04 Mar 2010 21:40:12 +0000

An official GMAT score report consists of four parts: a Verbal Scaled Score (on a scale from 0 to 60), a Quantitative Scaled Score (on a scale from 0 to 60), a Total Scaled Score (on a scale from 200 to 800) and an Analytical Writing Assessment Score (on a scale from 0 to 6). For each of these four scores, you will receive a percentile rank. Each rank shows the percentage of test-takers who scored below you based on the scores for the most recent three-year period. The percentile rankings change from year to year but your scaled score is fixed. To see how the score report looks, you can download a sample score report at www.mba.org.

The GMAT scores the multiple choice and the writing sections differently. There are a total of 78 multiple choice questions: 41 in the verbal section and 37 in the quantitative section. To compute the scaled score for each section, the GMAT uses an algorithm that takes into account the total number of questions answered, the number of questions answered correctly, and the level of difficulty of the questions answered.

The AWA score is an average of the scores given to both essays. Each essay is given two independent ratings. According to the official GMAT website, one of which may be determined by an automated essay-scoring engine. If the two ratings differ by more than one point, another evaluation is required to determine the final score. Once both essays have been scored, the four scores are averaged to provide the overall score. If for any reason you believe your AWA score is inaccurate, you may request that your essays be rescored using the Essay Rescore Request Form.

When you schedule your GMAT appointment you will be asked to indicate if you wish to access your Official Score Report online or in the mail. Unofficial scores for the multiple-choice section are available immediately after the test. You will receive your official scores within 20 days of testing. It is usually faster to receive them online and if you opt to do so keep your authorization number from your unofficial score report. You will need this number to access your online score report.

As for the multiple-choice sections, at the beginning of each section the computer will present a question in the middle range of difficulty. If the question is answered correctly, the next question will be harder and the score will adjust upwards. If the question is answered incorrectly, the next question will be easier and the score will adjust downwards. The computer is constantly recalculating the scaled score as the student progresses through the section to determine the precise ability of the test-taker.

This is why questions at the beginning of a section count much more than questions at the end of a section. While the total scaled score ranges from 200 to 800, approximately two-thirds of test takers score between 400 and 600. The most recent percentiles can be found in the chart to the left.

Rectangular Solids and Cylinders

amanda chen — Mon, 01 Mar 2010 16:35:58 +0000

Questions involving rectangular solids, particularly data sufficiency questions, test whether you understand the concept of volume and surface area. You essentially need to remember that you need three different values to find volume and surface area (the length, the width and the height). If the prompt and statements 1 and 2 are lacking some these values or some way to find them, neither of the statements will be sufficient.

A rectangular solid is formed by 3 pairs of similar rectangular faces. In other words, 6 rectangular faces in total.

The formulas you need to remember for a rectangular solid are

Volume = Length (l) x Width (w) x Height (h)

Surface Area = (2 x Length x Width) + (2 x Length x Height) + (2 x Width x Height)

If length = width = height, that means that the rectangular solid is, in fact, a cube.

Other vocabulary that might be important is the terms vertex and edge. A vertex is a mathematical way of referring to the corner of any figure. The rectangular solid above has 8 vertices (plural of vertex), can you identify them? The edge is simply the lines you see in the diagram above: the line where two surfaces meet.

Questions involving cylinders are similar and perhaps easier because there are only two values you need to know to solve cylinder-problems – the radius (r) and the height (h).

If you don’t know the radius, anything that enables you to determine the radius, such as the diameter (radius = diameter / 2) or the circumference (radius = circumference / 2pi) will suffice.

Regarding cylinders, the formulas you need to know are

Volume = area of the base circle x height = pi x (radius)² x height

Surface Area = (2 x pi x (radius)² )+ (pi x (diameter) x height)

Let’s try a problem: A cylindrical water tank has a stripe painted around its circumference, as shown in the figure provided. What is the surface area of this stripe?
(1) y = 0.7
(2) The height of the tank is 2 meters.

To find the surface area of the stripe, you need to know the circumference of the cylinder, but there is not data in the question that gives you the radius or diameter to let you find the circumference. Hence the answer should be that neither statement together is sufficient.

Weighted Averages On The GMAT

jake becker — Fri, 26 Feb 2010 17:29:24 +0000

This post will introduce weighted average questions you’ll see on the GMAT. There is one main formula you need to solve simple GMAT Average questions:

Average = SUM / # of observations

Note that this can be rearranged to read:

SUM = Average x (# of obs)
# obs = SUM / Average

So, if you are given ANY 2 of the 3 values, you should be able to find the 3rd. For example:

John drinks an average of 1.5 cups of water/day. After how many days has he drank 3 gallons of water? (1 gallon = 16 cups.)

In this case, we are looking for the number of days (or observations) such that we total 48 cups (3 gallons) of water.

# = SUM / Average

# days = 48 cups / 1.5 cups/day
# days = 32 days

NEVER AVERAGE AVERAGES!

Class A has 15 students and an average height of 60”. Class B has 20 students. What is class B’s average height if the average height of both classes is 65”.

One might say: (A + B) / 2 = 65”; A = 60”; so B must be 70”. However, keep in mind:

TOTAL AVERAGE = TOTAL SUM / TOTAL OBS

CLASS A + CLASS B = BOTH

15 students + 20 students = 35 students
60” average + 68.75” average = 65” average
900” total in A + 1375” total in B = 2275” total in Both

The given information is in black. The necessary intermediate steps are in blue, and the red is your answer. Note that the average of the averages ≠ total average. We must calculate each average separately, and to do this we need the SUM and # of observations for each category. This brings us to the idea of WEIGHTED AVERAGES.

A WEIGHTED AVERAGE is needed when you are taking average of a large group in which there are subgroups with a different number of observations in each. Take a look at this generalized formula, assuming there are 3 groups A, B and C.

(Average of A x Obs in A) + (Average of B x Obs in B) + (Average of C x Obs in C)

(Obs in A) + (Obs in B) + (Obs in C)

Think of weighted averages like a tug of war between numbers. The “stronger” one side (dog) is, the more that weighted average (tennis ball) will be “pulled” in that direction.

In the previous question, we had:

CLASS A + CLASS B = BOTH

15 students + 20 students = 35 students
60” average + 68.75” average = 65” weighted average

Note that the weighted average is CLOSER to B’s average than it is to A’s. This is because there are 20 students in Class B compared to only 15 students in Class A.

Two More Examples

At a certain restaurant, the average (arithmetic mean) number of customers served for the past x days was 75. If the restaurant serves 120 customers today, raising the average to 90 customers per day, what is the value of x?

A. 2

B. 5

C. 9

D. 15

E. 30

WITHOUT using the formula, we can see that today the restaurant served 30 customers above the average. The total amount ABOVE the average must equal total amount BELOW the average. This additional 30 customers must offset the “deficit” below the average of 90 created on the x days the restaurant served only 75 customers per day.

30/15 = 2 days. Choice (A).

WITH the formula, we can set up the following:

90 = (75x + 120)/(x + 1)
90x + 90 = 75x + 120
15x = 30

x = 2 Answer Choice (A)

Use whichever makes more sense to you!

Anita spent a total of $780 on 52 bottles of wine for her wedding. She then decided to buy 8 bottles of sparkling wine for the toasts, as well. Was the average (arithmetic mean) price per bottle of wine less than $20?

(1) Each bottle of sparkling wine cost more than $15.

(2) Each bottle of sparkling wine cost less than $40.

Take another look at what exactly the question is calling for: the TOTAL average price of all the wine at the wedding. We should look at the suggested average ($20) and use that as our threshold amount.

60 bottles * $20/bottle = $1200 total
$1200 total – $780 (given) = $420 (left for sparkling wine)
$420 / 8 bottles = $52.50/bottle of sparkling wine (for the total average to equal $20)

Which of the answer choices are conclusively above or below $52.50/bottle of sparkling wine? Only (2). With (1), we can be below OR above the threshold, so (1) is not sufficient.

Answer Choice (B)

Now, you’re score will be above average! Please visit the Grockit forum or leave a comment here if you have more questions on weighted averages.

Good luck!