A school administrator will assign each student in a group of N GMAT Explanation

A school administrator will assign each student in a group of N students to one of M classrooms.        If 3 < M < 13 < N, is it possible to assign each of the N students to one of the M classrooms so that each classroom has the same number of students assigned to it?

(1) It is possible to assign each of 3N students to one of M classrooms so that each classroom has the same number of students assigned to it.

(2) It is possible to assign each of 13N students to one of M classrooms so that each classroom has the same number of students assigned to it.

This is a super tricky GMAT number properties/divisibility question from the GMAT official guide. Oddly we don't see this one a lot in GMAT tutoring sessions but maybe we can credit the video explanation for that. As always let's start by reading carefully and figuring out the question: is it possible to assign each of the N students to one of the M classrooms so that each classroom has the same number of students assigned to it? OK, we know that if it has a question mark at the end it's a question BUT what does that actually mean? Especially on Data Sufficiency you really need to do a step further. In this case the question is, does M divide evenly into N. Or, is N divisible by M. Or is N/M an integer. However way you want to look at it. The point is: dig into Data Sufficiency information and try to make an inference, organize, or transform in some way. Dig!!!

OK, so now we've got the question, and if we had the values for M and N it would be very simple. Of course, we don't have the values but in the statements we'll be given constraints which will potentially narrow things down to sufficiency. We do have a constraint in the questions stem though: M is between 3 and 13 and N is greater than 13. For M that's quite a narrow range so I might just write values out: 4, 5, 6, 7, 8, 9, 10, 11, 12. Boom! Easy organization.

Let's get to the statements:

(1) It is possible to assign each of 3N students to one of M classrooms so that each classroom has the same number of students assigned to it.

What does that mean??? 3N/M is an integer. How does that relate to N/M. Let's try to do a Yes/NO.

M = 4, 5, 6, 7, 8, 9, 10, 11, 12  N > 13

Yes

Pick a value for M and then pick something for N in the numerator that will cancel out that M (pick an N that's divisible by the M).

M = 4

N = 40

Make sure you're keeping N and M constraints in mind.

3*40/4 = integer

40/4 = integer

 

No

For the no we'll do something similar but this time pick an N not divisible by the M. Let's pick an M that will be at least partially cancelled out by the 3. So 6 would be a good M. And then just pick an even N (to cancel the 2 in 6) not divisible by 6. So N could be 14.

3*14/6 = integer

14/6 is not equal to an integer.

For statement one we have a yes and a no so it's insufficient.

(2) It is possible to assign each of 13N students to one of M classrooms so that each classroom has the same number of students assigned to it.

M = 4, 5, 6, 7, 8, 9, 10, 11, 12  N > 13

Statement two looks very similar BUT just because statements look similar doesn't mean they''ll have the same outcome. The details matter. Still, we can use the same procedure setting up a yes/no to test sufficiency.

Yes

So 13N/M = integer

So M could be 7 and N 14.

13*14/7 = integer

No

Given the constraint this is impossible. 13/M will never be an integer because M can only be 4, 5, 6, 7, 8, 9, 10, 11, 12. Eh? So what! Well, if M can't be cancelled out by the 13 then how is it going to disappear from the denominator to leave an integer? The N has fully cancel the M. Aha! So, given this constraint N/M must be an integer. Statement two is sufficient.

B.

A school administrator will assign each student in a group of N students to one of M classrooms GMAT Explanation

Additional GMAT Data Sufficiency Divisibility Practice Questions

GMAT Data Sufficiency Divisibility Question of the Day 1

GMAT Data Sufficiency Divisibility Question of the Day 2

GMAT Data Sufficiency Divisibility Question of the Day 3

 

At a certain fruit stand, the price of each apple is 40 cents - GMAT Explanation

At a certain fruit stand, the price of each apple is 40 cents and the price of each orange is 60 cents. Mary selects a total of 10 apples and oranges from the fruit stand, and the average (arithmetic mean) price of the 10 pieces of fruit is 56 cents. How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is 52 cents?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Here's another word problem from the GMAT official guide (Video solution below). It's a basic weighted average/system of equations but often people get tripped up defining the equations or stall mid-way because it all feels too complicated. Let's just take it one step at a time!

First step: read carefully. Don't rush. Don't skim. Don't trail off at the end of sentences. If something doesn't make sense re-read it. If you think about it for a little and it still doesn't click you can keep reading to see if more context helps to shed some light. If the lightbulb still isn't going off that's OK, once you start getting things organized it's possible that the puzzle pieces will fall into place.

So now define the question: How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is 52 cents?

We're not solving for the number of oranges but the number that Mary must put back in order to decrease the average price to 52 cents. So we're changing the number of oranges to manipulate the price. OK, so now let's get our system of equations defined. Start with whatever is easiest for you.

Mary selects a total of 10 apples and oranges from the fruit stand (this looks easiest to me)

A + O = 10

The average (arithmetic mean) price of the 10 pieces of fruit is 56 cents

(40A + 60O)/10 = 56 We can simplify the fraction. 4A + 6O = 56

Now you have a basic system of equations. You can solve via substitution or by adding/subtracting equations. Usually adding/subtracting is easier/faster than substitution. We're going to subtract:

4A + 6O = 56

4A + 4O = 40

2O = 16

O = 8

We have 8 oranges and 2 apples. So we need to set up an equation for a weighted average that equals 52. We're going to define the oranges Mary must put back as X. That has to be subtracted from the oranges in the numerator AND the total in denominator. That's a major pain point in this question, forgetting the remove to oranges from the total on the bottom.

(2*40 + (8 - X)60)/(10 - X) = 52

Now we solve for X.

80 + 480 - 60X = 520 - 52X

40 = 8X

X = 5

For the average price of the pieces of fruit that she keeps to be 52 cents Mary must put 5 oranges back.

We can also error check that. (2*40 + 3*60)5 = 260/5 = 52

At a certain fruit stand, the price of each apple is 40 cents and the price of each orange is 60 cents. GMAT Explanation

I solved it slightly differently in the video. I prefer the solution here but the video is still a good reference.

 

Additional Weighted Average Practice

Here's another weighted average question from GMAT Question of the Day

 

Alex deposited x dollars into a new account that earned 8 percent annual interest GMAT Explanation

Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually. One year later Alex deposited an additional x dollars into the account. If there were no other transactions and if the account contained w dollars at the end of two years, which of the following expresses x in terms of w?

A. w/(1+1.08)

B. w/(1.08+1.16)

C. w/(1.16+1.24)

D. w/(1.08+1.08^2)

E. w/(1.08^2+1.08^2)

We need to get x in terms of w. So, what does that mean? This is important because it comes up often on the GMAT. x in term of w or anything in terms of anything else means: x = w (multiplied by, divided by, added to, subtracted from something). So x on one side by itself. w on the other side with some sort of operation which, of course because it's an equation, makes w equal to x. Make sure you understand that! OK, on with Alex and his dollars. Video solution and diagram below!

Take your time reading and understanding the question. When this one comes up in tutoring sessions the problem is often that the student has misread something. In this case Alex's x dollars are growing at 8 percent annual interest. How do you grow something at 8%? Well, add 1 to the decimal equivalent of the percent and multiply that by the money. So, 8 percent = .08. Add 1, 1.08. Multiply by x = 1.08x. So that's after our first year, we have 1.08x. Then we an additional x dollars. OK. so 1.08x + x is our total after year one. And then in the second year that amount grew by an additional 8 percent so multiply the whole expression by 1.08.

1.08(1.08x + x)

1.08^2x +1.08x = w

To get x in terms of w we need to get x by itself. So factor it out.

x(1.08^2 + 1.08) = w

Now just divide by (1.08^2 + 1.08)

x = w/(1.08^2 + 1.08)

D.

Alex deposited x dollars into a new account that earned 8 percent annual interest, compounded annually GMAT explanation

Additional related GMAT Practice

Here is another word problem with some algebra/arithmetic follow through from the GMAT prep tests: When a certain tree was first planted

 

Seven pieces of rope have an average (arithmetic mean) length of 68 centimeters and a median length of 84 centimeters. If the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope, what is the maximum possible length, in centimeters, of the longest piece of rope?

(A) 82
(B) 118
(C) 120
(D) 134
(E) 152

Here's another tough one that we see a lot in GMAT tutoring sessions. Video solution below! The biggest mistake I see on this one: creating 7 variables for the 7 pieces of rope. If you're ever working a GMAT question and feel the need to create 7 variable you might want to re-think your approach. I don't think I've ever seen a question that required more than 5 variables. I'm sure there's some exception out there but generally, again, be careful about creating an army of variables.

Instead of variables I'd plot out seven slots. Median is a spacial measurement so getting the numbers in a row is important. Then go ahead and pop the median, 84, in the middle slot. We know that the seven pieces of rope have an average length of 68 so the sum of the 7 divided pieces by 7 is 68. You can go ahead and divide the slots by 7 and set them equal to the average, 68.

Now you can also pick a variable for either the biggest or smallest piece of rope. I'd define the smallest as "S" and then derive the biggest as 4S + 14. Great. Now we've got 3 of 7 slots filled in. What to do with the rest? Well, we're looking the maximize the length of the biggest piece of rope. So what does that mean in terms of the rest of the rope? Minimize it!

Great - let's make the rest of the pieces 0!!! Wait a second. Can you do that? Especially if the median is 84? Nope! We've got some constraints. Let's use them. So, again, what's the smallest piece of rope? S! Yes. OK. So to the left of the median what's the smallest we can make those pieces? 0! Hmmm. What's the smallest piece of rope again? S! Fantastic. OK. So keeping in mind that the absolute smallest piece of rope is S what is the smallest we can make those two other pieces to the left of the median? S! YESSSS.

OK so now we need to figure out how small we can make the pieces of rope to the right of the median. So, how small can they be? S! Hmmm. Think constraints. Are there any constraints? Any difference being positioned to the right of the median? Oh. 84? Right! Why? Because to the right of the median must be equal to or greater than the median. Perfect. OK.

So now we have an equation with one variable and can solve. But before we do that let's give the equation a quick look to see if we can simplify anything. I'd group like terms. Once you do that you might see a bunch of things divisible by 7. That makes the arithmetic easier. Solve for S and then plug that into 4S + 14 and you end up with the maximum possible length 134. D.

These tough GMAT max/min questions do come up. Here's another challenging GMAT max/min question and solution for extra practice.

 

 

In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x?

In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x?

(A) 40
(B) 36
(C) 34
(0) 32
(E) 3

This is a GMAT geometry question from the GMAT official guide for quantitative review. We get this one a lot in GMAT tutoring sessions so went ahead and made an in depth explanation. Video solution below.

This is a good one to review because it brings up an important GMAT geometry rule: the exterior angle theorem! Eh? Sounds complicated but it's actually pretty simple. Angle ACB above is an "exterior angle" of triangle AOC. The exterior angle theorem states that the exterior angle ACB equals the sum of the "remote interior angles" OAC and AOC. Basically, the exterior angle equals the sum of the angles that it's not connected to, the other two angles of the triangle known as the "remote interior angles".

Always watch out for this rule if you're dealing with triangles that share sides because: if the triangles share sides it's not unlikely that you'll have an exterior angle and will be able to use (and will probably need to use) the exterior angle theorem.

For this question we don't have much to define. We need to find the value of x.

First step in this:

  1. Draw a diagram and make easy inferences. The first one to label is that OC = AC = AB.
  2. Once you have those equal sides you can also label angles that are equal as sides that are equal within the same triangle have angles opposite them that are also equal.
  3. Now you should be thinking, hmmm, triangles sharing sides maybe there's an exterior angle. Yes, of course there is! Angle ACB. ACB is equal to AOC + OAC.
  4. Now you should also be wondering why this triangle is in a circle. How can the circle help you make an inference? For GMAT circle questions always think about "helpful radii". There's usually a radius whether it's drawn or you have to draw it that will help. In this case OA and OB are both radii so they are equal boom!
  5. That's it. You should have all angles defined by x. Go ahead and sum all of the x's  and set that equal to 180 to solve.

In the figure above, point O is the center of the circle and OC = AC = AB. What is the value of x? Solution

GMAT ARTICLES

CONTACT


Atlantic GMAT Tutoring

405 East 51st St.

NY, NY 10022

(347) 669-3545

info@AtlanticGMAT.com