# Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m. By what percent did the ratio of price per share to earnings per share increase, in terms of k and m? GMAT Explanation, Video Solution, and More Practice!

Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m. By what percent did the ratio of price per share to earnings per share increase, in terms of k and m?

A. k/m
B. (k–m)
C. 100(k–m)/(100+k)
D. 100(k–m)/(100+m)
E. 100(k–m)/(100+k+m)

This is a very challenging percent change word problem with a ton of variables from the GMAT Official Guide. A few things to remember when working on word problems:

2. Define the question before starting calculations (use the nouns to define the question you don't have to fill in the numbers)
3. Solve thoughtfully. Avoid simply burrowing in. Zoom in/Zoom out. Pause. Think.

Since there are variables in the answer choices this is a good candidate for: picking numbers.

Now, let's also remember the practical picking numbers test. If the question would be easier if you simply had the numbers then go ahead and pick them! This question would just be a percent change. So I'd say that is a clear signal to pick numbers.

Alright, we're getting a little ahead of ourselves. Let's go ahead and define the question: By what percent did the ratio of price per share to earnings per share increase, in terms of k and m?

That's it, no more no less. In tutoring sessions students often struggle getting this question defined. The question is always just that last line and doesn't need interpretation. If you're having trouble defining it just read the last line to yourself word by word verbatim. Often, if you're confused it's because you missed a word or added a word or two. Avoid paraphrasing here.

What gets missed on this one? That you're comparing RATIOS. It's the percent change of the RATIO of price per share to earnings per share. It's not Price compared to Earnings. It's P/E last year compared to P/E now.

So let's put that information into the percent change formula:

(((New Price Per Share/New Earnings Per Share)/(Old Price Per Share/Old Earnings Per Share)) - 1)100. The general formula is ((New/Old) - 1)100

We have a bunch of variables. The New/Old price and earnings and k and m. Are there any constraints? k > m. But that's it. So you can pretty much pick whatever numbers you'd like.

Any thoughts on where to start? I'd start by picking the old price and earnings then k and m which we'll use to derive the new price/earnings.

Because we're dealing with percents let's use 100 because that makes things easy.

Can you make earning and price the same? Yes! Why not?

Old

Price 100

Earnings 100

Now let's pick an easy k and m. I'd go for 10 and 5. or 20 and 10. Something simple. Let's do 10 and 5.

New

Price 110

Earnings 105

Now let's put those numbers into our percent change formula.

((110/105)/(100)(100) - 1)

Notice that by making old price and earnings the same the denominator cancels. So after cancelling the denominator and reducing 110/105 we're left with:

(22/21 - 1 )100 = (22/21 - 21/21)100 = 100/21

So that's our percent change, 100/21. Now we need to do what we always do picking numbers: plug the numbers into the answer choices to yield our answer, 100/21.

If you follow our blog and have read our other GMAT explanations you probably know that we have a shortcut for this. We're going to use divisibility to avoid some of the calculations.

Let's look at the denominator of our answer: 21. So we know that the correct answer must have a denominator that is a multiple of 21. I'd break down 21 to primes, 7 and 3. I'd focus on three because it has easy divisibility rules. So we also know that the denominator must be a multiple of 3. So let's start eliminating answer choices based on that.

A. k/m

10/5 = 2. Not Div by 3.

B. (k–m)

10-5 = 5. Not Div by 3.

C. 100(k–m)/(100+k)

110. Not Div by 3.

D. 100(k–m)/(100+m)

105 is Div by 3 so this is possible.

E. 100(k–m)/(100+k+m)

115 is not Div by 3. ## More Challenging GMAT Word Problem Practice Questions

Here's an almost identical question from the GMAT Official Guide: Last Sunday a certain store sold copies of Newspaper A

Here's a work and rate example that's a little different but you try picking numbers and using the divisibility trick for eliminating answers: During a trip, Francine traveled x percent of the total distance

# If Q is an odd number and the median of Q consecutive integers is 120, what is the largest of these integers? GMAT Explanation

If Q is an odd number and the median of Q consecutive integers is 120, what is the largest of these integers?

(A) (Q - 1)/2 + 120
(B) Q/2 + 119
(C) Q/2 + 120
(D) (Q + 119)/2
(E) (Q + 120)/2

To me this screams: pick a value for Q. Just make sure that you follow any rules outlined in the question. Always take your time to get things straight. That doesn't mean sit and stare. Or obsessively re-read things. That means find the pace and level of detail that allows you to comprehend the question. Pause. Think. Digest.

## Let's get to work!

So the median is 120, Q is an odd number, and there are Q consecutive integers. We're looking for: the largest of these integers.

If 120 is the middle number (median) let's just add one above and once below: 119 120 121

So, since Q represents the number of numbers, Q is 3. That value follows the rule: Q is odd.

## Plug back into the answer choices and look out for shortcuts (choices that clearly won't match 121)

Now what? Now we do what we always do picking values and plug back into the answer choices. We need to sub in 3 for Q and output 121 (the largest of these integers).

You can rule out some of these without doing much since you can see fractions that clearly won't work (B, C, E).

Then work out A and D.

D you can also eyeball. It's way too small to equal 121.

It has to be A.

## Additional GMAT Statistics Example Questions!

Here's a GMAT statistics puzzle (median/range) which will teach you how to work with a while bunch of unknowns. Also a good one to work on maximizing a value.

A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer that could be in this set?

Here's a top shelf statistics, max/min question which we tackle in just about every GMAT preparation. it's a great question to work on organizing a ton of variables. Also, it will help you understand how deal with a max/min scenario.

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?

Bread and butter GMAT statistics question. Not the most difficult but still very important. Be ready to interpret a chart/graph on test day. What isn't challenging in practice can quickly become a swirling mess on GMAT day.

The table above shows the distribution of test scores for a group of management trainees. Which score interval contains the median of the 73 scores?

Great one to wrap your head around statistics concepts and how they can be tested on the GMAT. It's a PS (problem solving) question but having this square will also help you on DS statistics questions.

Last month 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was \$150,000 and the median sale price was \$130,000. Which of the following statements must be true?

Moderately challenging DS statistics range question from GMAT question of the day

And here's a very challenging GMAT statistics range question from GMAT question of the day that requires algebra and number properties skills.

Good luck and happy studies!

# Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails? GMAT Explanation, Video Solution, and Additional Practice Questions.

Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?

(A) x/(x+y)
(B) y/(x+y)
(C) xy/(x+y)
(D) xy/(x-y)
(E) xy/(y-x)

Working simultaneously at their respective constant rates. Ring a bell? It should. If machines/people/whatever are working simultaneously it is almost certain you are dealing with a cooperative rate question.

Ok, let's define the question: How many hours does it take Machine B, working alone at its constant rate, to produce 800 nails? So we want Machine B's hours.

Are there any complicating factors here? Well, sort of. You have a bunch of variables to deal with. Machine A's and total rate are defined by y and respectively. Don't panic. That's OK. Treat variables as you would treat numbers. They behave the same way in most circumstances.

Let's start solving this with the variables and then, because we have variables in the answers and it seems like it could be a good strategy here, let's try picking numbers.

Set up three T's. One for each machine and one for the machines working together. Fill in the 800 nails for all three T's and add the x and y hours into the time slots for the cooperative rate T and for Machine A's T. How can we derive Machine B's hours? You have the total rate/coop rate and A's rate. A's rate + B's rate = Total Rate/Coop Rate. So Total - A = B. Bingo! Subtract the part (A) from the total (A + B) to get the other part (B). Once we have B's rate we can solve for B's time (since we already have the job/work, 800 nails. All of the computation is in the diagram below. The other way to do this and my preferred solution is to pick numbers. Again, how do you decide whether it's worth picking numbers? Ask yourself: If I had the numbers (instead of the variables) would solving for Machine B's hours be easy? The answer here is a resounding YES! Why? Well, you would just have to plug the numbers into the rate T's and voila, you'd be done. OK - great!

So what numbers to pick... We need to define x, y, and B's hours keeping in mind that the hours it takes each machine to do the job is directly related to the hours it takes them to do the job together. To make life very easy I would make the time each machine takes to produce 800 nails the same. The numbers you pick don't really matter but choose integers and keep them on the smaller side because you'll plugging back into the answer choices.

What if y is 2. Machine B hours also 2. What is x (hours working together)? If they take the same time then they have the same rate. Meaning if you have two of them the rate is doubled and the time is halved. So x (total time with machines working together) is 1. That's it. Plug in 1 and 2 for x and y respectively and the correct answer will yield 1. It's that simple. Basically no calculations involved. ## Additional GMAT Cooperative Rate and Picking Numbers Practice Questions

This is a slightly oddball work and rate question from the GMAT Official Guide that feels very different than the above and is tougher but, in my mind, tests a lot of the same skills.

Here's another work and rate question but it's an average rate. Though you'll still use rate T's the setup is a bit different. That said, it's a good one to review for picking numbers practice: During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour

And a very typical but tough cooperative rate question from the GMAT Prep Tests. These come up so important to know the style:

Mary passed a certain gas station on a highway while traveling west at a constant speed of 50 miles per hour. Then, 15 minutes later, Paul passed the same gas station while traveling west at a constant speed of 60 miles per hour. If both drivers maintained their speeds and both remained on the highway for at least 2 hours, how long after he passed the gas station did Paul catch up with Mary?

This is a cooperative rate from GMAT question of the day. You can practice your rate T's and sharpen your understanding of cooperative rate.

And another cooperative rate example from question of the day. This one uses proportions instead of real numbers so gives you another opportunity to practice picking numbers.

This GMAT question of the day is a bit more of a work/rate puzzle but the basic components are the same. Though it seems a bit non-standard you do see this variation a bunch so it's important to practice.

And just for some balance here's a Data Sufficiency Cooperative Rate question.

# During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip? GMAT Explanation, Video Solution, and More Practice!

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip?

A. (180−x)/2

B. (x+60)/4

C. (300−x)/5

D. 600/(115−x)

E. 12,000/(x+200)

Average speed, distance, miles per hour... Yes, it's a GMAT work and rate question. In this case we have an average rate with variables in the question and in the answer choices. That's a signal for considering picking numbers. Why pick numbers? To make life easier. How do I judge if picking numbers will make life easier? If the question would be a simple 1-2-3 if you had the actual numbers that's a pretty good indicator. Here, if you just had to calculate the average speed, assuming you know how to do a weighted average, things would be smooth. Here's another GMAT word problem from the GMAT Official Guide on which you can pick numbers to make things much easier.

Let's get back to Francine and her trip! Let's define the question: what was Francine's average speed for the entire trip?

Now let's take inventory. What numbers are we missing? Distance! Yes, I'd go ahead and pick a distance for the trip. Pick something that works well with the speeds, 40 and 60. Why not go for 240 total distance and divide that into 120 and 120 for each leg of the trip? That way you've got easy division.

120 miles/40mph = 3 hours

120 miles/60mph = 2 hours

Total time = 5 hours

Total distance = 240 miles

240/5 = 48mph

Just as a sanity check for the 48. We spent more time (3 hours) traveling at the slower speed (40mph) so it makes sense that the average speed would be closer to 40mph than 60mph.

OK, so, with total distance at 240 the average speed is 48. Of course, that's not an answer. We have to plug in "x", the total distance at an average speed of 40 miles per hour, in order to yield 48. We can use our divisibility shortcut here. Since we know the correct answer is 48 we know that the numerator has to be a multiple of 3 (because 48 is a multiple of 3). So just focus on the numerator and eliminate answer choices not divisible by 3.

A. (180−x)/2.  180-50 = 130

B. (x+60)/4.  50 + 60 = 110

C. (300−x)/5.  300-50 = 250

D. 600/(115−x) Div by 3

E. 12,000/(x+200) Div by 3

You end up with D and E as possibilities. At this point you can just work out the entire expression for both choices. You could also:

1. Continue using divisibility. 48 has four 2's in it. 600 only has 3. So that's out.
2. Use Magnitude. It should be pretty clear that D. is way too small.

So, in terms of x, what was Francine's average speed for the entire trip? E. 12,000/(x+200)

I think picking numbers is a great way to go on this one. You can also do the algebra which could be a little faster depending on how speedy you are at setting it up but, at least in my mind, is less straight forward and I've seen many GMAT tutoring students screw it up. I'll do the algebra in the diagram and in the video for your reference.   Video Solution: During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip?

## Additional GMAT Work and Rate/Weighted Average Practice

Here's a work and rate question from GMAT Question of the day which has the same basic structure and explains the rate t's a bit more.

Here's a very challenging average rate question from the GMAT Prep Tests. Same basic premise but on this one the follow through is tougher because it involves a quadratic.

GMAT Work and Rate question from the Official Guide to practice picking numbers. It's a cooperative rate so the follow through is a bit different than the above "Francine" average rate but you can still use rate T's and again pick numbers: Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours

# Last Sunday a certain store sold copies of Newspaper A for \$1.00 each and copies of Newspaper B for \$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p? GMAT Explanation, Video Solution, and additional practice questions!

Last Sunday a certain store sold copies of Newspaper A for \$1.00 each and copies of Newspaper B for \$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?

A. 100p/(125–p)

B. 150p/(250–p)

C. 300p/(375–p)

D. 400p/(500–p)

E. 500p/(625–p)

OK, so this Newspaper Store question is a big one. We see this all the time in the course of GMAT tutoring. It's dense. No matter how you slice it's a bit time consuming. Understand it. Yes you should. Solve it on your GMAT? That's a maybe.

That does't mean run and hide whenever you see a dense word problem. Still, dense + variables in answer choices might be an indicator that you could be in for the long haul even if you ultimately succeed. Again: long word problem does not equal auto skip. Here's a wordy GMAT remainder/ divisibility question from the GMAT official guide that while considered tough can be solved pretty easily provided that you have a decent method.

Back to last Sunday at the newspaper store! First step: read carefully. Don't skim. Digest as you read BUT try not to solve as you read. We need to get the setup before we follow through.

Now let's go ahead and define the question. Again, let's not calculate or worry about the numbers but focus on the story (the nouns).

Here's the question: which of the following expresses r in terms of p?

Not necessarily a straightforward thing to define in a useful way. In this case I'd start by defining r and p separately and then see how we can connect them. You can start with either r or p but generally aim to start easy. I'd say p is slightly simpler so let's start there.

p = the percent of the newspapers that the store sold were copies of newspaper A

Great. Easy. So let's add some variables. #A and #B. So P = (#A/(#A + #B))*100. Why are we multiplying by 100? It's a percent! Without the 100, the fraction is just a decimal. To get the percent you need to multiply by 100. This step can range from unimportant to critical. If picking numbers and plugging back into answer choices multiplying by 100 for percents is very important. If you don't multiply by 100 your result won't match any of the answer choices.

r = the percent of the store’s revenues from newspaper sales was from Newspaper A. OK. Not bad. We don't need any additional variables for this because we have the per unit prices for Newspapers A and B.

r = (#A(1)/(#A(1) + #B(5/4)))*100

Now you could try to equate those and isolate r. r in terms of p just means, how do you transform p to equal r? That said, I'd consider picking numbers. Why? Well, if you have the numbers. Meaning, if you knew the values of r and p would it be easy to equate one with the other? Meaning, if r were 4 and p 2 can you relate those? Sure you could! r = 2p. If the question would be easier if you had the numbers then go ahead and pick some! Clearly you can't always pick numbers. There are generally two scenarios that make this possible:

1. You have variables in the answer choices
2. You have proportions in the answer choices

That doesn't mean if you are in scenario 1 or 2 that you should always pick numbers. It's just that those scenarios tend to make it possible and certainly suggest that you consider picking values.

Ok, so what variables do we have? #A and #B. We're just missing the number sold. Do we have constraints? Not really. Don't pick negative numbers or 0 because at least 1 newspaper was sold. You could pick any positive integer.

I'd stick to smaller numbers and potentially set up your expression so you create easy arithmetic. In this case we the 5/4 from the price of Newspaper B. I'd try to cancel the 4 in the denominator. Why not make #B = 4. And, while we're trying to make the GMAT easy, let's make #A also equal to 4 so we know p, the percent of the newspapers that the store sold were copies of newspaper A, is 50%. No calculation needed.

Now let's do r: (4/(4 + (5/4)*4))*100 = 400/9

Perfect. So r = 400/9. That's what we want to transform p into. Simpler way to put it: plug p (50) into the answer choices and the correct answer will yield 400/9. Now, you could do the arithmetic for all of the answer choices but there is a shortcut if you've picked numbers and are plugging back into the answers. Use divisibility. We know that the correct answer will simplify to 400/9. So I'd look at the denominator, 9. We know that the denominator if the correct answer, even before being simplified, must be a multiple of 9. So let's start by looking only at the denominators and eliminate any choice that doesn't have a denominator that is a multiple of 9.

A. 100p/(125–p). 125-50 = 75

B. 150p/(250–p). 250-50 = 200

C. 300p/(375–p) 375 - 50 = 325

D. 400p/(500–p) 500-50 = 450

E. 500p/(625–p) 625-50 = 575

Just because the denominator is divisible by 9 doesn't mean it is the correct answer. It could be that after using the divisibility trick that you're left with a couple of choices. That's fine. You can then plug in the rest for the ones you've got left. You can also use magnitude to decide between them. Sometimes choices that pass the divisibility test are either way too big or small to be the correct answer.

In terms of the algebra answer choice D means that r = 400p/(500–p).  ## Additional GMAT Word Problem, Percent Change, Picking Numbers, Practice Questions

Here's a very similar tough word problem with a bunch of variables from the GMAT Official Guide: Last year the price per share of Stock X increased by

And another question from the Official Guide that's similar-ish in that you can pick numbers and use the divisibility shortcut for eliminating answer choices. It's also one that tutoring students tend to mis-interpret. During a trip, Francine traveled x percent of the total distance

Here's another very challenging word problem from the GMAT Official Guide: Last year, a certain company began manufacturing product X and sold every unit of product X that it produced. It's a data sufficiency question and neither features percent change nor picking numbers but it has the same level of density and difficulty. It requires a great setup. It's also one that you might skip (even though you should understand 100% how to get it through it in practice).

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