Sucking All the Juice Out of GMAT Quant Problems (Part 2)
Grab your Official Guide as we walk through 3 GMAT Quant problems (Problem Solving), hoping to drink every drop of knowledge from the problem before we say, “Yo, that keg is kicked.”
In part 1, we discussed the process of maxing out the value of the GMAT Quant problems you do.
As you review them, classify your current level of mastery for that problem.
1: Basic exposure, little to no clue what to do
2: Partial clue, knew how to do some stuff (may have even guessed the correct answer), but didn’t know a “real way” to get a definite answer
3: Got the answer correct through a legitimate process, but it felt hard
4: Got it correct and felt totally in control and normal (you can even think of other problems it’s similar to or more than one way to do the problem)
Take an inventory of all the component parts of a problem and assess whether you could be better at any of those parts.
- I know how to simplify/reframe the question being asked (if possible).
- I know any underlying formulas/properties being tested.
- I am aware of all constraints in the problem and know what inferences (if any) could be drawn from them.
- I can make any applicable GMAT vocab translations.
- I know what clues signify the Topic, and what my First Move/First Thought (if any) is for that Topic.
- I can execute the arithmetic involved swiftly, accurately, and easily.
- I can recognize any answer choices that would be savvy to eliminate if I’m guessing.
- I can recognize the potential thinking traps or mechanical missteps involved in this problem.
- For Problem Solving, I can recognize multiple ways of doing this problem (if they exist).
- For Data Sufficiency, I can recognize any easy eliminations we get.
Follow up on what you’ve attempted to learn on GMAT Quant problems by testing yourself later.
For any little blip of content or “recognition → move” we’re trying to memorize, we should create a flashcard (and we should look at flashcards at least five times a week).
For any problem whose overall process felt heavy or halting, we should schedule at least one Redo Appointment between 2-20 days later (the lower our level of mastery was initially, the sooner we’d schedule it).
—–
If you have a copy of the 2018 Official Guide, let’s try a few GMAT Quant problems in Problem Solving and practice modeling what our inventory of each problem might look like. I’m going to randomly pick three without looking: PS 75, 150, 225.
Go give them a try and then compare your breakdown to mine.
PS 75
1. Reframe the question: find the four possible numbers for total, or use some property that a legal answer needs (like “multiple of 16” in this case) to spot an illegal number.
2. Do we know, given a real number for some part of a ratio, how to scale up the rest of the numbers by the same multiplier?
3. No real important constraints here, because the 30 does all the work for us. Ratios usually have an integer constraint, meaning that the real numbers the ratio scales up to usually have to be integers (because they usually represent a quantity like number of people or number of items). This ratio doesn’t have an inherent integer constraint, because it’s fine for hours to scale up to a decimal or fraction.
4. No vocab translations needed here.
5. Seeing the word ratio tells me it’s a Ratio problem, and my First Move/First Thought is, “Write the ratio horizontally, add a column for total and add up the ratio, and then look for a real number to pair up with a ratio number so that we can solve for the multiplier.”
6. I needed to add 2 + 3 + 5 + 6 = 16. I needed to be able to do 30/2, 30/3, 30/5, and 30/6 in order to get multipliers of 15, 10, 6, and 5. Then I needed to be able to do 16*15, 16*10, 16*6, and 16*5 to find the four possible numbers for the total.
7. None of these answers are obvious bad ones.
8. No obvious traps being laid here. Just seems to be a time-sucker in terms of executing four computations.
9. I could definitely work backwards. Say I test (C) and say the total number of hours was 160. That would allow me to compute a multiplier of 10 and know that the four staff members worked 20 hrs, 30 hrs, 50 hrs, and 60 hrs. Since one of them worked 30 hours, this is a valid number for a total and I could eliminate (C) and test another answer.
PS 150
1. J (now) = ?
2. No content needed other than Algebraic Translations.
3. No constraints to worry about other than the Algebraic Translations.
4. Two sentences to translate into algebra:
Jake loses 8 → “J – 8”
(the verb) he will weigh → “=”
twice as much as his sister → “2s”
Together they weigh → “sum”
5. It feels like Algebraic Translations because it’s a story with two unknowns (Jake’s weight and his sister’s weight), and the story is providing us with relationships between the two unknowns.
6. If we’ve gone the Algebraic route, then we should be able to solve the system of equations in 45 seconds or less.
J – 8 = 2s
J + s = 278
We can isolate one variable and then substitute the other side of that equation into the other equation. Or we can stack the two equations (possibly scaling them up in order to make the coefficients of a given variable match), and add or subtract the equations in order to eliminate one of those variables.
If we initially isolate s by saying s = 278 – J, then we’ll have the unenviable task of doing J – 8 = 2(278 – J). The burden of 2 * 278 should suggest to us that there’s probably a friendlier route.
If we’ve gone the Working Backwards route, we need to be able to subtract 8 from a three-digit number, divide that number by 2, and then add up a three-digit and two-digit number to see if it’s equal to 278.
7. Since they’re asking us to solve for Jake’s present weight, but the problem talks about Jake losing 8 pounds, there might be Evil Twin answers (one answer is Jake’s real weight, the other is 8 less, in case someone accidentally solves for that number). 131 and 139 are Evil Twins, because they’re 8 apart, as are 139 and 147. We’d be ditching the lower one, since that one would be solving for Jake – 8 rather than for Jake. So we wouldn’t guess 131. But 139 appears as the bigger twin and as the lower twin, so we’d probably not know what to do with that one.
8. The big potential errors on Algebraic Translation are always botched translations. People might accidentally do (J-8)*2 = s.
9. Since the answers are friendly integers, we could Work Backwards. If we started with (C), for instance, and said that Jake’s present weight is 139, we could calculate that if he loses 8 pounds, he’ll be 131. His sister would therefore be 65.5 lbs. That decimal weight is already enough to know this number is wrong, but going a step further and thinking, “together, they weigh about 130 + 70 = 200 lbs,” we know that this number was way too small. We’d eliminate (A) and (B) and (C) and probably jump to (E) since 137 was way too small and (D) is only somewhat bigger. We also might recognize that if J – 8 = 2R, then J – 8 = even (since 2 * anything is even). If J – 8 = even, then J = even + 8, which means that J = even. That makes (E) the only legal number to pick.
PS 225
1. Say whaaaaa? No, I don’t know how to simplify this dumpster fire. This question SCREAMS out, “Use one of your three free skips on me!” It’s clear from the size of the paragraph, the number of variables or functions being named and defined, the complicated constraints (1/3 of the decimals have a tenths digit that’s even), the cryptic question (“possible value”) and the three-case answer choices that this is a very hard problem.
2. Anyone, including a teacher, who thinks that they are at level 4 of mastery on this problem the first time they try it is lying to you or themselves. Now that we’ve established that it would take several times doing this problem to be able to answer these 10 questions, we’ll pretend like we’re already at THAT stage of understanding it.
There really isn’t one compact reframing for this dense a question, but whenever questions are asking which is a possible value, then the question is either asking:
-which of these values adheres to the number property that any possible value would have?
or
-which of these values is within the range of possible values?
In other words, you’re either going to know that “all possible values are multiples of 11” and pick accordingly, or you’re going to know “all possible values are between 4 and 18” and pick accordingly.
So we could reframe this question by asking ourselves, “Is this asking me to infer a potential number property of E – S, or is it asking me to find the min/max of E – S?”
2. There aren’t any formulas or properties at work here (for example, we AREN’T using the normal property we think of with rounding decimals: “If it’s .5 thru .9, round up… if it’s .4 thru .1, round down”).
3. Lots of vocab here.
“30 positive decimals, none of which is an integer” = must have some nonzero digit to the right of the decimal
“tenths digit is even” = tenths digit is .0, .2, .4, .6, or .8
“tenths digit is odd” = tenths digit is .1, .3, .5, .7, or .9
S = sum of the non-rounded numbers
E = sum of the rounded numbers
4. We need to apply the function rules to think about what’s happening with E.
“1/3 of the decimals have even tenths digits” = 1/3 of the numbers get rounded down
and we need to think about the complementary leftover
“2/3 of the decimals have odd tenths digits” = 2/3 of the numbers get rounded up
If we’ve figured out that we’re seeking the minimum/maximum value of E – S, we are saying:
MAX VALUE: E is as big as it can be, compared to S
MIN VALUE: E is as small as it can be, compared to S
Biggest E compared to S comes when the even tenths digits are all 0’s, because the function forces us to round up, so we get more upgrade going from something like 0.01 to 1 than we would going from 0.81 to 1.
Also, we need the odd tenths digits to be 1’s, because they will be rounded down, and we want to minimize the downgrade we get out of rounding down. We lose less when we go from 0.11 to 0 than when we go from 0.91 to 0.
Smallest E compared to S has the opposite logic. We’d want to have all the even tenths digits be 8’s so that we have as little rounding up as possible. And we’d want the odd tenths digits to be 9’s so that we have as much rounding down as possible.
5. The big topic clues are DECIMALS, ODD/EVEN, but the normal first moves don’t apply.
The normal first move with DECIMALS is to “clean it up into integers and powers of 10.” For example, change 0.0034 into 34 * 10-4. The normal first move for ODD/EVEN is to “think about the rules of adding/subtracting/multiplying two evens, two odds, or a mix.” For example, seeing 3x + 4y is odd and thinking “if 3x + [even] is odd, then 3x is odd, so x is odd.”
The real topic is FUNCTIONS, and the giveaway there is “The estimated sum, E, is defined as follows.”
Our first move on FUNCTION problems is usually just to think, “Should I bail on this problem? Do I typically do poorly on function problems? I know it’s a topic that skews hard, so it wouldn’t hurt me to miss this.” or “Calm down, FUNCTIONS problems are designed to be brain-numbing the first time we read them. Read this multiple times and think flexibly about what process it’s describing.”
6. It should be easy to take 1/3 of 30 and getting that 10 of these numbers have even tenths digits (i.e. will be rounded up) and that 20 of these numbers have odd tenths digits and will be rounded down.
When we’re calculating biggest E and smallest E, we will get to some ugly numbers that suggest we should switch over to approximating.
Biggest E: 10 of the numbers of a tenths digit of 0 and 20 of them have a tenths digit of 1. What’s the net gain/net loss, compared to what we would have gotten by just adding all 30 numbers (S)?
For each of our 10 round ups, we’re getting something as big as 0.9999… since we can go from a 0.01 to a 1. Let’s approximate and say that these 10 numbers are being rounded up by 1. So we get +10 from those. Meanwhile, we’re losing 0.10 from each of the 20 odd numbers, because we’re going from 0.1 to 0. We get a -2 from that (20 * 0.10).
For our Biggest E, then, we have a net gain of about +8.
E – S < 8
To calculate our Smallest E, we’re going from 0.89 to 1, and we’re going from 0.99 to 0. So we have 10 gains of roughly 0.11 = +1.1. We have 20 losses of roughly 1 = -20.
So E could potentially be about 19 smaller than S.
-19 < E – S
7. It would be weird for (D) to be right. It’s the only answer that thinks case 1 fails. Since case 1 is in four of the answers, it’s much more likely that it actually works.
It would be weird for the middle number NOT to work. Most of these “possible values” questions are basically testing whether we’ve solved for the mix/max of possible values. The middle number is usually safe and one or both of the extreme numbers have gone beyond the legal range. Since the middle number probably works, (B) or (E) would be a smart guess.
8. It’s possible that people would rule out “0” as one of the potential even tenths digits, although it mercifully doesn’t affect our getting the correct answer. People often forget to consider 0 when they think about even digits.
People might waste time trying to find a case where they land on E – S = 6, instead of realizing that 6 is just some arbitrary number they’re picking that’s somewhere within the legal range.
9. There aren’t really multiple ways of doing this problem. Subbing in each of the three cases as a possible value of E – S wouldn’t allow us to calculate anything, so Backsolving is off the table. We sort of do need to Make Up Numbers to get through this, but this problem largely hinges on two things:
-sensing that a problem asking “Which of these are possible values of E – S” is really asking us to figure out the min and max value of E – S
-understanding that finding the min/max of E – S means picking which odd or even tenths digit would achieve the most rounding up / least rounding down or the opposite.
Now’s the REAL practice:
We’ve dissected all three GMAT Quant problems. Which of those are you putting on your Redo Calendar and when? (Remember that PS 225 is a stupid problem for us to attempt on test day, so we should not prioritize getting better at it in practice.) Which little nuggets / moves / properties / translations we mentioned were ones that had escaped you? Go write some flashcard quizzes for each of those, so that you can reinforce those ideas a few more times in the next couple weeks. ?
Want some more GMAT tips from Patrick? Attend the first session of one of his upcoming GMAT courses absolutely free, no strings attached. Seriously.
Patrick Tyrrell is a Manhattan Prep instructor based in Los Angeles, California. He has a B.A. in philosophy, a 780 on the GMAT, and relentless enthusiasm for his work. In addition to teaching test prep since 2006, he’s also an avid songwriter/musician. Check out Patrick’s upcoming GMAT courses here!