Lately, I’ve been speaking with a few different students who are aiming for a 750+ score “ in other words, stratospheric! I’ve tried (and hope I’ve succeeded!) to impress upon these folks that getting such a score involves a lot more than studying the hardest questions.

What’s another crucial component? Finding faster / easier ways to answer questions that you can already answer now.

Why? The questions that you can do right now in the 650 or 700 range will need to turn into very easy for you questions in order to hit 750+. It isn’t enough that you can do them now in relatively normal time. You’ll actually need to turn these into I can answer this very quickly without making a mistake so that you can knock these out and have a little bit more time and mental energy to spend on the even-harder questions you’ll need to answer to hit 750+

Try this GMATPrep question:

* A certain bakery has 6 employees. It pays annual salaries of $14,000 to each of 2 employees, $16,000 to 1 employee, and $17,000 to each of the remaining 3 employees. The average (arithmetic mean) annual salary of these employees is closest to which of the following?

(A) $15,200

(B) $15,500

(C) $15,800

(D) $16,000

(E) $16,400

Done? Okay, how long did it take you? More than 30 seconds? Go look at the question again and see whether you can find any shortcuts or faster ways to approach it. Take all the time you need.

When you’re done, come back here and keep reading!

Okay, let’s do the real math solution first. Then we’ll look at the 30-second solution.

They’re asking for an average and they actually straight out give me the numbers that I need to average. Great “ I can just plug them into the average formula and crunch the numbers!

Let’s see. Average = sum / # of terms.

The sum is 14,000 + 14,000 + 16,000 + 17,000 + 17,000 + 17,000. That’s a little annoying. Oh, but I can save some time here by just adding 14 + 14 + 16 + 17 + 17 + 17 and then adding the three zeroes back in. Okay, 14 + 14 + 16 + 17 + 17 + 17 = 95. Add three zeroes to get 95,000.

There are 6 employee salaries, so the number of terms is 6. Now we’ve got 95,000 / 6 = ugh, another annoying calculation. I’m going to do 95/6 let’s see, long division, that’s 15.833 repeating. Then I have to add in the three zeroes that I chopped off, so it turns out to be 15,833.33.

Oh, I see. I should’ve watched the answers while I did that long division. As soon as I saw it was going to be 15.8, I could’ve stopped and picked the correct answer, C.

Got all that? Not too horrible as far as GMAT questions go, although there were several annoying calculations there.

Now, let’s get into the big leagues.

There are 6 salaries overall. Three of them are $17,000. What if the other three were all at the other end of the range, $14,000? What would the overall average be?

Because there would be three of each, the average would be halfway between 14,000 and 17,000, or 15,500.

What did we just learn?

(1) The answer is NOT 15,500 (answer B), because we don’t actually have three 14,000 salaries.

(2) The answer is also not A (15,200). Two of the three salaries are 14k but the third one is higher (16k), so the overall average also needs to be higher.

(3) This is a weighted average question in disguise

That last little realization was exactly what allowed me to figure out the rest of my 30-second solution.

I’ve got three answers left. One is prettier than the others: 16,000. What would get me an average of 16,000?

Well, if the top 3 are still 17,000, and if the bottom three averaged to 15,000, then the overall average would be 16,000.

Do the bottom three actually average to 15k? The bottom three are 14k, 14k, and 16k.

Once again, we’ve got another mini-weighted average. 14, 14, and 16 can’t average to 15. The values are skewed towards 14, so the average has to be less than 15.

Bingo. Answer E is definitely wrong because that would require a bottom three average greater than 15.

[Note: this next bit was added after original publication because I glossed over this math.] What about D? The mini-average is not 15, it’s true… but the question asks for the “closest” answer. So is the overall average closer to 15.8 (that is, less than 15.9) or closer to 16 (that is, more than 15.9)? Estimate how great the “skew” is.

14, 15, and 16 would average to 15 (in which case the overall average would be 16 and the answer would be D). We’ve actually got 14, 14, and 15. Only that one “off” number is skewing the average down.

That 14 is one of 6 numbers, so it has a weighting of 1/6 in the overall calculation. It is 1 “off” from what it would have to be (15) in order to get an overall average of 16. So that last 14 is going to “pull” the overall average down by 1 Ã— 1/6 = 1/6.

1/6 is larger than 1/10, so the overall average must be “pulled” from 16 to below 15.9. The correct answer has to be C.

Done “ with almost no calculation at all, let alone the incredibly annoying calculations from our first solution method.

Right now, many people reading this are thinking, Wow, I would never have thought of doing it that way. That’s perfectly fine if you’re not going for a 750+ score (or a 95+ percentile score on quant alone). Do it the old-fashioned math way as we first did above.

If you *are* going for crazy high scores, though, then our first solution method above is not going to be sufficient. You’re going to have to take some time to figure out how to answer this one far more efficiently (without making a mistake “ so you can’t just speed up!).

Finally, a word of caution and encouragement. Going for a super-high score is let’s call it unpredictable. Most people, by definition, will never make it that high. If you’re really determined to get there, then you’re going to have to try to learn how to do what I described above, and this is going to take time, patience, and hard work. You’re going to be slow at first “ that’s okay. If you stick with it and don’t try to rush things, you’ll make progress (though I can’t, of course, guarantee that you’ll progress all the way to 750+).

**Key Takeaways for Figuring Out Major Shortcuts**

(1) These are not just about doing the same work faster. You’re likely going to need to come at this from a completely different angle “ and this will usually involve a new way of *thinking through* what’s going on, not just different math.

(2) If you’re going for a super-high score, it’s not enough to be able to do the lower-level problems in normal to slightly-faster-than-normal time. You’ve got to knock those problems out of the park “ very quickly without reducing your accuracy.

(3) You might spend 5 or 10 minutes examining a problem (after trying it) to try to figure out better approaches. You can also google the problem to see whether someone else has come up with a great alternate method “ but try yourself first. If you figure it out for yourself, you are far more likely to understand why the alternate approach works the way it does, and this means you’ll be far more likely to recognize when you can use the same approach on different, future questions.

*Edited after original publication. See my comment posted 18 March (below) for more takeaways!*

* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

Better way – U have 3 salaries…. 14000 for 2, 16000 for 1 and 17000 for 3….. Just take (14+16+17)/3 will give u 15.88 faster……

Just a coincidence. You definitely should not do it this way.

No. Its not the correct way anand.

There is another better way that the blog –

1) We would do all calculations without 1000 multiple. So 14,000 means 14; 16000 means 16 and so on.

2) For calculation simplicity assume 15 as the delta number. Reason for chosing 15 as delta ==> quick glance on the numbers and as well as the answer choices suggest average is above 15

3) We can do all calculations as (salary – delta).

4) So average = 15 + ((14 – 15) * 2 + (16 – 15) + (17 – 5) * 3) / 6 = 15 + (-1 * 2 + 1 + 2 * 3) / 6 = 15 + (5/6) = 15 + 0.83 [this is standard division, one would remember] = 15.83

OR

We could have taken even 16 as the delta, but then you will get 16 – 1/6 (chances of errors are bit higher in decimal subtraction, then addition).

OR

We can taken 14 also as the delta, calculation would be equally simpler, but requires one more step of simplification. I would better take some cues from the answer choices.

* Small typo in point 4 – read “17 – 5″ as “17 – 15″

Anand – your method works only by coincidence. For example, if the numbers were 14,000 for 2, 16,000 for 1 and 1,000 for 3, the correct answer would be 7,833 but using your method you would get 10,333.

Shail – your method is EXCELLENT – well done! Most people, however, would find it difficult to implement and error-prone.

In your experience, what could be error prone in that approach. Would be glad to know – as it would help me adjust.

This is a great example of a problem that is not terribly difficult but is intended to waste time – they’re all over the GMAT. Another “imprecise” way to do this is to realize that the average of 14, 14, 16, 17, 17 and 17 is the same as the average of 14, 14, 16, 16, 17 and 18. Since the average of 14, 14, 16, 16, 18 and 18 would be 16, changing one 18 to a 17 will just subtract 1/6 from the average. Also, the fact that the question asks “is closest to” rather than “is” basically begs us to estimate rather than calculate.

That’s how I think but it’s quite dangerous to apply this kind of analysis. OK, we know that it’s greater than 15.5 and it’s not 16, what makes sure that it couldnt be 15.9000001, which is closer to 16 than to 15.8

Love all the ideas, guys – keep up the great work!

An, good point – I did gloss over how to realize that it’s closer to 15.8. If you’re not sure whether it’s weighted “enough” towards 15,800 (vs. 16,000), try sketching this out:

14, 14……….16…..17, 17, 17

avg two of the 14s and 17s to get 15.5; note that b/c you’re using 4 numbers to create these averages, they get 4 “weightings” (so you still end up with 6 numbers):

15.5, 15.5, 15.5, 15.5…16……17

avg 16 and 17:

15.5, 15.5, 15.5, 15.5…..16.5, 16.5

avg two pairs of 15.5 and 16.5

15.5, 15.5…..16, 16, 16, 16

so the final number is closer to 16 than to 15.5… but it’s not SO close that it’s over 15.9.

Some people will “see” that earlier in the chain and some people will see it later – take it as far as you need to in order to be confident that it’s closer to 15.8 than 16.

I used a very different method:

(1) Required: average of 6 salaries (3 sec)

(2) 16 + 28 + 51 = 95 (~10 sec)

(3) 15.8 * 6 = 94.8, 16 * 6 = 96 (~5 sec)

(4) No further calculations necessary: C

You can either select the middle answer (C) and move towards the answer that bounds the approximation (so here C=15.8 and the other answer that bounds the approximation should be larger than C as 15.8*6 = 94.8 < 95, thus D) OR if you're very comfortable with numbers take an answer you expect to be right.

Additional method, choose 14 (for simplicity) as the average and then see how much the remaining numbers deviate from it

EX.

$14,000 to each of 2 employees, (0)

$16,000 to 1 employee (2)

$17,000 to each of the remaining 3 employees (3*3 (because of the 3 employees)

0+2+9=11

divide by the 6 people and you get 1.8333 which you then use to adjust your dummy mean of 14.

14+1.833=15.8333*1000=15833 approx 15800

We have 14 14 16 17 17 17. Change the 16 to 14, we have 14 14 14 17 17 17 17, with an extra 2. Avg of that is 15.5. We know all we are doing now is sharing 2 over 6 items, which is 1/3. Avg is 15.5+0.33, 15.833. 15 secs, maybe? *shrugs*

Just started studying for the GMAT. It took me 20 secs to read and write the problem. I completed the problem in 40 seconds. When you say “Do this question…,” does that mean to read and answer the question or only to answer the question once read?

I used a more visual approach to solve this problem. I pictured the numbers as “pillars”. I started breaking pieces off and moving them around to make everything level. It looked like the average was around 15 to me; I ignored the zeros. I took 2 from each of the 17s. This gave me 6 in my “bank”. I took 1 off the 16. This gave me 1 more in my “bank”. I removed 2 from my “bank” to level off both the 14s to 15s. Now I have 6 15s and 5 left in my “bank” so I know the average is less than 16 but very close to it. Looking at the answers, I can immediately eliminate D and E because they are bigger than 16. Looking at B, I know immediately that it is wrong because I know the average is closer to 16 than to 15. This rules out A and B, leaving only C.

Maybe it’s just me, but doing the addition and division should be fairly quick for this question. 14*2=28. 28+16=44. 17*3=51, and 51+44=95. If you remember that 96=6*16, you know the answer is just below that.

Although the methods suggested here would certainly be very helpful problems with more difficult numbers!

For this problem I wouldn’t even use the tens place as the ones and tenths are most important for the answer. 4+4+6+7+7+7=35. 35/6=5.8 and you don’t need to go beyond 5.8 to have the right answer.

This is the same way I did it. I was reading all the replies to see if anyone else did it this way and sure enough!

To be truthful though in my head what I actually did was: 7*3 = 21 + 6 = 27 + 4*2 = 35 / 6 = 5 5/6 = 5.8, but it’s all the same.

I concur with Dima’s method – straightforward. Once you get the total to 95000, instead of dividing 95000 by 6, divide 96000 by 6, and then subtract 1000/6 (which is 166). It pays to learn beforehand the values of fractions such as 1/4, 1/5, 1/6, 1/7, 1/8 and 1/9/.

If you want to do this question in 30 secs including reading the question, understanding the requirements, locating the data, (it took me 35 to read and answer – the answer options were very close), then most of the methods described above will probably take you longer unless you know beforehand exactly what the question is about (and you dont – they dont put a heading on the GMAT questions). In fact trying to do a question in less than 30 seconds is not what you should aim for – you may end up messing up a relatively simple question. Aim for solving most questions in uncer 1 minute, and you should be ok.

Sum = 95 = 60 + 30 + 5 divide each by 6 gives 10 + 5 +5/6 = 15 + 5/6, something close to 15.9

if you have 2×14, 1×16, 3×17… I rewrote this as 1×15 and 5×16 by just moving “1,000′s” from all the numbers. This means the average has to be close to 16,000 but not exactly and answer choice C is the only option.

What a great article Stacey! For the other commentators it’s not about trying to find an easier shortcut to this problem it’s about the approach, the strategy. The strategy is the take away not the shortcut.

How is this? definitely 95/6 is either 15.5 or 15.8 from d options 15.5*6=93 so option is 15.8.. BINGO

An simplistic approach would be to take a median figure and divide the difference by the total number of items. In the above case the following methodology would be apt :

1.Lets assume 17 as the avg. Now the deviations are -3,-3,-1.

2.Add them and divide by 6(no. of items).

3.It turns out to be -1.16.

4.Now subtract it from the assumed mean and we have the answer to be 15.833 (17-1.16).

HI Raghav,,,,, pls advise some more dynamic shortcuts like this one … Thanks in Advance !!

LOVE all of the ideas, guys! Great work.

A couple of comments:

1. Anwar said “The strategy is the take away not the shortcut.” Bravo, Anwar. And when we say strategy, we don’t even mean “the strategy for the shortcut.”

The overall idea: just because we can do something doesn’t mean we can’t find a better or faster or computationally easier way to do it. Now, if you’re not going for a 750+, then you don’t have to spend as much time searching for better (combining efficency *and* effectiveness) ways to do the problem – but if you ARE going for a super-high score, you’ve got to push yourself *constantly.*

For example, somebody above said something like, “Yeah, these numbers aren’t that hard. But I could see how this would be useful if the numbers were larger.” But this IS EXACTLY when you learn to get better – when the problem wasn’t super hard for you!!! Figure this stuff out when the numbers / problems are NOT crazy hard – then it’s easier to brainstorm better ways! (Then you can just apply those better methods when you do see the crazy-hard numbers or whatever.)

2. Someone else mentioned that we shouldn’t be aiming to solve in 30 seconds… and I agree. In fact, I tell all of my students that, if they finish a quant problem in less than about 45-60 seconds, they need to check their work – just in case. But the headline sure got your attention, didn’t it?

The overall point stands: find better ways to do what you already know how to do. That will help you on even harder problems. (And, no, don’t feel pressured to spend literally only 30 seconds on a problem.)

3. The second takeaway (after point 1 above) is that there are a million different ways to approach this thing – look at all of the awesome methods people described above! Did you see anything that made you think, “Hmm, that’s interesting! I want to try that myself.” I certainly did. Learn from friends / fellow students – they might think of something that would never occur to you. (Of course, they may also think of perfectly valid ways that don’t work for your brain – that’s okay too.)

Now I’m going to go test out some of the approaches I just read about above and see which ones work well for me!

Thanks so Much !!

As we are very good at Tables up to 10×10…

4*2=8

6*1=6

7*3=21

Total…35

Divide by 6=5.8333

Add 10…15.8333 …That’s all…

Having a dynamic approach to every problem with few standard skills mastered will get us best scores…Saiprakash(GMAT 720; Q51 V35)

HI Saiprakash,,,,, pls advise some more dynamic shortcuts like this one … Thanks in Advance !!

To get avg=17 all 6 numbers should be 17 but this cannot be made here from the set 14,14,16,17,17,17.

To get avg=16 all 6 numbers should be 16 from the set 14,14,16,17,17,17–>16,16,16,16,16,15

avg 16-1/6=16-0.16=15.84

this is probably the best way to do it.

you have 14,14,16,17,17

visualize it as 16,16,16,16,16,15

you know to get 15,500 you need half and half, so the answer is 15,800

I also tried a visual approach:

I moved units around. Move 1 from the first two 17s to the 14s. Represent all the numbers as 15 + somethings

14 15 + 0

14 15 + 0

16 15 + 1

17 15 + 1

17 15 + 1

17 15 + 2

So the average is 15.something (so D and E are out). The something must be close to (1+1+1+2)/6 = 5/6 – which is very close to 1 (5/6 is almost 6/6 = 1). From the remaining answer choices the something choices are 0.2, 0.5 or 0.8 so something closer to 1 is .8 – so answer is C.

what about using baselines (technique from adv quant strategy guide/visual solutions chapter)?

make a line with 14, 16, and 17 on it and note the differentials.

using 14 as the starting point: +0 (+0 from 14), +2 (14 to 16, one time), +9 (14 to 17, three times) => 0+2+9=11.

11/6 =~1.8. 14+1.8 = 15.8, the answer.

Although I think I’d probably be able to just crunch this one out faster by using the traditional approach and dropping the 000s as many have done above.

I also used a similar method.

After dropping zeros, I “guessed” the closest integer average, that would be 16. Than I did the differentials: -2, -2, 0, +1, +1, +1. The sum is -1, that, divided by 6, gives about -0,17. 16 – 0,17 = 15,83.

Using the guessed average as baseline you keep numbers as small as possible.

PS: fun to see so many diffent approaches.

(14+14+16)/3 just <15

(17+<15)/2 just <16

looking at options

answer is C