Mental Math Practice for Consulting Case Interviews

I have consulting interviews coming up, and urgently need mental math prep. Can anyone suggest online (free) resources I can use to pick up my terrible mental math skills? What math strategies are helpful to acing the math part of the consulting interviews?

Other advice on mental math prep is welcome.

Consulting Case Interview Math Prep

The best way to work on your mental math skills is practice. You should follow the advice broken down below and practice some mental math problems during your spare time.

For complex multiplication problems: one effective way to solve problems is to round one of the numbers you are working with to make it a multiple of ten and then add or subtract the difference.

For example: 163 * 242

  1. Round 163 to 200
  2. Multiple 200 by 242 = 48400
  3. Since we rounded up we need to subtract away (200 - 163) * 242
  4. 37 * 242 can be rounded to 40 * 242 = 9680
  5. 48400 - 9680 = 38720
  6. Finally, we need to add back the additional amount that we subtracted so (40 - 37) * 242 = 726
  7. 38720 + 726 = 39446

This process becomes much more simple with smaller problems. Example: 63 * 13 = 60 * 13 = 780 + (3*13) = 780 + 39 = 819

For square root questions - @StudentLoanBackedSecurity offers good advice:

StudentLoanBackedSecurity:

Think of the nearest perfect squares. So for 2,000 I would look first at what the square root of 20 could be (obviously just approximate). The nearest perfect squares are 16 and 25. The square root of 20 would be in between 4 and 5, because obviously 4 squared is 16 and 5 squared is 25. Approximate and say the square root of 20 is 4.5. So now just move the decimal over, and I would have your answer as "about 45" and its basically very close and I would imagine that's all they expect.

User @Matrick, a hedge fund analyst, shared a good website for working on your mental math skills - Windhoff.net.

User @Proboscis, a consulting analyst, shared their advice:

Proboscis - Consulting Analyst:

There's an iPhone app called Mathemagics that you can practice with - focuses on math "tricks" though (ie. how to multiply 2 digit numbers by 11, etc.) - I found this less helpful overall.

I also memorized all of my fractions until 11/11. (ie., 1/1, 1/2, 2/2, 1/3, 2/3, 3/3, 1/4, 2/4, 3/4, 4/4, 1/5, 2/5,... etc.) Other fractions can then at least be ball parked or found exactly by basing it off of these memorized fractions. (ie. 1/12 is just half of 1/6)

User @Consultingrs shared their advice on handling the numerical portion of case interviews:

Consultingrs:
  • Make no mistakes - watch out for units, as they are often there to purposely trick you (i.e., is it in $M or $B?, is it 100M lbs or just individual lbs?)
  • Perform math quickly - cancel out extra zeros quickly. For example, recognize that $B divided by $M, equals $K
  • Ask for permission to round - if you need to multiply 29 cents by 15 million, ask if you can round to 30 cents - the interviewer will tell you if it's acceptable or not
  • Put the answer in context - once you get a result from your calculation, put this in the context of the case. For example, if you calculate the payback period for the new competitor to be 17 years, you could confidently state that this doesn't seem to be a very good return for them and they might not be a threat for too long. Instead of generically stating, "it takes 17 years for them to payback their investment."
  • Organize your notepad so that one piece of paper is devoted only to your calculations. Then you have a separate piece of paper where you put the output of your calculations onto
  • Simply fractions whenever possible. The easiest way to do this is to divide the numerator and denominator by two in your head. This will make some of the divisions much easier when you simplify the fractions

You will almost never be asked to do anything other than simple addition, subtraction, division, or multiplication. The biggest challenge often isn't doing the calculations themselves. Rather, it's organizing the information, so that your calculations don't get messy and make you likely to make mistakes.

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Comments (66)

Nov 11, 2011
Nov 11, 2011

https://www.caseinterview.com/math/home.php - developed by ex-McK Victor Cheng to help practice math skills

There's an iPhone app called Mathemagics that you can practice with - focuses on math "tricks" though (ie. how to multiply 2 digit numbers by 11, etc.) - I found this less helpful overall.

I also memorized all of my fractions until 11/11. (ie., 1/1, 1/2, 2/2, 1/3, 2/3, 3/3, 1/4, 2/4, 3/4, 4/4, 1/5, 2/5,... etc.) Other fractions can then at least be ball parked or found exactly by basing it off of these memorized fractions. (ie. 1/12 is just half of 1/6)

Good luck

Proboscis

Nov 11, 2011

Sometimes I personally found it helpful to write large numbers in scientific notation to help me deal with order of magnitude.

.....magnitude....pop pop!

Proboscis

Nov 11, 2011

all good tips.

also look into the trachtenberg system.

Nov 11, 2011

good stuff. thanks guys

Nov 11, 2011

quality

Nov 11, 2011

Downloaded an app called Math Gym for the iPhone, it's been helpful

Nov 11, 2011

Just know how to break complicated problems into easier components.. ex. 73 * 25 = 7020 + 705 + 320 + 35 = 1400 + 350 + 60 + 15 = 1825

This makes mental math a lot easier... other than that, practice with large numbers

    • 1
Jun 26, 2012

^^ (You could also do 73*25 as 7300/4 = 3650/2 = 1825).

I think a combination of (1.) practice and (2.) an awareness of different mental math tricks is the key. (For example: If you're asked to multiply 3743 during an interview, they're probably looking for the "trick" 3743 = 40^2 - 9 = 1591.)

For lists of tricks, both Wikipedia:
http://en.wikipedia.org/wiki/Mental_calculation and "Wikibooks":
http://en.wikibooks.org/wiki/Mental_Math have some.

I also recommend Ars Calcula:
http://arscalcula.com/mental_math_multiplication_g...

Mar 24, 2013

calculationrankings com

Oct 14, 2014

I can suggest a book and an app that were really helpful to me.

Bill Nye's Secrets of Mental Math Book: http://www.amazon.com/Secrets-Mental-Math-Mathemag... - I found this book to be superb. I read it twice and it changed the way I do mental math and at least doubled my speed.

Great free iPhone app to practice mental math on the go: http://www.bizmathtutor.com

Oct 14, 2014

Go to your uni library.

Look into the psychology section.
There will be books about mentalism.
They contain GOLD in terms of mental math/cool tricks/body language/etc.

Or the math section, should have some mental math books.
It's a great skill honestly which can show an illusion of great math comprehension even if you don't.

"It is better to have a friendship based on business, than a business based on friendship." - Rockefeller.

"Live fast, die hard. Leave a good looking body." - Navy SEAL

Oct 14, 2014

to get a great score on this dimension, you must:

Make no mistakes - watch out for units, as they are often there to purposely trick you (i.e., is it in $M or $B?, is it 100M lbs or just individual lbs?)
Perform math quickly - cancel out extra zeros quickly. For example, recognize that $B divided by $M, equals $K
Ask for permission to round - if you need to multiply 29 cents by 15 million, ask if you can round to 30 cents - the interviewer will tell you if it's acceptable or not
Put the answer in context - once you get a result from your calculation, put this in the context of the case. For example, if you calculate the payback period for the new competitor to be 17 years, you could confidently state that this doesn't seem to be a very good return for them and they might not be a threat for too long. Instead of generically stating, "it takes 17 years for them to payback their investment."
Interestingly, the case interview math is not supposed to be purposely hard. You will almost never be asked to do anything other than simple addition, subtraction, division, or multiplication. The biggest challenge often isn't doing the calculations themselves. Rather, it's organizing the information, so that you're calculations don't get messy and make you likely to make mistakes.

One easy trick to do is to organize your notepad so that one piece of paper is devoted only to your calculations. Then you have a separate piece of paper where you put the output of your calculations onto. That way, your notes don't get as messy. You can also write the letter A where you did your calculation, and then tie that to the letter A where you placed the output from your calculation. Then, if you make a mistake, you are easily able to find the calculation you did to try to quickly fix the error.

Performing math quickly is one of the key things to differentiate yourself on. The easiest way to do this is to always cancel out all excess zeros at every opportunity. If you are given numbers that are in the billions, just write the number and then put B after it. Likewise, if you are given numbers that are in the millions, just write the number and then put an M after it. Then you can put B over M and delete both to be left with K, which represents thousands. This trick will allow you to work though large numbers very quickly.

Another key trick that allows you to perform math quickly is to simply fractions whenever possible. The easiest way to do this is to divide the numerator and denominator by two in your head. This will make some of the divisions much easier when you simplify the fractions.
- Craig (PM me for more info)

Mar 7, 2016

There are many of them but most of them teach the same thing. I've been reading "Secrets of Mental Math" by Arthur benjamin and Michael Shermer this summer, and some of the techniques are very helpful. My mental division has definitely gotten a lot better and I can now calculate 29/35 = 5.8/7 = 0.8285714 within 10 seconds. It's astonishing reading about some of the techniques out there which aid in breaking down complex calculations. Mental calcs use to be one of my major weaknesses and I can't wait to flex my newly found prowess.

edit: Just realised the book you linked to is the same one I'm reading.
Tip: I found mine free online (ebook)

Mar 7, 2016

"29/35 = 5.8/7 = 0.8285714"

Very intriguing. Explain how you would do this in your mind within 10 seconds Brown.

Mar 7, 2016
lawschool121:

"29/35 = 5.8/7 = 0.8285714"

Very intriguing. Explain how you would do this in your mind within 10 seconds Brown.

One way it teaches you to divide, is by reducing a division to a base lower than or equal to 11, and the books already taught easy ways to memorize the fractions from 1/2 to 10/11/

For divisions by 7, it's always a reccuring decimal with the numbers 142857, but the order just changes. To find the decimal, multiply the number you're dividing by 7 by, by 14 to get the first two decimals

so 29/35, reduced to a base lower than 11, you divide by 5 (as it's always easier to divide by 5 than 7), which lowers it to 5.8/7, which in turn is equal to 0.8 remainder 2/7
to get the remaining decimals, multiply two by 14, you get 28, so you start the 142857 sequence with 28, hence 285714 recurring
the answer is then 0.8285714 with the last 6 digits recurring.

Takes practice to get used to some of the techniques, but they're truly amazing.

division by 7 is the only base <= 11 that has an awkward decimal sequence so I thought I'd use that as an example as it's the most difficult one.

you can try this with other divisions by 7 keeping the sequence 142857 in mind

3/7 (3x14 = 42) = 0.4285714 recurring (time to calculate, less than 3 secs)

6/7 (6x14 = 84, round up) = 0.857142 recurring (time to calculate, less than 5 secs)

Mar 7, 2016

Thats horseshit brownie....even if you could do that calculation its not that impressive.

Mar 7, 2016
IAmTensai:

It's good enough to say

29 = approx 47
35 = 5
7
29/35 = approx 4/5 = 0.8
So it's just above 80%

I don't think it's necessary to go to the 5th decimal if I asked that question in an interview. With that question I'm just looking to see how good you are at approximating quickly.

If it takes the same amount of time to get an exact answer, would it hurt?

Also, I've heard people ask give me x/y to 3 decimal places kind of questions. In that case, "just above 80%" won't cut it.

Mar 7, 2016

I only practice the basic stuff, just to keep my speed up, try to calculate as many sums as possible in 120sec at calculationrankings.c

Mar 7, 2016

Most of the finance and calculations performed in IB/PE/HF are fairly straight-forward so you should review and practice basic math concepts. Most mental math is pure arithmetic and I cant think of any way of practicing beyond rote learning and case study. You dont have to be a mathematician to do well in most areas of finance but if you cant calculate a simple annual increase of 20% on a $120 million investment you may want to rethink your chosen field or crack open a book.

Mar 7, 2016

Mental math isn't really that important, but there's absolutely no harm in getting good at it.

There are actually small things you can do to get better. This may sound a little silly but just perform random calculations in your head when you're walking or bored etc. As you keep crunching, you will get better and see tricks and patterns. For instance;

when you're squaring any number that ends in 5; 65^2 = 67 and 55, put them together and you'll get 4225

works with 85^2 = 89 and 55 = 7225.

Mar 7, 2016

Do you have any other tips? That's a great one.

Mar 7, 2016

great tip!

Mar 7, 2016

i think the most mental math i got in a IBD interview was... 'what's 7 cubed'

Mar 7, 2016

One time in an ER interview, I was asked, "Whats 3% of 3B?"

Mar 7, 2016

yah my friend was asked 'whats 4% of 9' in an interview... its obviously not hard but they do ask it

Mar 7, 2016

Remember that (x-1)*(x+1)=x^2-1.

So..., 1416=225-1, 2325=576-1, etc

In general, memorize squares up to like 30.

Mar 7, 2016

Whenever I have to use the john and there's none around, I do mental math to distract myself. Double win.

Mar 7, 2016

I had 19x17. Again straight forward as long as you take the simple approach.
I also had a few Physics 101...two cars travelling towards each other, one @10mph, the other....when will they meet.

If if don't make dollars, it don't make sense.

Mar 7, 2016

Isn't the answer to that last one... 1?

Mar 7, 2016

Wow, you guys are f'ing maniacs (read: nerds) but I guess whatever works. Luckily, you almost always have a BAII+ or excel nearby.

Mar 7, 2016

you can learn a lot of tricks like that by studying for the GMAT (for example, the square of any odd number is going to be 1 greater than a multiple of 4)

The reason for that is an odd number is 2n+1 so (2n+1)^2 = 4n^2 + 4n + 1

Mar 7, 2016

insight's trick is from a book called "secrets of mental math" (benjamin and shermer). buy it for 10 bucks on Amazon, read it, and then practice the art whenever you can. i checked it out recently, it's pretty good.

i agree about trying to develop that skill every chance you get. i have always thought it natural to calculate in my head whenever possible. it gets easier and quicker over time. i think it's a lot about memory, being able to store lots of digits in your head. also give memorizing your phone numbers a go. i didn't use the phonebook on my cell for the longest time, and it became natural for me to remember 200+ phone numbers. or, even if you store numbers, try actually typing them out when you make calls to develop your memory.

bah i'm rambling.

Mar 7, 2016

sure buddy^^ and i can remember the last 3000 lottery ticket winning combinations...its so easy if you just practice!

Mar 7, 2016

3640 + 236

Mar 7, 2016

Yeah, there's no reason you need to make both numbers multiples of 10.

You should be able to do 36*40 mentally (30*40+6*40).

    • 1
Mar 7, 2016
BTbanker:

Yeah, there's no reason you need to make both numbers multiples of 10.

You should be able to do 3640 mentally (3040+6*40).

I would have done it by going 360x2x2 = 720x2 = 1440; 36x2 = 72. 1440 + 72 = 1512. Most of this would probably depend on how you could do it faster.

Mar 7, 2016

36*42

34 + 32 + 64 + 62

=1200+60+240+12

Mar 7, 2016

if you expect alot of these i would advise getting trachtenbergs book of speed maths, its pretty useful (but I forgot most of it lol, I think their method is similar to the one you posted)

I personally would just do 3640+236

    • 1
Mar 7, 2016

for numbers between 10 and 19

eg. 1817 -
i) 18 +7 =25 (must add the largest to the right of the smallest)
ii) 25 *10 =250
iii) 8
7 = 56
iv) 250+56 =306

this will work for any number in the teens

Mar 7, 2016

Get Math Gym if you have an iPhone.

Mar 7, 2016

Do brain squats.

Mar 7, 2016

I'm curious for sum tips as well. Also what type of math is actually done as an analyst

Mar 7, 2016
The Black Gordon Gekko:

I'm curious for sum tips as well. Also what type of math is actually done as an analyst

Nice pun in the first sentence. No math is done in your head.

Mar 7, 2016

I bought this book, but haven't taken the time to start it yet. Looks like a good purchase though:

http://www.amazon.com/Secrets-Mental-Math-Mathemag...

Mar 7, 2016

You're better off just learning the software.

Mar 7, 2016

32 squared?

    • 1
Mar 7, 2016
bondtradercu:

I was quite good with mental math when I was younger (think junior, senior HS) but as might be the case with many of you guys out there, calculator and technology kind of dampen the skills quite a bit.
What are the best websites or books or techniques to improve mental math?

There are several books including: Secrets of mental math.
For practice... try to solve as many in 120 sec on: calculationrankings com

Mar 7, 2016

piece of advice, at least for flow equity derivs, ive never really "had" to be amazing at mental maths, calculator always there.

Mar 7, 2016

The math gym iPhone app is good.
Victor Cheng has some good stuff on his website, too.

Mar 7, 2016

Mental math is know the shortcuts and repetition. Just keep doing stuff and try and calculate shit in your head and double check it w/ a calculator, that's how I got better.

Mar 7, 2016

Either figure out tricks on your own or look them up online. Then practice them by doing all non-complex calculations in your head and then double checking with a calculator. You can get good with practice.

Mar 7, 2016
LHDan:

Either figure out tricks on your own or look them up online. Then practice them by doing all non-complex calculations in your head and then double checking with a calculator. You can get good with practice.

Thanks LHDan, great advice!!!!!!

Mar 7, 2016

or just be smart. thats what i did

    • 2
Mar 7, 2016
jktecon:

or just be smart. thats what i did

http://images.google.com/imgres?q=genius&biw=1280&...

Mar 7, 2016

try www.calculationrankings.com, solve as many mental math questions within 120 sec with 4 different levels.

Apr 4, 2017
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