Tricks for fast Addition of numbers

Here is another post on basic quantitative aptitude techniques, particularly concerning shortcut techniques in integer  addition Some of these are very easy to follow, still if you are not able to get something, please use the comments box to start the discussion on that particular topic.

Trick 1 : Adding Consecutive Numbers

Any set of consecutive numbers can be added by this simple relation-
Sum = (Last Number – First Number + 1) * (First Number + Last Number)/2

For example, you need to add numbers from 30 to 40, then…
Using the above relation, we get, Sum = (40-30+1) * (40+30)/2 = 11*70/2 = 385

Trick 2 : Sum of Odd numbers from 1 to n

This one is very easy. Sum of odd numbers from 1 to n is ((n+1)/2)2.
For example, 1+3+5+7+9+11+13+15+17+19+21 = ((21+1)/2)2 = 112

Trick 3 : Sum of Even numbers from 2 to n

To find this, use this relation-
Sum = (n/2)*((n/2)+1)
For example, to find sum up to 100, use this relation-
Sum = (100/2) * ((100/2)+1) = 50 * 51 = 2550

However these tricks are pretty obvious and very small, but when used in competitive exams like CAT and UPSC, these can be valuable resource in reducing the time required per question.

Check out a few more related posts here-

  1. Quantitative Aptitude Techniques
  2. Syllogism Cheat Sheet

Syllogism Cheat Sheet

Dealing with syllogism had always been a big problem to me. Even if I do learn it how to deal with it, I often forget how to use them when required, I guess it needs constant revision and great logical and deductive reasoning capabilities to deal with.

Logical deductions

To make things easier, I keep this small cheat sheet which contain basic formulas to deal with syllogism in verbal reasoning. I will try to explain here, how to use this cheat sheet to solve the problems which often appears in competitive exams like CAT, XAT, IIFT, and nowadays even CSAT and other job oriented exams are also looking for reasoning capabilities in candidate.


First Premise Second Premise Conclusion
All All All
All No No
All Some No Conclusion
Some All Some
Some No Some Not
Some Some No Conclusion
No All “Some Not” – Reversed
No Some “Some Not” – Reversed
No No No Conclusion
“Some Not” or “Some Not” – Reversed Anything No Conclusion


You can use above table for your reference while solving problem on syllogism. Here is an example for same-

[Q] Premise Statements-
[a]. All cities are town.
[b]. Some cities are villages.
Conclusion Options-
[i]. All villages are town.
[ii]. No village is a town.
[iii]. Some villages are town.

Answer Options –

  1. Only conclusion [i] follows.
  2. Only conclusion [ii] follows.
  3. Only conclusion [iii] follows.
  4. None of these.

This is clearly a combination of All + Some in premise statements. Hence no conclusion can be drawn. So, option 4 is correct. None of these.

I had took this question from Combined Graduate Level Exam conducted by SSC in 2013, questions like this could be a matter of 2 or 3 seconds if you can remember this table. Here is another quick example of how to use this cheat sheet.

[Q] Premise Statements
(a). All papers are books.
(b). All books are pages
(c). All pages are material.
Conclusions Options:
I. Some material are pages.
II. All books are material.
III. All papers are pages.
IV. Some books are papers.

Answer Options –

  1. All the four follow.
  2. Only II, III follow.
  3. Only I, III and IV follow.
  4. Either I or III and II follow.
  5. None follows

This one is a bit tricky question. While it seems All + All = All, should give us II & III as correct statements, but we should also look what other statements are telling us. Conclusion statement I & IV are mere implications of premise statement (a) and (c). Hence the correct answer option is [1.] All the four follows.

And last but not the least, a very simple problem, which would generally come in medium difficulty level exam-

[Q]. Premise Statements-
Statement 1. Some dogs are bats.
Statement 2. All bats are cats.

You probably don’t need options for problem like this one, as you can refer the table above, which says Some + All = Some. So, cancelling out bats, we get Some dogs are cats as our answer statement…

There are plenty of various types of questions which one can practice based on Syllogism and deductive reasoning these generally include conditional statements, possibility cases, premise – conclusion etc. Sometimes conventional way of using Venn Diagram might help in solving Syllogism but not always a good practice in competitive exams. It would always be better if you could remember the various possibility cases, and arrive to conclusion in seconds instead of working out a problem in minutes. An advanced level cheat sheet for syllogism is available here.

Quantitative Aptitude Techniques

If you are a bit slow in Quantitative Aptitude problems in Mathematics, which generally appears in many competitive exams like CAT, AFCAT, SNAP, XAT, IIFT etc… These tricks will only be helpful if you at least remember Tables up to 20 and can do addition of two numbers very fast. If you are not at that level, then practice first, come back here latter.

Various techniques which I use in problem solving are-

  • Digital Sum Technique
  • Divide and Rule Technique
  • Using Algebraic formulas
  • Percentages
  • Vedic Mathematics
  • Russian Peasant Mathematics
  • Base Method
  • Duplex Method

And there is no end to this list as you get deep into mysteries of mathematics. I will be briefing about these methods in this post, with the links to original source articles for further explanation. If you got any doubts, please let me know through comments section.

  1. Digital Sum Technique-
    As we work on decimal number system, it is inherent property of a number to retain its digital sum even when it is multiplied, divided, added or subtracted.
    Digital sum is sum of digits with which a number is made, till we get a single digit number, for example
    753729026 has digital sum = 7+5+3+7+2+9+0+2+6 = 41 = 4 + 1 = 5. Now if we do any operation with this number our basic mathematics operations can be easily verified with this technique, Example-
    753729026 X 2132 = 1609650283432
    Writing digital sum of each number-
    753729026 => 5
    2132 => 8
    1609650283432 => 4
    Multiplying digital sums – 8 X 5 = 40 (digital sum = 4)
    So this method might help you in quick verification of calculation or answer option elimination.
  2. Divide and Rule Technique-
    Suppose you have to perform a lot of calculation, you can arrive at an approximate answer using this technique-
    Example- Find 13.8 X 6.3 + 173% of 600
    Now, divide and rule by breaking them into parts
    This can be written as-
    13 X 6 + 0.8 X 6.3 + 0.3 X 13.8 + 175% of 600 – 2% of 600 = 72 + 5 + 4 + 7/4 of 600 – (2 x 1% of 600)
    = 83+ 1050 – (2×6) = 1121 (approximately)
    1124.94 (Exactly)
    Hence using this we got an error of 0.35% and calculation can be done mentally.
  3. Using algebraic formulas-
    Keep some of the algebraic formulas in your mind like-
    a2 – b2 = (a-b) (a+b)
    Example – 222 – 132 = 9 x 35 = 315
    There are lot more other formulas too, lookout for them.!
  4. Percentages
    Well, if you know all the percentages and corresponding fractions, a lot of hard work can be reduced!
    Try learning the common percentages, some of these are shown below-
    Percentage
  5. Vedic Mathematics
    I use this to do faster multiplication. See example below-
    Multiply and write the numbers as shown.
    Multiply
  6. Russian Multiplication
    To use this you need to be very fast in doubling the numbers and arriving at halves as well. Here is a quick demo how to perform such multiplication.
    Multiply
  7. Base Method
    This one is applicable to digits near same base, like here is an example-
    Multiply
  8. Duplex Method
    This one is used in calculating squares as shown below-
    Square

 

References-

  1. http://mathworld.wolfram.com/RussianMultiplication.html
  2. http://en.wikipedia.org/wiki/Digital_root
  3. http://www.sciencenews.org/view/generic/id/8203/title/Math_Trek__Divide-and-Conquer_Multiplication
  4. http://en.wikibooks.org/wiki/Vedic_Mathematics/Techniques/Multiplication
  5. Bonus – Japanese Multiplication