Quicker Maths » Speedy Calculation

No More Carrying Over

admin — Thu, 19 Jan 2012 18:46:44 +0000

"Carry" in Addition - Can it be Avoided?

Carrying over is a concept taught to us at a very early stage of our life. However, it has never been an easy thing to do. In adding two or more numbers, most of us face problem while “carrying over”. Larger the digits, involved in the numbers to be added, more likely it is to involve carrying. More the carrying over involved, more likely are we to make mistakes.

Friends, remember the most basic and effective rule of making arithmetic fast and quick is to break difficult calculations into simpler, easily manageable small calculations.

Any digit when added to 9 (except 0) makes carrying over mandatory. On the contrary, anything added to 0 can’t produce a two digit number. Even if 9 is added to 0, no carrying is needed.

So, to make things simpler let us create some zeroes.

Example:

Add 38 + 86

First make 38 to 40 by adding 2. Now obviously adding 86 to 40 is definitely easier than adding 86 to 38.

86 + 40 = 80 + 40 + 6 = 126

I am sure you must be concerned about the 2 we added out of nowhere. Well you must be, but if you can balance out this extra 2 by subtracting 2 from the answer (126), the final answer will be the same.

You can create 0 towards the end of both the numbers to be added. Try to understand this with an example,

187 + 139

Add 140 (=139+1) to 190 (=187+3)

140 + 190 = 330

Now deduct back ‘1’ and ‘3’ added to the respective numbers. Hence to balance out subtract 1 and 3 from 330 = 330 – 1 – 3 = 326 is the final answer.

The effect of the above trick can be remarkable. Like any other quick calculation tricks, this also requires a lot of practice to master it.

Check out yourself by adding the following numbers -

37 +54 =?

79 + 23 =?

While adding decimals, this can be a very powerful trick. Try these questions -

12.97 + 1.34 = ?

14.95 + 11.60 = ?

Trick for Adding Time

admin — Wed, 09 Nov 2011 16:42:16 +0000

Have you ever faced any problem in adding time?

If you have ever have faced the slightest difficulty in adding time or duration expressed in hours and minutes, this trick is meant for you.

Say you have to add 4 hours 55 minutes and 2 hours 40 minutes.

Make 4 hours 55 minutes into one number, which will give us 455 and do the same for the other number, 2 hours 40 minutes, giving us 240.

Shortcut Trick to Add Two Numbers

Simply add these two numbers together:

455

240

------

695

What you need to do is add 40 to the above result. No matter what the hours and minutes are, just add the constant 40 to the total obtained by adding the time in above manner.

695 + 40 = 735; this means 7 hours 35 minutes.

Be Careful

If there is no carryover from minutes to hours and the minutes digits are less than 60, don't add 40. What you get after first step is your final answer. Say for example, if you have to add 2 hours 20 minutes and 3 hours 35 minutes:

220

335

------

555

So you straight away get your answer as 5 hours 55 minutes.

Another Illustration

Similarly, if someone is coming after 3 hours 55 minutes and right now the time is 1:35PM, you can use the above method to find the time of arrival.

355

135

-----

490

Adding 40 + 490, we get 530. Hence the person will arrive at 5:30PM.

More than two duration

This method can easily be extended for adding more than two time/duration.

So say for example, if you have to add 3 hours 40 minutes, 2 hours 25 minutes and 1 hour 55 minutes, we can do it like this:

340

225

155

------

720

Adding 40 twice to 720, we get 800. Hence, the answer is 8'o Clock.

For more information on math classes, consider looking into online universities

Base Method of Multiplication

admin — Wed, 12 Oct 2011 03:27:30 +0000

Base method of multiplication derived from Vedic Mathematics can be applied for multiplication of two numbers close to 100.

This post in is in continuation of an earlier post named "Vedic Multiplication of two numbers close to hundred". Though you can understand this post stand alone, yet I'll recommend you to read the linked post before reading this one.

In this post I'll explain how to multiply two numbers lesser than the base (in this case 100). In the earlier post it was about both numbers more than 100.

Multiplication where both numbers are lesser than 100

Rule: Still remains the same as earlier. Here you go -

You will get the answer in two parts

First part, to get left hand side of the answer: Add the difference between 100 and either of the numbers to the other number

Second part, to get right hand side of the answer: multiply the difference from 100 of both the numbers

Example

93 x 94

First part: 93 - 100 = - 7; Add this to the other number, thus 94 + (- 7) = 87

Or you can start with the other number 94;

94 - 100 = - 6; Add this to the other number, thus 93 + (- 6) = 87

Result will be same in both the cases

Second part:

Multiply the difference from 100 of both the numbers.

Hence, (93 - 100) x (94 - 100) = -7 x -6 = 42

Combined effect: 87 | 42 = 8742

*| is just a separator.

Example

92 x 86

Step 1: 92 + (86 - 100) = 78

Step 2: (92 - 100) x (86 - 100) = -8 x -14 = 112

Combined effect will look like this: 78 | ₁12

Step 3: Add the 1 (digit at 100s place) of 112 to 78

Answer: 78 + 1 | 12 = 79 | 12 = 7912

When One number is lesser than 100 and the other is more than 100

Same Rule as Above

Example

96 x 103

First part: 96 - 100 = - 4; Add this to the other number, thus 103 + (- 4) = 99

Or you can start with the other number 103;

103 - 100 = 3; Add this to the other number, thus 96 + 3 = 99

Result will be same in both the cases

Second part:

Multiply the difference from 100 of both the numbers.

Hence, (96 - 100) x (103 - 100) = -4 x 3 = - 12

Combined effect: 99 | -12 = 8742

Now to remove negative sign from the right side, we have to take one from the left hand side. 1 when shifted from left to right becomes 100. Thus we’ll have:

Combined effect: 99 – 1 | 100 - 12 = 9888

*| is just a separator.

Example

89 x 113

= 89 + 13 | -11 x 13

= 102 | -143

In this case, right side number is greater than 100, so we need to subtract it from next higher 100, i.e. 200. Hence, we’ve to take 2 from left hand side, so that we get 200 on the right hand side.

= 102 - 2 | 200 - 143

= 100 | 57

= 10057

Does this method makes such multiplications simpler for you? Leave a comment below to express your opinion.

Simplify Multiplication with the Lattice Method

admin — Fri, 16 Sep 2011 07:08:03 +0000

Simplify Multiplication using Lattice Method

Multiplication tables are a pillar of growing up no matter where you are in the world. Spending most of fourth grade learning how to multiply up to 12 x 12 was a fun and exciting time, but I was never a fan of how long it took to multiply larger numbers. I didn’t learned the lattice method until later but as a fan of matrices in calculus, this alternative method of multiplication appealed to me. Here’s how it works:

Step 1) Draw a. grid 8 x 5 should give you enough space, and make sure it’s large.

Step 2) Reserve the top right of this grid for a 4 x 4 grid. Then draw diagonal lines as the image below shows. You should have many of those squares divided in half.

Step 3) Enter the numbers you want to multiply in the grid.

Step 4) Multiply number by their respective places: hundreds by hundreds, tens by tens, etc. In this example, it would be 3 x 4, 7 x 9, and 1 x 2. Take the products of each of these and enter them into the corresponding square, placing the tens digit in the left triangle and the ones digit on the right triangle. If there is no tens digit as is the case with 1 x 2, use 0 as a placeholder.

Step 5) Starting from the right (important), add up the numbers in each diagonal column and place them at the bottom on said diagonal column. Don’t forget to carry!

Step 6) Voilá! The product is 182532!

Is this an easier or more tedious multiplication method for you?

Danielle, a busy college student, likes to solve word and math problems as a stress reliever. With a daily bombardment of information to process via email, billboards, and direct mail, she finds cerebral activity that requires logic to be relaxing and satisfying

Memory Tricks for Trigonometry

admin — Sun, 04 Sep 2011 18:13:25 +0000

In the post titled Trigonometry Formula Memorization Trick, I agreed to write about a simple memory trick for memorizing the value of all major angles of different trigonometry ratios like sin30, cos45, tan60, etc. So here you go –

Values of Trigonometric Angles

Let’s start with most commonly used angles of Sin. The angles are 0°, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2). For these angles we’ve to make fractions for which we’ve to write 0, 1, 2, 3 and 4 in the numerators and write 4 in the denominator of each fraction. After that take the square root of each of these fractions and there you are. Refer to the table below for better understanding.

For Cosine we simply have to write the results of Sin in reverse order. Refer to the table below. The values in the Cos row is in reverse order to that of Sin row.

Tan is very simple. You just have to remember that Tan=Sin/Cos. Hence, to get the value of Tan we’ve to divide values of angles of Sin from SinA row by values of angles of Cos from CosA row.

I am glad that of late, many QuickerMaths.com readers have started participating actively in discussions (comments) under various posts. A quick calculation trick/technique/method from your side can also be posted on QuickerMaths.com. You just have to write it properly and send it to me at vineetpatawari@gmail.com. Don't forget to write a brief description (2-3 lines) of yours, to be added at the end of the post.

Quickly Multiply by 21

admin — Fri, 02 Sep 2011 13:57:27 +0000

Rule for multiplying any number by 21

Start from left. Double the first digit and add it to left side neighboring digit. Repeat the steps for subsequent digits. The last number will be same as the last number of the multiplied number.

This rule is very much like the shortcut for multiplying by 11. Since 21 is sum of 11 and 10, it does belong to the same family of short cuts.

Let’s understand the whole concept with an example. Let’s multiply 5392 by 21.

The first digit of the answer will be equal to twice the first digit of 5392. To make the rule consistent assume there is a zero before the number.

So it looks like 05392

0 + (5 x 2) = 10

As stated in the rule above, next, add the first digit of the given number, 5, to twice the second digit, 3.

5 + (2 x 3) = 11

Since we must have a single digit at each step, the tens place of the result above will be carried over and added to the previous number.

1 | (0 +1) | 1 = 111

The first 3 digits up to this point are 111

The next digit is obtained by adding 3 to twice of 9

3 + (2 x 9) = 21

Thus the first four digits of the answer are -

1 | 1 | (1 + 2) | 1 = 1131 (carried over 2 added to the last digit of 111 )

The next digit is obtained by adding 9 to twice of 2

9 + (2 x 2) = 13

Thus the first five digits of the answer are -

1 | 1 | 3 | (1+1) | 3

The last digit of the answer will be same as the last digit of the number itself.

Hence, in this case last digit will be 2.

Therefore the answer is 113232

Note: " | " is used as a partition between two digits.

Shortcut for Addition of Consecutive Numbers

admin — Thu, 25 Aug 2011 18:05:47 +0000

In this post I’ll share with you a useful shortcut maths trick for “finding out the sum of consecutive numbers”. For example, this trick I am talking about can help you in finding the sum of all the numbers from 23 to 31 or any other set of numbers.

Shortcut Addition Trick

Add the smallest number to the largest number of the given set of consecutive numbers. Then multiply the result by the number of numbers in the set. Finally divide the result by two.

Solving the above example, let’s find: 23+24+25+26+27+28+29+30+31

Step 1: Add the smallest and the largest number from the above set of numbers:

23 + 31 = 54

Step 2: Multiply the result by the number of numbers in the above set. In the above set there are 9 numbers from 23 to 31.

Therefore, multiply 54 by 9

54 x 9 = 486

Step 3: Finally, divide the above result by 2

486/2 = 243

Hence, 23+24+……..+31 = 243

So now, since you know this simple calculation trick, you don’t have to add up each number individually to get the answer. With a little practice, this trick might become a good tool to save lot of your time.

If you want to suggest some additions or modification in the above method, feel free to post your suggestion as comment below.

I am glad to include the suggestions posted as comment below, for the benefit of everyone.

Suggestion by Sagar Shah -

"If there are odd number of terms then multiply the middle term with number of terms and you get the answer.

We will take the same example. 23+24+25+26+27+28+29+30+31

Here the middle term = 27 and the num of terms is 9.
Therefore the answer is 9*27 = 243

If there are even number of terms then take the mean of the two middle terms
eg 23+24+25+26+27+28+29+30

Here there are 8 terms and the two middle terms are 26 and 27. So mean is 26.5. Multiply it with num of terms i.e 8
Solution is 26.5 * 8 = 212"

One very simple formula is used to deduce this addition shortcut. If you could identify that, post it as comment below.

Polish Hand Magic

admin — Tue, 26 Jul 2011 17:45:31 +0000

This is a guest post by Danielle

I learned this problem from The Puzzler’s Elusion (flipkart link) by Dr. Dennis E. Shasha. It’s called Polish Hand Magic. It’s not a method of counting faster, but it is a fun little trick to show young kids (and adults) who know their multiplication tables.

In this Magical Polish tradition, a closed fist equals 5. Let’s say you want to multiply 7 x 8.

7 is represented by …||, or three fingers down and two up. 8 is represented by ..|||, or two fingers down and three up. Find the sum of the fingers that are up, in this case, the amount of vertical lines. Then multiply the number of finger down. So:

…|| and ..|||

2 + 3 fingers up = 5

3 × 2 fingers down = 6

Bada bing! The answer: 56

Here’s another:

If you want to multiply 6 x 9 you have ….|(six) and .||||(nine)

1 + 4 fingers up = 5

4 × 1 fingers down = 4

The answer is 54!

Okay, so it works! But do you know how? Try thinking about it for five minutes before scrolling down to the answer.

Got it?

10 [(x-5) + (y-5)] 1 [(10 – x) × (10-y)]

_______________ + _________________

10 times sum of 1 times product

Values of down values

= 10x + 10y – 10x – 10y + x – y

= X × Y

Try doing this problem with larger and smaller numbers by imagining extra fingers and negative fingers.

Danielle is a young business litigation student who spends a lot of time on web sites like Tanga.com seeking deals to save her money and math games to keep her entertained. In doing so, she stumbled across Polish Hand Magic and tried it with her eight-year-old niece, Abbe. Her response? “Why don’t you just do it the easy way instead?”

Mentally Multiply by 5, 25, 50, 250

admin — Sat, 04 Jun 2011 16:02:56 +0000

This is a simple quicker math trick but it can be very useful for young students to solve seemingly difficult calculations. I will be glad to get your feedback on this.

Mental multiplication by 5, 25, 50, 250, 500 and so on.

Any number can be expressed in different ways. For example, 5 can be expressed as 10x(1/2).

Trick: Multiplication by 5

Step 1: Multiply the number by 10, i.e. simply place a zero after the number.

Step 2: Halve the resultant number.

Example 1:

5 × 136 = ten times of 136 i.e. 1360 should be divided by 2 = 1360/2 = 680

Example 2:

5 × 343, half of 3430 is 1715

Also check out, how to mentally multiply by 111?

Trick: Multiplication by 50, 500, 5000 and so on..

The above logic can also be applied to 50 which is half of 100 i.e. 1/2x100

Instead of multiplying the number by 10 we have to multiply by 100. So we have to place two zeroes after the number.

Example 1:

50 × 136 = hundred times of 136 i.e. 13600 should be divided by 2 = 13600/2 = 6800

Example 2:

50 × 647 = hundred times of 647 i.e. 64700 should be divided by 2 = 64700/2 = 32350

Same logic can be extended to numbers like 500, 5000 and so on.

Trick: Multiplication by 25

For multiplication by 25, we have to go one step further, 25=100 x (1/2) x (1/2)

Step 1: Multiply the number by 100, i.e. simply place two zeroes after the number.

Step 2: Halve the resultant number.

Step 3: Again halve the resultant number.

Example 1:

25 × 136 = hundred times of 136 i.e. 13600 should be divided by 2 = 13600/2 = 6800. Again halve this number, 6800/2 = 3400

Example 2:

25 × 343, half of 34300 is 17150. Again half of 17150 is 8575.

The above logic can also be extended to numbers like 250, 2500 and so on.

Let me know if you liked the above post by posting a comment below. You can also like it on facebook by using the button below or on the right side of the page.

How to convert from decimal to other number systems

admin — Sat, 12 Mar 2011 15:26:43 +0000

This post will be of special interest for people who are regularly in touch with mathematics. Students preparing for competitive examinations usually have Base System (Number Systems) in the list of their topics under quantitative aptitude. You can suggest any addition to the post below by posting a comment or mailing me at vineetpatawari[at]gmail[dot]com. If you have any queries post it as comment.

Conversion from decimal to binary and other number bases

In order to convert a decimal number into its representation in a different number base, we have to be able to express the number in terms of powers of the other base. For example, if we wish to convert the decimal number 100 to base 4, we must figure out how to express 100 as the sum of powers of 4.

100 = (1 x 64) + (2 x 16) + (1 x 4) + (0 x 1)

= (1 x 4^3) + (2 x 4^2) + (1 x 4^1) + (0 x 4^0)

Then we use the coefficients of the powers of 4 to form the number as represented in base 4:

100 = 1 2 1 0 base 4

Take another example; convert 117 into binary system –

Now since we have to convert 117 into binary we have to express 117 as the sum of the powers of 2. Obviously all the powers need to be less than 128 (=2^7)

117 = (1 x 64) + (1 x 32) + (1 x 16) + (0 x 8 ) + (1 x 4) + (0 x 2) + (1 x 1)

117 in decimal = 1110101 in binary

This method is less of calculation and more of application of mind and needs a lot of practice to master.

The other way to do this, which is more frequently used, is to repeatedly divide the decimal number by the base in which it is to be converted, until the quotient becomes zero. As the number is divided, the remainders - in reverse order - form the digits of the number in the other base.

Example: Convert the decimal number 82 to base 6:

Solution: 82/6 = 13 remainder 4

13/6 = 2 remainder 1

2/6 = 0 remainder 2

The answer is formed by taking the remainders in reverse order: 214 in base 6

In my next post, I will write about converting other number bases to decimal number system.

Author – Vineet Patawari