PREFACE

PREFACE

DEAR FRIENDS,                               

                                WE DID WITH SHORT CUTS, TRICKS IN SOLVING MATHEMATICAL CALCULATION IN THE PROJECT MUCH OF THE INFORMATION PUBLISHED HERE ARE KNOWN TO MANY OF YOU, BUT DUE TO SHORT TERM MEMORY WE FORGET MANY THINGS. OUR ATTEMPT HERE IS TO BRING SUCH FORGOTTEN FACTS TO THE STUDENT COMMUNITY FOR DOING THE MATHEMATICAL CALCULATION IN A BETTER WAY.
                                                                                                                                                                                 THANK YOU

MEMBERS

1. ASHWIN

2. PRATHULYAN

3.SHASWAT 

4.SURAJ

5.MANO

MULTIPLICATION BY 9, 99, 999, ETC.

MULTIPLICATION BY 9, 99, 999, ETC.

There is another way to multiply fast by 9 that has an analogue for multiplication by 99, 999 and all such numbers. Let's start with the multiplication by 9.

To multiply a one digit number a by 9, first subtract 1 and form b = a - 1. Next, subtract b from 9: c = 9 - b.Then just write b and c next to each other:

9a = bc.

For example, find 6×9 (so that a = 6.) First subtract: 5 = 6 - 1. Subract the second time: 4 = 9 - 5. Lastly, form the product 6×9 = 54.

Similarly, for a 2-digit a:

bc	= 100b + c
	= 100(a - 1) + (99 - (a - 1))
	= 100a - 100 + 100 - a
	= 99a.

Do try the same derivation for a three digit number. As an example,

543×999	= 1000×542 + (999 - 542)
	= 542457.

MULTIPLIPLY, THEN SUBTRACT.

MULTIPLIPLY, THEN SUBTRACT.

When multiplying by 9, multiply by 10 instead, and then subtract the other number. For example,

23×9 = 230 - 23 = 207.

The same applies to other numbers near those for which multiplication is simplified:

	23×51	= 23×50 + 23
		= 2300/2 + 23
		= 1150 + 23
		= 1173.
	87×48	= 87×50 - 87×2
		= 8700/2 - 160 - 14
		= 4350 - 160 - 14
		= 4190 - 14
		= 4176.

FASTER ADDITION.

FASTER ADDITION.

Addition is often faster in two steps instead of one.

For example,

487 + 38 = (487 + 13) + (38 - 13) = 500 + 25 = 525.

A generic advice might be given as "First add what's easy, next whatever remains". Another example:

1049 + 187 = 1100 + (187 - 51) = 1200 + 36 = 1236.

FASTER SUBTRACTION.

FASTER SUBTRACTION.

Subtraction is often faster in two steps instead of one.

For example,

427 - 38 = (427 - 27) - (38 - 27) = 400 - 11 = 389.

A generic advice might be given as "First remove what's easy, next whatever remains". Another example:

1049 - 187 = 1000 - (187 - 49) = 900 - 38 = 862.

PRODUCT OF NUMBERS CLOSE TO 100

PRODUCT OF NUMBERS CLOSE TO 100

Say, you have to multiply 94 and 98. Take their differences to 100: 100 - 94 = 6 and 100 - 98 = 2. Note that94 - 2 = 98 - 6 so that for the next step it is not important which one you use, but you'll need the result: 92. These will be the first two digits of the product. The last two are just 2×6 = 12. Therefore, 94×98 = 9212.

PRODUCT OF TWO 2-DIGIT NUMBERS.

PRODUCT OF TWO 2-DIGIT NUMBERS.

The simplest case is when two numbers are not too far apart and their difference is even, for example, let one be 24 and the other 28. Find their average: (24 + 28)/2 = 26 and half the difference (28 - 24)/2 = 2. Subtract the squares:

28×24 = 26² - 2² = 676 - 4 = 672.

The ancient Babylonian used a similar approach. They calculated the sum and the difference of the two numbers, subtracted their squares and divided the result by four. For example,

	33×32	= (65² - 1²)/4
		= (4225 - 1)/4
		= 4224/4
		= 1056.

PRODUCT OF TWO ONE-DIGIT NUMBERS GREATER THAN 5.

PRODUCT OF TWO ONE-DIGIT NUMBERS GREATER THAN 5.

This is a rule that helps remember a big part of the multiplication table. Assume you forgot the product 7×9. Do this. First find the excess of each of the multiples over 5: it's 2 for 7 (7 - 5 = 2) and 4 for 9 (9 - 5 = 4). Add them up to get 6 = 2 + 4. Now find the complements of these two numbers to 5: it's 3 for 2 (5 - 2 = 3) and 1 for 4(5 - 4 = 1). Remember their product 3 = 3×1. Lastly, combine thus obtained two numbers (6 and 3) as63 = 6×10 + 3.

SQUARES OF NUMBERS THAT END WITH 5.

SQUARES OF NUMBERS THAT END WITH 5.

A number that ends in 5 has the form A = 10a + 5, where a has one digit less than A. To find the square A² of A, append 25 to the product a×(a + 1) of a with its successor. For example, compute 115². 115 = 11×10 + 5,so that a = 11. First compute 11×(11 + 1) = 11×12 = 132 (since 3 = 1 + 2). Next, append 25 to the right of 132 to get 13225!

SQUARES OF NUMBERS FROM 51 THROUGH 99.

SQUARES OF NUMBERS FROM 51 THROUGH 99.

If A is between 50 and 100, then A = 50 + x. Compute a = 50 - x. Then A² = a² + 200x. For example,

63² = 37² + 200×13 = 1369 + 2600 = 3969.

Any Square.
Assume you want to find 87². Find a simple number nearby - a number whose square could be found relatively easy. In the case of 87 we take 90. To obtain 90, we need to add 3 to 87; so now let's subtract 3 from 87. We are getting 84. Finally,

87² = 90×84 + 3² = 7200 + 360 + 9 = 7569.

SQUARES OF NUMBERS FROM 26 THROUGH 50.

SQUARES OF NUMBERS FROM 26 THROUGH 50.

Let A be such a number. Subtract 25 from A to get x. Subtract x from 25 to get, say, a. Then A² = a² + 100x. For example, if A = 26, then x = 1 and a = 24. Hence

26² = 24² + 100 = 676.

CALCULATE ANY PERCENTAGE FAST!!

So now you can solve 5%, 10%, 15%, and 50% percentage problems with no trouble at all. Congratulations, you are already far ahead of most people! We can now take this skill and use it to calculate any common percentage! Most percentages are either multiples of 10 or 5. You’ve seen the stores that offer 75% off or even 90% off, now we’ll take our skill and use it to quickly solve these easy percentage problems.

95% of 2500 =

• Break 95% into 100% – 5%
• 100% of any number is itself, so 2500
• 5% is half & 10% -> 2250 -> 225
• Put them back together: 100% – 5% –> 2500 – 225 = 2375

75% of 880 =

• Split 75% into 50% + 25%
• 50% is half, so 880 ÷ 2 = 440
• 25% is half of 50%, so take your previous answer and half it again!  440 ÷ 2 = 220
• Put them back together: 440+220=660

This simple technique will work for any  percentage problem that ends in 0 or 5, even those greater than 100%!! The greatest part is that we can now use this same breaking up skill to calculate ANY percentage problem.

USING 5’S AND 10’S TO SOLVE 15% – THE TRADITIONAL ‘TIP’

Now we are getting into the meat of this technique. After this example we’ll start learning how to solve ANY percentage. I’ve included this example first, because it will likely be the one you use most in everyday life. Not only that, but it makes a great segue into the full technique.

What is 15%? It’s nothing more than 10% +5%!

15% of Rs 24 =

• Split 15% into 10%+5%
• Take 10% of 24 (by moving the decemal one place) = 2.4
• Take 5% of 24 (which is the same as half of 10%) = 1.2
• Add the 5% and the 10% together to get 3.6

15% of Rs 24 = Rs 3.60 EASY!!

CALCULATING PERCENTAGES FOR TIP – MENTALLY.

10% is by far the simplest, so let’s start there. Don’t worry, we’ll get to hard ones soon!

10% of 285 = 28.5

All we did was move the decimal from 285. to 28.5!

HOW TO CALCULATE 5%

Calculating 5% is almost the same as finding 10%, except the number is half the size. There are only two steps, and they can be done in either order!

• Find 10% (by moving the decimal place one digit over)
• Half the number (by dividing by 2)

5% of 264 = 13.2

• 10% of 264 = 26.4
• half of that is 13.2

PERCENTAGE & DECIMAL REVIEW

Before we jump into this exciting class, we need to fully understand how to work with decimals.  As you may recall, percentages are nothing more just numbers behind a decimal.

• 50% is the same as 0.5
• 25% is the same as 0.25
• 1% is the same as 0.01
• 125% is 1.25

100% is the whole thing. If you eat an entire pie, you eat 100% of the pie. If you eat half the pie, you eat 50% of the it. It’s pretty simple really.

Moving decimals is just as simple. Take 10% of 100 and you get 10. Why? Well, 10% is one decimal place. 1% of 100 is 1, because you moved the decimal place two places over.  

DIVISIBILITY RULES

DIVISION TRICKS

DIVISION BY 2

WHEN YOU WANT TO DIVIDE A NUMBER BY 2 JUST MULTIPLY THE NUMBER BY 5 THEN INSERT A DECIMALPOINT LEAVING ONE DIGIT
    FOR EXAMPLE  453WHEN DIVIDED BY 2 YOU GET 227.5(453X5=2275 AND JUST INSERT A DECIMAL LEAVING ONE DIGIT)



DIVISION BY 5
                                   WHEN YOU WANT TO DIVIDE ANY NUMBER BY 5 JUST MULTIPLY THE NUMBER BY 2 AND THEN INSERT A DECIMAL POINT LEAVING ONEDIGIT
  FOR EXAMPLE 543WHN DIVIDED BY 5 YOU GET 108.6 (543X2=1086 AND JUST INSERT A DECIMAL LEAVING ONE DIGIT)

DIVISION BY 20

                                  WHEN YOU WANT TO DIVIDE ANY NUMBER BY 20 JUST MULTIPLY THE NUMBER BY 5 AND THEN INSERT A DECIMAL POINT LEAVING TWO DIGITS
  FOR EXAMPLE 543WHN DIVIDED BY 20 YOU GET 27.15 (543X5=2715 AND JUST INSERT A DECIMAL LEAVING TWO DIGITS

DIVISION BY 25
                  

                     WHEN YOU WANT TO DIVIDE ANY NUMBER BY 25 JUST MULTIPLY THE NUMBER BY 4 AND THEN INSERT A DECIMAL POINT LEAVING TWO DIGITS
  FOR EXAMPLE 543WHN DIVIDED BY 25 YOU GET2172 (543X4=2172 AND JUST INSERT A DECIMAL LEAVING TWO DIGITS)

DIVISION BY50
                  

                     WHEN YOU WANT TO DIVIDE ANY NUMBER BY 50 JUST MULTIPLY THE NUMBER BY2AND THEN INSERT A DECIMAL POINT LEAVING TWO DIGITS
  FOR EXAMPLE 543WHN DIVIDED BY 50YOU GET10.86 (543X2=1086AND JUST INSERT A DECIMAL LEAVING TWO DIGITS)