INTRODUCTION
CUTTING CORNERS
Whether due to curiosity or sheer laziness , man has always been expeimenting , searching for and stumbling up on ways of making work easier for himseif . That anony mous caveman who chipped the corners off a flat rock and invented the wheel started this tradition .
Most of man’ s efforts in the past were directed at con- serving or increasing his muscle power’ but as time went on some were aimed at saving weat and teat on another vital organ; his brain. It followed naturally that his attention turned to reducing such laborious tasks as calculating.
WHAT SHORT CUTS ARE
Short cuts in mathematics are ingenious little tricks in calculating that can save enormous amounts of time and labour _ not to mention paper – in solving otherwise complicated problems. There are no magical powers connected with these tricks: each is based on sound mathematical principals growing out of the very properties of numbers themselves .The results they produce are absoulutly accurate and inflallible when applied correctly .Short-cut methods are by no means of recent origin:they were known even to the ancient Greeks. The supply of Short-cuts is unlimitted. Many are known ,and many are yet to be discovered. The all shortcuts included in this page have been selected because they are easy to learn,simple to use and can be applied to the widest range of calculating problems.
PUTTING NUMBERS IN THEIR PLACE
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 are called digits. Integers are numbers consisting of one or more digits. For example, 72,958 is an integer consisting of five digits, 7,2, 9, 5, and 8. In practice, the word number is applied to many different combinations of digits ranging from whole numbers, to fractions, mixed numbers, and decimals. The word integer, however, applies only to whole numbers.
Each digit in a number has a name based on its position in the number. The number system we are accustomed to dealing with is based on the number 10. Each number position in this system is named for a power of 10. The position immediately to the left of the decimal point of a number is called the units position. In the number 1.4 the digit 1 is in the units position and is called the units digit. In fact, any digit that occupies that position is called the units digit. The next position to the left of the units position is called the tens position, and any digit occupying that space is called the tens digit. In the number 51.4 the 5 is the tens digit. Continuing to the left, in order, are the hundreds, thousands, ten-thousands, hundred-thousands, millions positions, and so on.
The positions of the digits to the right of the decimal point also have names similar to those to the left. The position immediately to the right of the decimal point is called the tenths position. Notice that the name is tenths and not tens. In fact, all positions to the right of the decimal point end in ths. The next position to the right of the tenths position is the hundredths position, then the thousandths position, and, in order, the ten-thousandths, the hundred-thousandths, the millionths.
A. When multiplying by 11 there are certain steps you need to follow:
1. Write down the last digit of the number.
2. Add the last digit to the number to its left. Write this number down, carry if necessary.
3. Moving left, keep adding the digit to the digit to its left. Write down the numbers, carrying if necessary
4. When you reach the first digit write this number down.
Ex 1] 1752x 11 =_________
a) Write down the 2.
b) 2 + 5 = 7. Write down 7.
c) 5 + 7 = 12. Write down 2 and carry the *1.
d) 7 + 1 = 8+ *1 =9. Write down the 9.
e) Write down the 1 since there is nothing to carry.
f)The answer is 19272.
A. You can multiply by 12, 13, 14, 15, 16, 17, 18, or 19 the same way. For case of writing, these numbers will be referred to as la, where a is the last digit.
1. When multiplying "n" x la, first multiply a by the last digit of "n" . Write this number down, carry if necessary.
2. Now multiply the remaining digits by a and add back to "n". Write this result.
Ex [1] 52 x 17 = ---------
a) 7 x 2 = 14 so write 4 and carry 1.
b) 7 x 5 = 35 so add 35 + 52 = 87 + *1 = 88. Write 88.
c) The answer is 884.
Ex 12] 84 x 16 = ----------
a) 6 x 4 = 24 so write 4 and carry *2.
b) 6 x 8 = 48 + *2 = 50. 50 + 84 = 134. Write 134.
c)The answer is 1344.
Ex 13] 113 x 12 = ---------
a) 2 x 3 = 6 so write 6.
b) 2 x 11 = 22. 22 + 113 = 135. Write 135.
c) The answer is 1356.
B. Sometimes, as in Ex [|2], it is easier to add the carried
number first, before adding back to "n"
4)
Multiplying By 25:
A. Multiplying by 25 is one of the basic multiplication methods in number sense.
1. You can rewrite 25 to be 100/4. So the first step is to divide by 4 and write
this number down.
2. Depending on the remainder,referred to as (n MOD 4), the last numbers are
the following:
a) If (n MOD 4) = 0 then the last numbers are 00.
b) If (n MOD 4) = I then the last numbers are 25.
c) If (n MOD 4) = 2 then the last numbers are 50.
d) If (n MOD 4) = 3 then the last numbers are 75.
B. Examples:
Ex [1] 25 x 84 = -----------
a) 84/4 = 21. Write 21.
b) Since there is no remainder the last numbers are 00.
c) Tne answer is 2100.
Ex [2] 113 x 25 = ----------
a) 113/4 = 18 with a remainder of 1. Write 18.
b) The last numbers are 25.
c)The answer is 1825.
5)
Multiplying by 50
A. Multiplying by 50 is the same as multiplying by 5, except if the number is even you
write 00, and if the number is odd you write 50. (See Multiplying by 5)
1. When multiplying by 50, it is easier to divide by 2 and multiply by 100.
2. If the number you are multiplying is odd, then the last numbers will be a 50
(see Ex [2), otherwise the last number is 00.
B. Examples:
Ex [1] 126 x 50 = -------
a) 126 / 2 = 63. Write 63.
b) Since 126 is even the last numbers are 00.
c)The answer is 6300.
Ex [2] 321 x 50 = --------
a) 321 / 2 = 160 with a remainder of 1. Write 160.
b) Since there is a remainder of 1 the last numbers are 50.
c) The answer is 16050.
6)
Multiplying By 75
A. Many people use this method to multiply by 75. However, some people prefer to use the FOlL Method or the Double and Half Method. It is up to the individual's preference.
1. Think of 75 as being (3/4) x 100.
2. Divide by 4.
3. Multiply by 3, and then multiply by 100
4. If there is a remainder (r). then acd 75r to the answer.
B. Examples
Ex [ 1 ] 84 x 75 = ------
a) 84 / 4 = 21.
b) 21 x 3 = 63. 63 x 100 = 6300.
c) The answer is 6300.
Ex [2] 123 x 75 = ------
a) 123 / 4 = 30 with a remainder of 3.
b) 30 x 3 = 90. 90 x 100 = 9000.
c) Since 3 is the remainder multiply 75 x 3= 225 and add to 9000 or 9225.
d) The answer is 9225.
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