Now You Can Square Any Number Whose Tens Digit is 5 Without Using a Calculator
To begin with, I shall first introduce my canny friend, the
Rules.
It will be necessary to examine these rules in order to make your way
about this method simpler. Since the tens digit has already been
determined and defined to be 5, the next factor that would really affect
the result of the square would be the value of the hundreds digit. Now
this is where the rule comes in.
Step 1) Assuming the three-digit number to be squared is written in algebraic form,
100a +10b+c, where
b is defined as
5 and both
a and
c are any number from
1 - 9 and
0 - 9 respectively. First, we shall deal with
a as in the following:
a x
(a + 1); {this will give us the first one or two digits of the answer}
Step 2) Next we shall obtain the following third digit and so on by performing the following:
25 + (2a + 1) x
c; {this will give us the following digits of the answer. You may also assume that (
2a +1) is the same as adding the number that comes right after a to
a and multiplying that sum by the last the digit of the number to be squared}
Step 3) Finally, we determine the last two digit of the answer simply by obtaining the square of
c and read the results obtained in step1 and 2 followed by the one obtained in step 3 and that is the complete answer.
c2;
{gives us the last two digits of the answer, hence, it is better to
think that the last two digits are as easy in obtaining as finding the
square of the last digit of the number to be squared}
Example:
a) Find the square of 253.
1) 2 x (2+1) =
6
2) 25 + (2 x 2 +1) x 3 =
40
3) 3
2 =
09
4) Read
64009 (the answer!)
b) What is the square of 457?
1) 4 x 5 =
20 2) 25 + (4 x 2 +1) x 7 =
88
3) 7
2 =
49
4) Read
208849 (the answer!)
c) What is 959
2?
1) 9 x 10 =
90
2) 25 + (9 + 10) x 9 =
196
{note that the result is a three digit number, hence the value in the
hundreds digit is a carry digit that is added to the result in step 1}
3) 9
2 =
81 4) Read
919681 (the answer!)
Additional Tip: A shortcut to multiplying any single digit number by 19 would be as
simple as the following example showing 19 x 7:
Step1) Take the double of 7 and subtract it by one
That gives us,
14 - 1 = 13
Step 2) Subtract 7 from 10.
Which is,
10 - 7 = 3
Step 3) Read the result in step 1 followed by the result in step 2 and you will get
133, which is the answer.
Hence the general rule for
a x
19, would be:
(2a -1) x 10 + (10 - a) {this simply eliminates the hassle of having to deal with carries}
Or alternatively, you may also try; multiplying the number to be
multiplied by 19, by 20 instead and subtract that number again from the
product,
i.e: 7 x 20 - 7 = 133
