Probability Shortcuts
Squaring the numbers - Shortcuts
If there are two digits then,
Step-1: we write the first two natural numbers (i.e., 1 and 2)
Step-2:then we write reverse
Example 112= 12 (step-1,we write the first two natural numbers)
= 121 (step-2,then we write reverse)
If there are three digits then we write first three natural numbers and then we write reverse.
Example 1112 = 123 (step-1,we write the first three natural numbers)
= 12321 (step-2,then we write reverse)
If there are four digits then we write first four natural numbers and then we write reverse.
11112= 1234 321
Squaring a number with unit digit as 5:
Let’s say the number is R5
Now, the trick is R52 = R (R+1)/25
(here, the sign '/' does not mean division, instead it means that we have to always write 25 on the right hand side of the answer)
252 = 2 (2+1)/25
= 625
352 = 3 (3+1)/25
= 1225
1152 = 11 (11+1)/25
= 11*12/25
= 13225
Let’s say the number is R5
Now, the trick is R52 = R (R+1)/25
(here, the sign '/' does not mean division, instead it means that we have to always write 25 on the right hand side of the answer)
252 = 2 (2+1)/25
= 625
352 = 3 (3+1)/25
= 1225
1152 = 11 (11+1)/25
= 11*12/25
= 13225
Squaring a number nearer to 10n:
Let’s say you have to find the value of 962
First, find how far is 96 from 100. It is 4
digits far.
Now, write it as
962 = (96+4) (96-4)
+ 42
= 100*92 + 16
= 9200 + 16
= 9216
If you practice this squaring trick well, the whole calculation
can be done mentally, without using a pen.
Let’s take few more examples……………………
1022 =
(102 + 2) (102– 2) + 22
= 104 * 100 + 4
= 10404
9952 = (995
+ 5) (995– 5) + 52
= 1000 * 990
+25
= 990000 + 25
= 990025
10082
= (1008 + 8) (1008 - 8) + 82
= 1016 *
1000 + 64
= 1016064
Universal rule of squaring any number:
(ab)2 = a2
/ 2ab/b2
342 = 32 / 2*3*4/42
= 9/24/16
= 1156
More solved problems based on Ages
Ex.2
At present the age of father is 5 (Y) times that of
the age of his son. After 3 (t) years, the
father’s age would be 4 (x) times the age of his son. Find the present
ages of the father and the son?
Solution by conventional method
Let the present age of son = x yrs.
then, the present age of father = 5x yrs.
After, 3 years,
4(x+3) = 5x+3
or, 4x+12=5x-3
x = 9 yrs.
So, the son's age is 9 yrs.
Father's age = 45 yrs.
Shortcut Method:
Son's age = t(x-1) / (x-y)
The value of x can be identified easily as it will be given it the same sentence where you find value of t.
In the above formula the denominator must results in a positive value. If you think the value will come negative then reverse the denominator as (y-x)
I mean you have to always subtract the greater no. from the smaller no. so that the value obtained is always positive.
In the above example
t= 3 yrs.
x = 4
y= 5
Putting the values in the above equation
Son’s age = 3 (4-1) / 5-4
=9
and father’s age = 45 yrs.
Ex-3
Three years (t1) earlier
the father was 7 times (x) as old as his
son. Three years (t2) hence, the father’s age would be 4
times (z)
that of his son. What is the present age of the son?
Solution by conventional method
Let the present age of son = x yrs.
then, the present age of father = y yrs.
3 years earlier,
7(x-3) =y-3
or, 7x-y=18 ………. (1)
After, 3 years,
4(x+3) = y+3
or, 4x+12=y+3
or, 4x-y=-9………...(2)
Solving (1) and (2) we get, x = 9 yrs. and y = 45 yrs.
Shortcut Method
Son’s age = [t2 (z-1) + t1 (x-1)]/ (x-z)
= [3(4-1) + 3(7-1)]/ (7-4)
= (9+18)/3
= 27/3
= 9
Try yourself:
1. The age of a man is 4 times that of his son. Five years
ago the man was 9 times as old as his son was at that time. What is the present
age of the man? (Answer- 42 yrs.)
2. After 5 years the age of a father will be thrice the age
of his son, whereas 5 years ago, he was 7 times as old as his son was. What are
their present ages? (Son’s age=10 yrs. and Father’s age= 40 yrs.)
If you have any doubt/suggestion, feel free to ask in the below comment box.
Shortcuts for Age based problems
Have a look at the following questions
Let the present age of son = x yrs.
then, the present age of father = 5x yrs.
3 years ago,
7(x-3) = 5x-3
or, 7x-21=5x-3
or, 2x=18
x = 9 yrs.
So, the son's age is 9 yrs.
Father's age = 45 yrs
Son's age =t(x-1) / x-y
In the above example
t= 3 yrs.
X = 7
Y = 5
Putting the values in the above equation
Son’s age = 3 (7-1) / 7-5
= 18/2
=9
And father’s age = 45 yrs.
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