Video on Problems on Numbers - shortcuts, tips and tricksProblems on Numbers
Basic Formulae: (Must Remember) 1) (a - b)
2 = (a
2 + b
2 - 2ab)
2) (a + b)
2 = (a
2 + b
2 + 2ab)
3) (a + b) (a – b) = (a
2 – b
2 )
4) (a
3 + b
3) = (a + b) (a
2 – ab + b
2)
5) (a
3 - b
3) = (a - b) (a
2 – ab + b
2)
6) (a + b + c)
2 = a
2 + b
2 + c
2 + 2 (ab + bc + ca)
7) (a
3 + b
3 + c
3 – 3abc) = (a + b + c) (a
2 + b
2 + c
2 – ab – bc – ac)
Question VarietyType 1: Find the numbers, if we are given
a) Sum of squares/sum of products
b) Sum/average of consecutive numbers
If the sum two numbers is 31 and their product is 240, then find the absolute difference between the numbers.
a. 1
b. 3
c. 4
d. 5
View solution Correct Option: (a)
Let two numbers be x and y
We are given that, sum of two numbers x + y = 31 and product = xy = 240
Therefore,
x – y = (x + y)2 – 4xy
Substituting the values, we get
x – y = (31)2 – 4 × y
= 961 – 960
= 1
= 1
The required difference between the numbers is 1.
Find the three consecutive odd numbers whose sum of the squares is 2531.
a. 19, 21, 23
b. 23, 25, 27
c. 27, 29, 31
d. 31, 33, 35
View solution Correct Option: (c)
Let three consecutive odd numbers be x, x+2, x+4.
x2+(x+2)2+(x+4)2=2531
Simplifying we get,
x2+4x-837=0
27 x 31 = 837 and also the difference between 27 and 31 is 4
Therefore,
x2+31x-27x-837=0
(x + 31) (x – 27)
X = 27 or x = –31
Hence, the value of
x = 27
(x+2) = 27 + 2 = 29
(x+4) = 27 + 4 = 31
The sum of squares of three numbers is 138 and the sum of their products taken two at a time is 131. Find their sum.
a. 35
b. 42
c. 20
d.18
View solution Correct Option: (c)
Let the three numbers be x, y and z.
Given: Sum of squares of three numbers is 138 and sum of their products taken two at a time is 131
Therefore,
x2+y2+z2=138
xy+yz+zx=131
Formula: (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)
This formula can be used to easily find the sum of three numbers. Substituting the values, we get
(x + y + z) 2 = x2 + y2 + z2 + 2 (xy + yz + zx)
(x + y + z ) 2 = 138 + 2(131)
(x + y + z ) 2 = 400
Hence, (x + y + z) = 20
The average of 3 consecutive even numbers is 18, find the largest of these numbers.
a. 15
b. 16
c. 20
d. 26
View solution Correct Option: (c)
Let the numbers be x, x +2, and x + 4
Hence, sum of these numbers = (average x no. of numbers) = (18 x 3) = 54
x + (x +2) + (x + 4) = 54
3x = 48
x = 16
Largest number = (x + 4) = (16 + 4) = 20
