Here are some solved examples of Problems on Numbers :
Example. 1 : The sum of a rational number and its reciprocal is 13/6. Find the number?
Solution :
Let the number be, x.
Then, according to the question
x + 1/x = 13/6
or, (x2 + 1)/x = 13/6
or, 6x2 + 6 = 13x
or, 6x2 - 13x + 6 = 0
or, 6x2 - (9+4)x + 6 = 0
or, 6x2 - 9x - 4x +6 = 0
or, 3x (2x - 3) - 2 (2x - 3) = 0
or, (2x - 3) (3x - 2) = 0
We know, if multiplication of two numbers is 'zero' then the value of each number is equal to 'zero'.
so, (2x - 3) = 0 | (3x - 2) = 0
or, 2x = 3 | or, 3x = 2
or, x = 3/2 | or, x = 2/3
The required number is 3/2 or 2/3
Example. 2 : The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number ?
Solution :
Let the unit's digit be a and ten's digit be b
Then, the number = (10b + a)
After interchanging the position the number = (10a + b)
Now, according to the question (10b + a) - (10a + b) = 36
or, (10b + a - 10a - b) = 36
or, 9b - 9a = 36
or, 9 (b - a) = 36
or, b - a = 4
Example. 3 : If one-third of a number is 48. Then find 70 % of that number ?
Solution :
Let, the number be x
so, x/3 = 48
or, x = 48 × 3 = 144
therefore, 144 of 70% = (144 × 7)/10 = 1008/10 = 100.8
Example. 4 : The ratio between a two-digit number and the sum of the digits of that number is 3 : 1. If the digit in the unit's place is 5 more than the digit in the ten's place, then find the number ?
Solution :
Let, the ten's digit be x. then unit's digit = (x + 5)
.·. Number = 10x + (x + 5) = 10x + x + 5 = 11x + 5
and sum of digits = x + (x + 5) = 2x + 5
According to the question (11x + 5) : (2x + 5) = 3 : 1
or, 1 (11x + 5) = 3 (2x + 5)
or, 11x + 5 = 6x + 15
or, 11x - 6x = 15 - 5
or, 5x = 10
or, x = 2
.·. required number = 10 × 2 + (2 + 5) = 20 + 7 = 27
Example. 5 : The difference between a number and its three-fifth is 50. Find the number ?
Solution :
Let the number be x. Then its three-fifth = 3x/5
.·. according to the question, x - 3x/5 = 50
or, (5x - 3x)/5 = 50
or, 2x = 50 × 5
or, x = 125
Example. 6 : A two-digit number is such that the product of the digits is 8. When 18 is added to the number, then the digits are reversed. Find the number ?
Solution :
Let, the ten's digit be x and unit's digit be 8/x
.·. The number = (10x + 8/x)
According to the question, (10x + 8/x) + 18 = 10 × 8/x + x
or, 10x2 + 8 + 18x = 80 + x2
or, 9x2 + 18x - 72 = 0
or, x2 + 2x - 8 = 0
or, x2 + 4x - 2x - 8 = 0
or, x(x + 4) -2(x + 4) = 0
or, (x + 4)(x - 2) = 0
We know, if multiplication of two numbers is 'zero' then the value of each number is equal to 'zero'.
so, (x + 4) = 0 | (x - 2) = 0
or, x = -4 | or, x = 2
Here we take x = 2
.·. The required number = 20 + 8/2 = 20 + 4 = 24
Example. 7 : Twenty times a positive integer is less than its square by 96. Find the integer ?
Solution :
Let the integer be, X.
According to the question, 20 × X = X2 - 96
or, X2 - 20X - 96 = 0
or, X2 - (24 - 4) - 96 = 0
or, X2 - 24X + 4X - 96 = 0
or, X (X - 24) + 4 (X - 24) = 0
or, (X - 24) (X + 4) = 0
We know, if multiplication of two numbers is 'zero' then the value of each number is equal to 'zero'.
(X - 24) = 0 | (X + 4) = 0
or, X = 24 | or, X = -4
Here we take X = 24
So, the required integer is 24.
Example. 8 : The difference between two numbers is 7. If their product is 1548. Then find the sum of two numbers ?
Solution :
Let, the two numbers be X and Y.
According to question (X - Y) = 7 and XY = 1548.
We know that, (X + Y)2 = (X - Y)2 + 4XY
or, (X + Y)2 = (7)2 + 4 × 1548 = 49 + 6192 = 6241
or, (X + Y) = √(6241) = 79
.·. sum of two numbers is 79
Example. 9 : If the difference between two numbers is 3 and the difference between their squares is 39, then find the larger number ?
Solution :
Let, the larger number be X.
So, the smaller number = (X - 3)
According to the question, (X)2 - (X - 3)2 = 39
or, X2 - (X2 - 6X + 9) = 39
or, X2 - X2 + 6X - 9 = 39
or, 6X = 39 + 9
or, 6X = 48
or, X = 48/6 = 8
.·. The larger number is 8
Example. 10 : In a two-digit number, the digit in the unit's place is four times the digit in ten's place and sum of the digits is equal to 10. Find the number ?
Solution :
Let the the digit in ten's place be X.
Then the digit in unit's place = 4X
So, the number = 10X + 4X = 14X
According to the question, X + 4X = 10
or, 5X = 10
or, X = 2
.·. The number = 14 × 2 = 28
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