Problem 1 :
The sum of three consecutive even natural numbers is 48. Find the greatest of these numbers.
Solution :
Let the three consecutive even natural numbers be x, x + 2, and x + 4.
As per the question,
x + x + 2 + x + 4 = 48
3x + 6 = 48
3x = 48 - 6
3x = 42
x = 14
Now,
The three consecutive even natural numbers are
14, 14 + 2, 14 + 4
14, 16, 18
So, the greatest of these number is 18.
Problem 2 :
The sum of three consecutive odd natural numbers is 69. Find the prime number out of these numbers.
Solution :
Let the three consecutive odd natural numbers be x, x + 2, and x + 4.
As per the question,
x + x + 2 + x + 4 = 69
3x + 6 = 69
3x = 69 - 6
3x = 63
x = 21
Now,
The three consecutive odd natural numbers are
21, 21 + 2, 21 + 4
21, 23, 25
So, the prime number among these three is 23.
Problem 3 :
The sum of three consecutive numbers is 156. Find the number which is a multiple of 13 out of these numbers.
Solution :
Let the three consecutive numbers be x, x + 1, and x + 2.
As per the question,
x + x + 1 + x + 2 = 156
3x + 3 = 156
3x = 156 - 3
3x = 153
x = 51
Now,
The three consecutive odd natural numbers are
51, 51 + 1, 51 + 2
51, 52, 53
52 is exactly divisible by 3.
So, 52 is the multiple of 13 among these three numbers.
Problem 4 :
Divide 54 into two parts such that one part is 2/7 of the other.
Solution :
Let the one part be x.
And the other part will be 2/7x.
By given question,
x + 2/7x = 54
(7x + 2x)/7 = 54
7x + 2x = 54(7)
9x = 378
x = 42
Now,
One part = x = 42
Other part = 2/7x = 2/7(42)
Other part = 12
So, 54 is divided into 42 and 12.
Problem 5 :
Sum of the digits of a two-digit number is 11. The given number is less than the number obtained by interchanging the digits by 9. Find the number.
Solution :
Let the digit at one's place = x
Then, the digit at ten's place = y
Given, the sum of the digits of a two-digit number is 11.
x + y = 11 ----(1)
The number with digits x and y represented as 10y + x.
Now, Interchange the number with digits x and y represented as 10x + y.
By given question,
(10x + y) - (10y + x) = 9
10x + y - 10y + x = 9
9x - 9y = 9
Dividing by 9 on both sides,
x - y = 1 ----(2)
Add (1) and (2), we get
x + y + x - y = 11 + 1
2x = 12
x = 6
By applying x = 6 in equation (1), we get
x + y = 11
6 + y = 11
y = 11 - 6
y = 5
Now,
Digit at one's place = x = 6
Digit at ten's place = y = 5
So, the required number is 56.
Problem 6 :
Two equal sides of a triangle are each 4m less than three times the third side. Find the dimensions of the triangle, if its perimeter is 55m.
Solution :
Let the third side of triangle be x m.
Then, two equal sides of triangle = (3x - 4)m
Given, perimeter of a triangle is 55 m.
Formula for the perimeter of a triangle,
P = a + b + c
55 = (3x - 4) + (3x - 4) + x
55 = 7x - 8
55 + 8 = 7x
63 = 7x
x = 9
Now,
Third side = x = 9 m
Two sides = (3x - 4) m
= (3(9) - 4) m
= 23 m
So, the dimensions of the triangle are 9m, 23m and 23m.
Problem 7 :
After 12 years, Kanwar shall be 3 times as old as he was 4 years ago. Find his present age.
Solution :
Let Kanwar present age be x year.
After 12 yrs, Kanwar age is (x + 12)yr
And 4 years ago, Kanwar age is (x - 4)yr
As per the question,
x + 12 = 3(x - 4)
x + 12 = 3x - 12
x - 3x = -12 - 12
-2x = -24
x = 12
So, the Kanwar present age is 12 year.
Problem 8 :
If 1/2 is subtracted from a number and the difference is multiplied by 4, the result is 5. What is the number ?
Solution :
Let x be the number.
As per the question,
(x - 1/2)4 = 5
4x - 2 = 5
4x = 5 + 2
4x = 7
x = 7/4
So, the number is 7/4.
Problem 9 :
The sum of four consecutive integers is 266. What are the integers ?
Solution :
Let the four consecutive integers be x, x + 1, x + 2, x + 3
As per the question,
x + x + 1 + x + 2 + x + 3 = 266
4x + 6 = 266
4x = 266 - 6
4x = 260
x = 65
Now,
x, x + 1, x + 2, x + 3
65, 65 +1, 65 + 2, 65 + 3
65, 66, 67, 68
So, the integers are 65, 66, 67, 68.
Problem 10 :
Find a number whose fifth part increased by 30 is equal to its fourth part decreased by 30.
Solution :
Let x be the number.
Assume one fifth part of the number is x/5.
And one fourth part of the number is x/4.
As per the question,
x/5 + 30 = x/4 - 30
(x + 150)/5 = (x - 120)/4
By cross multiplication,
4(x + 150) = 5(x - 120)
4x + 600 = 5x - 600
4x - 5x = -600 - 600
-x = -1200
x = 1200
So, the number is 1200.
May 21, 24 08:51 PM
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