Chapter 6 Linear Inequalities Ex 6.1 Solutions
Question - 21 : - Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.
Answer - 21 : -
Let us assume, x be the marks obtained by Ravi in his third unit test
According to question, the entire students should have an average of at least 60 marks
(70 + 75 + x)/3 ≥ 60
= 145 + x ≥ 180
= x ≥ 180 – 145
= x ≥ 35
Hence, all the students must obtain 35 marks in order to have an average of at least 60 marks
Question - 22 : - To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.
Answer - 22 : -
Let us assume Sunita scored x marks in her fifth examination
Now, according to the question in order to receive A grade in the course she must have to obtain average 90 marks or more in her five examinations
(87 + 92 + 94 + 95 + x)/5 ≥ 90
= (368 + x)/5 ≥ 90
= 368 + x ≥ 450
= x ≥ 450 – 368
= x ≥ 82
Hence, she must have to obtain 82 or more marks in her fifth examination
Question - 23 : - Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.
Answer - 23 : -
Let us assume x be the smaller of the two consecutive odd positive integers
∴ Other integer is = x + 2
It is also given in the question that, both the integers are smaller than 10
∴ x + 2 < 10
x < 8 … (i)
Also, it is given in the question that sum off two integers is more than 11
∴ x + (x + 2) > 11
2x + 2 > 11
x > 9/2
x > 4.5 … (ii)
Thus, from (i) and (ii) we have x is an odd integer and it can take values 5 and 7
Hence, possible pairs are (5, 7) and (7, 9)
Question - 24 : - Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.
Answer - 24 : -
Let us assume x be the smaller of the two consecutive even positive integers
∴ Other integer = x + 2
It is also given in the question that, both the integers are larger than 5
∴ x > 5 ….(i)
Also, it is given in the question that the sum of two integers is less than 23.
∴ x + (x + 2) < 23
2x + 2 < 23
x < 21/2
x < 10.5 …. (ii)
Thus, from (i) and (ii) we have x is an even number and it can take values 6, 8 and 10
Hence, possible pairs are (6, 8), (8, 10) and (10, 12).
Question - 25 : - The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.
Answer - 25 : -
Let us assume the length of the shortest side of the triangle be x cm
∴ According to the question, length of the longest side = 3x cm
And, length of third side = (3x – 2) cm
As, the least perimeter of the triangle = 61 cm
Thus, x + 3x + (3x – 2) cm ≥ 61 cm
= 7x – 2 ≥ 61
= 7x ≥ 63
Now divide by 7 we get
= 7x/7 ≥ 63/7
= x ≥ 9
Hence, the minimum length of the shortest side will be 9 cm
Question - 26 : - A man wants to cut three lengths from a single piece of board of length 91cm.
The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second?
Answer - 26 : -
Let us assume the length of the shortest piece be x cm
∴ According to the question, length of the second piece = (x + 3) cm
And, length of third piece = 2x cm
As all the three lengths are to be cut from a single piece of board having a length of 91 cm
∴ x + (x + 3) + 2x ≤ 91 cm
= 4x + 3 ≤ 91
= 4x ≤ 88
= 4x/4 ≤ 88/4
= x ≤ 22 … (i)
Also, it is given in the question that, the third piece is at least 5 cm longer than the second piece
∴ 2x ≥ (x+3) + 5
2x ≥ x + 8
x ≥ 8 … (ii)
Thus, from equation (i) and (ii) we have:
8 ≤ x ≤ 22
Hence, it is clear that the length of the shortest board is greater than or equal to 8 cm and less than or equal to 22 cm.