Question 1

The value of k for which kx + 3y - k + 3 = 0 and 12x + ky = k, have infinite solutions, is

Accepted Answer

For infinite solution a1/a2 = b1/2 = c1/c2 ⇒ K/12 = 3/K = (-K + 3)/ -K ⇒ K/12 = 3/K ⇒ K2 = 36 ∴ K = √36 = 6

Question 2

In an examination, a student scored 4 marks for every correct answer and losses 1 mark for every wrong answer. A student attempted all the 200 question and scores 200 marks. Find the number of questions, he answer correctly.

Accepted Answer

Let the number of correct answer be x and number of wrong answer be y. Then, 4x - y = 200 ..(i) and x + y = 200 ..(ii) On adding Eqs. (i) and (ii), we get 4x - y = 200 x + y = 200 ---------------- 5x = 400 ∴ x = 80

Question 3

The sum of third , fourth and fifth part of a number exceeds half of the number by 34. Find the number?

Accepted Answer

Let us assume N be the given number. According to the question, Sum of third , fourth and fifth part of a number exceeds half of the number by 34. N/3 + N/4 + N/5 - 34 = N/2 N/3 + N/4 + N/5 - N/2 = 34 (20N + 15N + 12N - 30N)/60 = 34 17N/60 = 34 N = 34 x 60/17 N = 2 x 60 N = 120

Question 4

Krishna has some hens and some goats. If the total number of animal heads are 81 and the total number of animal legs are 234. How many goats does Krishna have?

Accepted Answer

Let us assume the number of Hens are H and Number of Goats are G. According to question, The total number of animal heads are 81. Since one animal has one head. Number of head of animals = Number of total animals H + G = 81..........................(1) The total number of animal legs are 234. Since hen's have 2 legs and goat's have 4 legs. 2H + 4G = 234...................(2) Multiply 2 with equation (1) and subtract from equation (2), we will get. ⇒ 2H + 4G - 2 x (H + G ) = 234 - 2 x 81 ⇒ 2H + 4G - 2H - 2G = 234 - 162 ⇒ 2G = 72 ⇒ G = 36 Put the value of G in equation (1), we will get. H + 36 = 81 H = 81 - 36 H = 45 Krishna have number of goats = G = 36

Question 5

The age of the father 5 years ago was 5 times the age of his son. At present the father's age is 3 times that of his son . What is the present age of the father?

Accepted Answer

Let us assume the present age of father = F year and Son's present age = S years According to question, 5 years ago, Father's age = F - 5 and Son's age = S - 5. According to the question, The age of the father 5 years ago was 5 times the age of his son. F - 5 = 5(S - 5) F - 5 = 5S - 25.....................(1) At present the father's age is 3 times that of his son. F = 3S.................................(2) Put the value of F from equation (2) in equation (1), we will get ⇒ 3S - 5 = 5S - 25 ⇒ 25 - 5 = 5S - 3S ⇒ 20 = 2S ⇒ 10 = S ⇒ S = 10. Put the value of S in Equation (2). we will get, F = 3S = 3 x 10 = 30 So the present Age of Father = 30.

Question 6

The number obtained by interchanging the two digits of a two-digit number is lesser than the original number by 54. If the sum of the two-digit number is 10, then what is the original number?

Accepted Answer

Let us assume the digits of the original number are unit's digit a and ten's digit b. The Original Number will be 10a + b. After interchanging the digits the new number will be 10b + a. According to question, The number obtained by interchanging the two digits of a two-digit number is lesser than the original number by 54. New Number = Original Number - 54 10b + a = 10a + b - 54 ⇒ 10b + a - 10a - b = -54 ⇒ 9b - 9a = -54 ⇒ a - b = 6....................................(1) Again according to question, Sum of the digits of original number = 10 a + b = 10..................................................(2) Add the equation (1) and (2), we will get a - b + a + b = 10 + 6 2a = 16 a = 8 Put the value of a in Equation (2) , we will get 8 + b = 10 b = 10 - 8 b = 2 Put the value of a and b for original number, we will get 10a + b = 10 x 8 + 2 = 80 + 2 = 82

Question 7

Smita was asked to multiply a certain number by 36. She multiplied it by 63 instead and got an answer of 3834 more than the correct one. What was the number to be multiplied?

Accepted Answer

Let us assume the certain Number is A. According to question, 63A = 3834 + 36A 63A - 36A = 3834 ⇒ 27A = 3834 ⇒ A = 142

Question 8

Ravi has spent a quarter (1/4) of his life as a boy , one fifth (1/5) as a youth , one third (1/3) as a man and thirteen (13) years in old age . What is his present age?

Accepted Answer

Suppose that the Ravi present age is A years. According to the given question, A/4 + A/5 + A/3 + 13 = A ⇒ (15A + 12A + 20A)/60 = A - 13 ⇒ 47A = 60A - 780 ⇒60A - 47A = 780 ⇒13A = 780 ∴ A = 780/13 = 60 years

Question 9

The ratio of the present ages of a mother and daughter is 7:1 . Four years ago the ratio of their ages was 19:1 . What will be the mother's age four years from now?

Accepted Answer

Method 1 Let us assume the present age of mother be M and daughter be D. According to question, Ratio of present age of mother and daughter = 7 : 1 M / D = 7 : 1 ⇒ M / D = 7 / 1 ⇒ M / D = 7 ⇒ M = 7D ..........................(1) Four years ago the age of mother = M - 4 Four years ago the age of daughter = D - 4 Again according to question, Four years ago ratio of mother age and daughter age = 19 : 1 ⇒ (M - 4 ) / (D - 4) = 19 : 1 ⇒ (M - 4 ) / (D - 4) = 19 ⇒ (M - 4 ) = 19(D - 4) ⇒ M - 4 = 19D - 76 .....................(2) Put the value of M from equation (1) in above equation (2), we will get 7D - 4 = 19D - 76 ⇒ 76 - 4 = 19D - 7D ⇒ 72 = 12D ⇒ 12D = 72 ⇒ D = 72/12 ⇒ D = 6 Put the vale of D in equation (1), we will get the present age of mother. Present age of Mother M = 7D = 7 x 6 Present age of Mother M = 42 ∴ mother's age after 4 yrs = 42 + 4 = 46 yrs Method 2 Let us assume the ratio factor is x. According to question, Ratio of present age of mother and daughter = 7 : 1 Then present age of mother = 7x and present age of daughter = x Four years ago the age of mother = 7x - 4 Four years ago the age of daughter = x - 4 Again according to question, Four years ago ratio of mother age and daughter age = 19 : 1 ⇒ (7x - 4 ) / (x - 4) = 19 : 1 ⇒ (7x - 4 ) / (x - 4) = 19 ⇒ (7x - 4 ) = 19(x - 4) ⇒ 7x - 4 = 19x - 76 ⇒ 76 - 4 = 19x - 7x ⇒ 72 = 12x ⇒ 12x = 72 ⇒ x = 6 Present age of Mother = 7x = 7 x 6 = 42 ∴ mother's age after 4 yrs = 42 + 4 = 46 yrs

Question 10

The ratio of two numbers is 4:7. If each of those numbers increased by 30, their ratio will become 5:8 . What is the average of these two numbers?

Accepted Answer

Let the numbers be x and y. According to question, The ratio of two numbers is 4:7. ⇒ x/y = 4/7 ⇒ 7x = 4y ∴ 7x - 4y = 0 ........................(1) Again According to question, If each of those numbers increased by 30, their ratio will become 5:8 (x + 30) / (y + 30) = 5/8 ⇒ 8(x + 30) = 5(y + 30) ⇒ 8x + 240 = 5y + 150 ⇒ 8x - 5y = -90 ..........................(2) Multiply 5 with equation (1) , we will get. 35x - 20y = 0 ..................(3) Multiply 4 with equation (2), we will get. 32x - 20y = -360 ............(4) Subtracts the Equation (4) from Equation (3). we will get, 35x - 20y - (32x - 20y) = 0 - (-360) ⇒ 35x - 20y - 32x + 20y = 360 ⇒ 3x = 360 ⇒ x = 120 Put the value of x in equation (1) to get the value of y. 7x - 4y = 0 ⇒ 7(120) - 4y = 0 ⇒ 840 - 4y = 0 ⇒ 4y = 840 ⇒ y = 210 ∴ Average of the numbers = (x + y)/2 put the vale of x and y. ∴ Average of the numbers = (120 + 210)/2 ∴ Average of the numbers = 330/2 ∴ Average of the numbers = 165

Question 11

A number of two digits has 3 for its unit's digit, and the sum of digits is 1/7 of the number itself, The number is

Accepted Answer

Let the ten's digit be x. Then, number = 10x + 3 and sum of digits = ( x + 3 ) So ,(x+3) = 1/7(10x + 3) ⇔ 7x + 21 = 10x + 3 ⇔ 3x = 18 ⇔ x=6 Hence, the number = 10x + 3 = (10 x 6) + 3 = 63.

Question 12

A number is doubled and 9 is added. If the resultant is tripled, it becomes 75. What is the number?

Accepted Answer

Let the number be N. According to question, A number is doubled and 9 is added then 2N + 9 Again According to question, If the resultant is tripled, it becomes 75. 3 ( 2N + 9 ) = 75 ⇒ 2N + 9 = 25 ⇒2N =16 ⇒ N = 8

Question 13

If the numerator of a fraction is increased by 200% and the denominator is increased by 200% then resultant fraction is 14/5.What is the original fraction?

Accepted Answer

Let us assume the original fraction be x/y. According to question, 200x/100 / 200y/100 = 14/5 2x/2y = 14/5 ⇒x/y = 14/5

Question 14

The sum of the two digits of a two-digit number is 15 and the difference between the two digits of the two digit number is 3. What is the product of the two digits of the two digit number?

Accepted Answer

Let the number be 10x + y where x > y According to the question. x + y = 15 ............(1) and x - y = 3 .........(2) Add the Equation (1) and (2) , We will get x + y + x - y = 15 + 3 ⇒ 2x = 18 ⇒ x = 9 Put the value of x in equation (1). We will get 9 + y = 15 ⇒ y = 15 - 9 ⇒ y = 6 So we have digits of number x = 9, y = 6 ∴ x * y = 9 * 6 = 54

Question 15

Joel purchased 40 notebooks at the rate of ₹18 per notebook and 55 pencils at the rate of ₹ 8 per pencil. What is the total amount that he paid to the shopkeeper?

Accepted Answer

According to question, 40 notebooks at the rate of ₹ 18 per notebook and 55 pencils at the rate of ₹ 8 per pencil. Amount paid = ₹ (40 x 18 + 55 x 8) Amount paid = ₹ (720 + 440) Amount paid = ₹ 1160

Question 16

Divide 50 into two parts so that the sum of their reciprocals is 1/2.

Accepted Answer

Let us assume the first part is A then second part will be 50 - A. According to question, Sum of their reciprocals is 1/2. 1/A + 1/(50 - A) = 1/12 ⇒ (50 - A + A)/(50 - A)A = 1/12 ⇒ (50)/(50 - A)A = 1/12 ⇒ (50) x 12 = 1 x (50 - A)A ⇒ 600 = (50 - A)A ⇒ 600 = 50A - A2 ⇒ A2 - 50A + 600 = 0 ⇒ A2 - 30A - 20A + 600 = 0 ⇒ A(A - 30) - 20(A - 30) = 0 ⇒ (A - 30)(A - 20) = 0 ⇒ (A - 30) = 0 or (A - 20) = 0 ⇒ A = 20 , 30

Question 17

The linear equation such that each point on its graph has an ordinate four times its abscissa is :

Accepted Answer

Let us assume the abscissa coordinate be x and Ordinate be y. According to question, ordinate four times its abscissa y = 4x ⇒ y - 4x = 0

Question 18

The cost of 2 sarees and 4 shirts is ₹ 16000 while 1 saree and 6 shirts cost the same. The cost of 12 shirts is

Accepted Answer

Let cost of one saree and shirt be x and y, respectively. 2x + 4y = 16000 ....(i) x + 6y = 16000 ....(ii) On multiplying Eq. (ii) by 2 and subtracting from Eq. (i). we get 2x + 4y = 16000 2x + 12y = 32000 ------------------------- -8y = -16000 ∴ y = 2000 Putting the value of y in Eq. (ii), we get x +6 x 2000 = 16000 ∴ x = 4000 ∴ Cost of 12 shirts = 12y = 12 x 2000 = ₹ 24000

Question 19

The system of equations 3x + y - 4 = 0 and 6x + 2y - 8 = 0 has

Accepted Answer

Given equations of system 3x + y = 4 ...(i) x + 2y = 8 ...(ii) Here, a1 = 3 , b2 = 2 and c2 = B ∵ a1/a2 = b1/b2 = c1/c2 = 1/2 So, the system of equations has infinite solutions, because it represents a parallel line.

Question 20

If 6x - 10y = 10 and x / (x + y) = 5/7, then (x - y) =?

Accepted Answer

Given, 6x - 10y = 10 ..........(i) and x/(x + y) = 5/7 ⇒ 7x = 5x + 5y ⇒ 2x - 5y = 0 ...(ii) On multiplying Eq. (ii) by 2 and subtracting from Ed.(i), we get 6x - 10y = 10 4x - 10y = 0 --------------------- 2x = 10 ∴ x = 5 Putting the value of x in Eq. (i), we get 30 - 10y = 10 ⇒ 10y = 20 ⇒ y = 2 ∴ (x - y) = 5 - 2 = 3

Linear Equation Questions