Question 1

The auto-rickshaw fare consists of a fixed charge together with the charge for the distance covered. For a journey of 10 km, the charge paid is ₹85 and for a journey of 15 km, the charge paid is ₹120. The fare for a journey of 25 km will be

Accepted Answer

Let use assume the fixed charge = ₹ a and charge for 1 km is ₹ = b According to question, for 10 KM journey charge paid = 85 a + 10 x b = 85 a + 10b = 85 .........................(1) for 15 KM journey charge paid = 120 a + 15b = 120.........................(2) Subtract the equation (1) from equation (2). we will get, a + 15b - a - 10b = 120 - 85 5b = 35 b = 7 Put the value of b in equation (1). we will get a + 10 x 7 = 85 a = 85 - 70 a = 15 Charges for 25 km = a + 25 x b Put the value of a and b in above equation. Charges for 25 km =15 + 25 x 7 = 15 + 175 = 190 Charges for 25 km =₹190

Question 2

The present ages of Vikas and Vishal are in the ratio 15:8. After ten years , their ages will be in the ratio 5:3. Find their present ages

Accepted Answer

Method 1 Let us assume the ratio factor is x. Therefor the present ages of Vikas and Vishal be 15x years and 8x years. After 10 years vikas's age = 15x + 10 and Vishal' age = 8x + 10 According to question, (15x+10)/(8x+10) = 5/3 ⇒ 3(15x + 10) = 5(8x + 10) ⇒ 45x + 30 = 40x + 50 ⇒ 5x =20 ⇒ x = 20/5 ⇒ x = 4 Therefore Present age of Vikas = 15x = 15 x 4 = 60 years and Present age of Vishal = 8x = 8 x 4 = 32 years Method 2 Let us assume the present age of Vikas = x years and Vishal's present year = y years According to question, Present age of Vikas/ Present age of Vishal = 15/8 ⇒ x/y = 15/8 ⇒ 8x = 15y ⇒ 8x - 15y = 0 ⇒ x = 15y/8 .................................(1) After 10 years Vikash age = x + 10 and vishal age = y + 10 Ratio of age after 10 years = 5/3 (x + 10)/(y + 10) = 5/3 ⇒ 3(x + 10) = 5(y + 10) ⇒ 3x + 30 = 5y + 50 ⇒ 3x - 5y = 50 - 30 ⇒ 3x - 5y = 20 ................................(2) Put the value of x from equation (1) in above equation (2). ⇒ 45y/8 - 5y = 20 ⇒ 45y - 40y = 20 x 8 ⇒ 5y = 20 x 8 ⇒ y = 4 x 8 = 32 Put the value of Y in equation (1) x = 15 x 32/8 = 15 x 4 = 60 Therefore Present age of Vikas = x = 60 years and Present age of Vishal = y = 32 years

Question 3

The sum of three consecutive multiples of 3 is 72. What is the largest number?

Accepted Answer

Method 1 Let us assume the number be 3p , 3(p+1) and 3(p+2) According to question, 3p + 3(p+1) + 3(p+2) = 72 ⇒ 3p + 3p + 3 +3p + 6 = 72 ⇒ 9p +9 = 72 ⇒ 9p = 72 - 9 ⇒ 9p = 63 ⇒ p = 63/9 = 7 ∴ Largest number = 3(p + 2) Put the value of p in above equation. ⇒ Largest number = 3 x ( 7 + 2 ) ⇒ Largest number = 3 x 9 ⇒ Largest number = 27 Method 2 Let us assume the numbers P , P + 3 ,P+ 6 According to Question, sum of three numbers = 72 P + P + 3 + P + 6 = 72 ⇒ 3P + 9 = 72 ⇒ 3P = 72 - 9 ⇒ 3P = 72 - 9 ⇒ P = 63/3 ⇒ P = 21 So largest Number = P + 6 = 21 + 6 = 27

Question 4

The sum of the ages of a father and his son is 4 times the age of the son. If the average age of the father and the son is 28 years, What is the son's age?

Accepted Answer

Let us assume father age is F and his son age is S. According to question, The sum of the ages of father and his son is 4 times the age of the son, F + S = 4S F = 3S.............. (1) Average age of father and son is 28. (F + S)/2 = 28 F + S = 56 Put the value of F from equation from (1), 3S + S = 56 4S = 56 S = 14 years Age of Son = 14 years.

Question 5

The product of two numbers is 192 and the sum of these two numbers is 28. What is the smaller of these two numbers?

Accepted Answer

Let the two numbers be x and y . According to question, ∴ xy = 192...................... (1) x + y = 28........................(2) As we know that, (x - y)2 = (x + y)2 - 4xy Put the value of xy and x + y from equation (1) and (2), we will get (x - y)2 = (28)2 - 4 x 192 ⇒ (x - y)2 = 784 - 786 ⇒ (x - y)2 = 16 ⇒ x - y = 4......................(3) Add the equation (2) and (3), we will get x + y + x - y = 28 + 4 2x = 32 x = 16 Put the value of x in equation (2), we will get 16 + y = 26 y = 28 - 16 y = 12 So numbers are x = 16 and y = 12. Smaller number is 12.

Question 6

The sum of three consecutive even numbers is 14 less than one-fourth of 176. What is middle number?

Accepted Answer

Let the middle number = x , then first number = x - 2 and last number = x + 2 (x - 2) + x + (x + 2) = 176/4 - 14 x - 2 + x + x + 2 = 44 - 14 3x = 30 x = 10 So middle number = x = 10

Question 7

The sum of three numbers is 300. If the ratio between first and second be 5:9 and that between second and third be 9:11, then second number is

Accepted Answer

Let us assume the ratio factor is x. Then the three numbers be 5x , 9x and 11x respectively. Then according to question, 5x + 9x + 11x = 300 ⇒ 25x = 300 ⇒ x = 12 So, the second number is 9x = 9 x 12 =108

Question 8

When 20 is subtracted from a number , it reduces to seven- twelve of the number. What is the sum of the digit of the number?

Accepted Answer

Let the number be x. According to question, x - 20 = 7x/12 ⇒ x - 7x/12 = 20 ⇒ (12x - 7x)/12 = 20 ⇒ 5x/12 = 20 ⇒ x = 20 x 12/5 ⇒ x = 4 x 12 ⇒ x = 48 Sum of the digits of 48 = 4 + 8 = 12

Question 9

Cost of 36 pens and 42 pencils is ₹ 460/- . What is the cost of 18 pens and 21 pencils?

Accepted Answer

Let us assume 1 pen cost be ₹ x and 1 pencil cost be ₹ y. According to question, The Cost of 36 pens and 42 pencils is 460. 36x + 42y = 460 ...................(1) then the cost of 18 pens and 21 pencils = 18x + 21y. Now divide the Equation (1) by 2. We will get, 18x + 21y = 230 So the Cost of 18 pens and 21 pencils = 18x + 21y = 230

Question 10

The cost of 5 pens and 8 pencils is ₹31.What would be the cost of 15 pens and 24 pencils?

Accepted Answer

Cost of 5 pens + 8 pencils = ₹31 On multiplying by 3 15 pens + 24 pencils = 3 x 31 = ₹15 pens + 24 pencils = 93

Question 11

If 2x + 3y = 29 and y = x + 3, what is the value of x?

Accepted Answer

2x + 3y = 29 ...(i) and y = x + 3 ...(ii) Putting the value of y from Eq. (ii) to Eq. (i), we get 2x + 3y = 29 2x + 3(x + 3) = 29 2x + 3x + 9 = 29 ⇒ 5x = 20 ∴ x = 4

Question 12

If 3x + 7y = 75 and 5x - 5y = 25, what is the value of x + y?

Accepted Answer

3x + 7y = 75 ..(i) 5x - 5y = 25 ∴ x - y = 5 ...(ii) On multiplying Eq. (ii) by 7 and adding to Eq. (i), we get 3x + 7y = 75 7x - 7y = 35 ------------------ 10x = 110 x = 11 ∴ Putting the value of x in Eq. (ii), we get 11 - y = 5 ∴ y = 6 ∴ x + y = 6 + 11 = 17

Question 13

If x + y - 7 = 0 and 3x + y - 13 = 0, then what is 4x 2 + y 2 + 4xy equal to?

Accepted Answer

we have, x + y - 7 = 0 ⇒ x + y = 7 ...(i) and 3x + y - 13 = 0 ⇒ 3x + y = 13 ..(ii) By subtracting Eq. (i) from Eq. (ii), we get 3x + y = 13 x + y = 7 ---------------------- 2x = 6 ...(ii) ∴ x = 3 On putting the value of x in Eq. (i), we get 3 + y = 7 ∴ y = 4 Now, 4x2 + y2 + 4xy = 4 x (3)2 + (4)2 + 4 x 3 x 4 = 4 x 9 + 16 + 48 = 36 + 16 + 48 = 100

Question 14

if 3 x + y = 81 and 81 x - y = 3, then what is the value of x?

Accepted Answer

Given, 3x + y = 81 ⇒ 3x + y = 34 ⇒ x + y = 4 ...(i) and 81x - y = 3 or (34)x - y = 3 ⇒ x - y = 1/4 On solving the Eqs. (i) and (ii), we get 2x = 17/4 ⇒ x = 17/8

Question 15

solve 6x + 3y = 7xy and 3x + 9y = 11xy.

Accepted Answer

Given equation are 6x + 3y = 7xy ⇒ (6/y) + (3/x) = 7 .....(i) And 3x + 9y = 11xy ⇒ (3/y) + (9/x) = 11 .....(ii) On multiplying Eq. (ii) by 2 and subtracting from Eq. (i), we get (6/y) + (3/x) = 7 (6/y) + (18/x) = 22 ------------------------- - 15/x = - 15 ∴ x = 1 On putting this value in Eq. (i), we get (6/y) + 3 = 7 ∴ 6/y = 4 ∴ y = 6/4 = 3/2

Question 16

Deepak has some hens and some goats. If the total number of animal heads is 90 and the total number of animal feet is 248, what is the total number of goats Deepak has?

Accepted Answer

Let hens = H, goats = G According to the question, H + G = 90 ...(i) 2H + 4G = 248 ...(ii) On multiplying Eq. (i) by 2 and subtracting from Eq. (ii) , we get 2H + 2G = 180 2H + 4G = 248 ------------------- -2G = -68 ∴ G = 34

Question 17

The sum of the two digits is 15 and the difference between them is 3. What is the product of the digits?

Accepted Answer

Let the number be x and y. Then, according to the question, x + y = 15 ...(1) x - y = 3 ...(ii) on adding Eqs. (i) and (ii), we get 2x = 18 ⇒ x = 9 On putting the value of in Eq. (i), we get y = 6 ∴ Product = xy = 54

Question 18

The numerator of a fraction is 6x + 1 and the denominator is 7 - 4x, x can have any value between -2 and 2, both included. The values of x for which the numerator is greater

Accepted Answer

6x + 1 > 7 - 4x ⇒ x > 3/5 ∴ 3/5 < x ≤ 2

Question 19

If (x/4) + (y/3) = 10/24 and (x/2) + y = 1, then the value of x + y is

Accepted Answer

Given, (x/4) + (y/3) = 10/24 ⇒ (3x + 4y)/12 = 10/24 ∴ 3x + 4y = 5 ...(i) and (x/2) + y = 1 x + 2y = 2 ...(ii) On multiplying Eq. (ii) by 2 and subtracting from Eq. (i). 3x + 4y = 5 2x + 4y = 4 - - - -------------------------- x = 1 On putting the value of x in Eq. (ii), we get x + 2y = 2 ⇒ 1 + 2y = 2 ⇒ y = 1/2 ∴ x + y = 1 + 1/2 = 3/2

Question 20

The cost of 21 pencil and 9 clippers is ₹ 819. What is the total cost of 7 pencils and 3 clippers together?

Accepted Answer

Let cost of 1 pencil and 1 clipper be p and c, respectively. Now, according to the question, 21p + 9c = ₹ 819 ⇒ 3(7p + 3c) = ₹ 819 ₹ 7p + 3c = ₹ 273 Cost of 7 pencil and 3 clippers = ₹ 273

Linear Equation Questions