Question 1

Solve: 2x + 3x = 35

Accepted Answer

Step 1: Combine like terms on the left side: 2x + 3x = 5x. Step 2: Now the equation is 5x = 35. Step 3: Divide both sides by 5: x = 7. Therefore, x = 7.

Question 2

If 5x + 12 = 37, find the value of x.

Accepted Answer

Step 1: Start with 5x + 12 = 37. Step 2: Subtract 12 from both sides: 5x = 37 - 12 = 25. Step 3: Divide both sides by 5: x = 25 ÷ 5 = 5. Therefore, x = 5.

Question 3

Solve: 3(2x + 1) = 27

Accepted Answer

Step 1: Start with 3(2x + 1) = 27. Step 2: Divide both sides by 3: 2x + 1 = 9. Step 3: Subtract 1 from both sides: 2x = 8. Step 4: Divide by 2: x = 4. Therefore, x = 4.

Question 4

If 3x + 7 = 22, find the value of x.

Accepted Answer

Step 1: Start with 3x + 7 = 22. Step 2: Subtract 7 from both sides: 3x = 22 - 7 = 15. Step 3: Divide both sides by 3: x = 15 ÷ 3 = 5. Therefore, x = 5.

Question 5

If 3x + 7 = 2x + 15, then the value of x is:

Accepted Answer

Step 1: Start with the equation 3x + 7 = 2x + 15. Step 2: Subtract 2x from both sides: 3x - 2x + 7 = 15, which gives x + 7 = 15. Step 3: Subtract 7 from both sides: x = 15 - 7 = 8. Therefore, the answer is 8.

Question 6

If 6x - 4 = 2x + 12, find x.

Accepted Answer

Step 1: Start with 6x - 4 = 2x + 12. Step 2: Subtract 2x from both sides: 4x - 4 = 12. Step 3: Add 4 to both sides: 4x = 16. Step 4: Divide by 4: x = 4. Therefore, x = 4.

Question 7

If 4(x - 3) = 20, what is the value of x?

Accepted Answer

Step 1: Start with 4(x - 3) = 20. Step 2: Divide both sides by 4: x - 3 = 5. Step 3: Add 3 to both sides: x = 8. Therefore, x = 8.

Question 8

Solve for y: 3(y - 4) = 18

Accepted Answer

Step 1: Start with 3(y - 4) = 18. Step 2: Divide both sides by 3: y - 4 = 18 ÷ 3 = 6. Step 3: Add 4 to both sides: y = 6 + 4 = 10. Therefore, y = 10.

Question 9

The sum of two numbers is 48 and their difference is 14. What is the larger number?

Accepted Answer

Step 1: Let the two numbers be x and y where x > y. So x + y = 48 and x - y = 14. Step 2: Add both equations: 2x = 62, so x = 31. Step 3: y = 48 - 31 = 17. The larger number is 31. Therefore, the answer is 31.

Question 10

If 2x - 5 = x + 3, what is the value of x?

Accepted Answer

Step 1: Start with 2x - 5 = x + 3. Step 2: Subtract x from both sides: 2x - x - 5 = 3, which gives x - 5 = 3. Step 3: Add 5 to both sides: x = 3 + 5 = 8. Therefore, x = 8.

Question 11

Solve for y: 5y - 12 = 2y + 9

Accepted Answer

Step 1: Start with 5y - 12 = 2y + 9. Step 2: Subtract 2y from both sides: 3y - 12 = 9. Step 3: Add 12 to both sides: 3y = 21. Step 4: Divide by 3: y = 7. Therefore, y = 7.

Question 12

A number when multiplied by 6 and then reduced by 8 equals 40. Find the number.

Accepted Answer

Step 1: Let the number be x. According to the problem: 6x - 8 = 40. Step 2: Add 8 to both sides: 6x = 40 + 8 = 48. Step 3: Divide both sides by 6: x = 48 ÷ 6 = 8. Therefore, the number is 8.

Question 13

The sum of two consecutive even numbers is 54. What is the larger of the two numbers?

Accepted Answer

Step 1: Let the two consecutive even numbers be x and x + 2. Step 2: Set up the equation: x + (x + 2) = 54. Step 3: Simplify: 2x + 2 = 54. Step 4: Solve for x: 2x = 52, so x = 26. Step 5: The two numbers are 26 and 28. The larger number is 28. Step 6: Verify: 26 + 28 = 54 ✓

Question 14

If 5(2x - 3) = 3(x + 4), then x equals:

Accepted Answer

Step 1: Expand the left side: 5(2x - 3) = 10x - 15. Step 2: Expand the right side: 3(x + 4) = 3x + 12. Step 3: Set up the equation: 10x - 15 = 3x + 12. Step 4: Move all x terms to the left: 10x - 3x = 12 + 15. Step 5: Simplify: 7x = 27. Step 6: Solve for x: x = 27/7. Step 7: Verify: 5(2(27/7) - 3) = 5(54/7 - 21/7) = 5(33/7) = 165/7 and 3(27/7 + 4) = 3(27/7 + 28/7) = 3(55/7) = 165/7 ✓

Question 15

If 3x + 7 = 2x + 15, then the value of x is:

Accepted Answer

Step 1: Rearrange the equation by moving all x terms to the left and constants to the right. 3x - 2x = 15 - 7. Step 2: Simplify both sides. x = 8. Step 3: Verify: 3(8) + 7 = 24 + 7 = 31 and 2(8) + 15 = 16 + 15 = 31. Both sides equal 31, so x = 8 is correct.

Question 16

Two numbers are in the ratio 5:7. If their sum is 120, what is the larger number?

Accepted Answer

Step 1: Let the two numbers be 5k and 7k. Step 2: Their sum: 5k + 7k = 120. Step 3: Simplify: 12k = 120. Step 4: Solve for k: k = 10. Step 5: The larger number = 7k = 7 × 10 = 70.

Question 17

Rajesh has some money. If he spends ₹240, he will have 3/5 of his original amount left. How much money does Rajesh currently have?

Accepted Answer

Step 1: Let Rajesh's original amount be x. Step 2: After spending ₹240, he has (x - 240) left. Step 3: According to the problem, x - 240 = (3/5)x. Step 4: Rearranging: x - (3/5)x = 240. Step 5: (5x - 3x)/5 = 240, so (2x)/5 = 240. Step 6: 2x = 1200, therefore x = 600. Step 7: Verification: 600 - 240 = 360, and (3/5) × 600 = 360 ✓. Therefore, Rajesh has ₹600.

Question 18

A man is 4 times as old as his son. After 8 years, he will be 3 times as old as his son. What is the son's current age?

Accepted Answer

Step 1: Let son's current age = y. Then father's current age = 4y. Step 2: After 8 years, son's age = y + 8 and father's age = 4y + 8. Step 3: Set up equation: 4y + 8 = 3(y + 8). Step 4: Expand: 4y + 8 = 3y + 24. Step 5: Subtract 3y: y + 8 = 24. Step 6: Subtract 8: y = 16.

Question 19

A man is 4 times as old as his son. If the sum of their ages is 60 years, how old is the son?

Accepted Answer

Step 1: Let the son's age be x years. Step 2: The man's age is 4x years. Step 3: Set up the equation: x + 4x = 60. Step 4: Simplify: 5x = 60. Step 5: Solve for x: x = 12. Step 6: The son is 12 years old. Step 7: Verify: Man's age = 4(12) = 48. Sum = 12 + 48 = 60 ✓

Question 20

Three times a number decreased by 8 equals 19. What is the number?

Accepted Answer

Step 1: Let the number be x. Step 2: Translate the statement: 'Three times a number' = 3x, 'decreased by 8' means subtract 8. Step 3: Set up the equation: 3x - 8 = 19. Step 4: Add 8 to both sides: 3x = 27. Step 5: Divide by 3: x = 9. Step 6: Verify: 3(9) - 8 = 27 - 8 = 19 ✓

Question 21

If (2x - 3)/5 = (x + 1)/3, then x equals:

Accepted Answer

Step 1: Cross-multiply: 3(2x - 3) = 5(x + 1). Step 2: Expand: 6x - 9 = 5x + 5. Step 3: Subtract 5x from both sides: x - 9 = 5. Step 4: Add 9 to both sides: x = 14.

Question 22

If 3(x + 2) = 2(x + 5) + 4, then the value of x is:

Accepted Answer

Step 1: Expand both sides: 3x + 6 = 2x + 10 + 4. Step 2: Simplify right side: 3x + 6 = 2x + 14. Step 3: Subtract 2x from both sides: x + 6 = 14. Step 4: Subtract 6 from both sides: x = 8.

Question 23

If 3x + 4y = 18 and 4x + 3y = 17, find the value of (x + y)² + (x − y)².

Accepted Answer

Step 1: Add both equations: 7x + 7y = 35, so x + y = 5. Step 2: Subtract second from first: (3x + 4y) − (4x + 3y) = 18 − 17 → −x + y = 1, so y − x = 1, meaning x − y = −1. Step 3: (x + y)² + (x − y)² = (5)² + (−1)² = 25 + 1 = 26. Therefore, the answer is 26.

Question 24

A man is 4 times as old as his son. In 6 years, the man will be 3 times as old as his son. If the man's current age is M years and his son's current age is S years, find the value of M + S.

Accepted Answer

Step 1: Set up equations from the given conditions. Currently: M = 4S. Step 2: In 6 years: M + 6 = 3(S + 6). Step 3: Substitute M = 4S into the second equation: 4S + 6 = 3S + 18. Step 4: Solve for S: S = 12. Step 5: Find M: M = 4(12) = 48. Step 6: Calculate M + S = 48 + 12 = 60.

Question 25

A fraction becomes 1/2 when 1 is subtracted from both numerator and denominator. The same fraction becomes 2/3 when 8 is added to both numerator and denominator. What is the original fraction?

Accepted Answer

Step 1: Let the fraction be p/q. Step 2: From first condition: (p − 1)/(q − 1) = 1/2 → 2(p − 1) = q − 1 → 2p − 2 = q − 1 → q = 2p − 1. Step 3: From second condition: (p + 8)/(q + 8) = 2/3 → 3(p + 8) = 2(q + 8) → 3p + 24 = 2q + 16 → 3p + 8 = 2q. Step 4: Substitute q = 2p − 1 into 3p + 8 = 2q: 3p + 8 = 2(2p − 1) → 3p + 8 = 4p − 2 → p = 10. Step 5: Find q: q = 2(10) − 1 = 19. Step 6: Original fraction = 10/19.

Question 26

Two numbers are in the ratio 5:7. If 8 is subtracted from each, the resulting numbers are in the ratio 2:3. Find the sum of the original two numbers.

Accepted Answer

Step 1: Let the two numbers be 5x and 7x (since they are in ratio 5:7). Step 2: After subtracting 8 from each: (5x − 8) and (7x − 8). Step 3: These are in ratio 2:3, so: (5x − 8)/(7x − 8) = 2/3. Step 4: Cross-multiply: 3(5x − 8) = 2(7x − 8). Step 5: Expand: 15x − 24 = 14x − 16. Step 6: Solve: x = 8. Step 7: Original numbers are 5(8) = 40 and 7(8) = 56. Step 8: Sum = 40 + 56 = 96.

Question 27

Solve for y: 5y - 12 = 2y + 9

Accepted Answer

Step 1: Start with 5y - 12 = 2y + 9. Step 2: Subtract 2y from both sides: 3y - 12 = 9. Step 3: Add 12 to both sides: 3y = 21. Step 4: Divide by 3: y = 7. Therefore, the answer is 7.

Question 28

If 3x + 7 = 2x + 12, then the value of x is:

Accepted Answer

Step 1: Start with the equation 3x + 7 = 2x + 12. Step 2: Subtract 2x from both sides: 3x - 2x + 7 = 12, which gives x + 7 = 12. Step 3: Subtract 7 from both sides: x = 12 - 7 = 5. Therefore, the answer is 5.

Question 29

If 6x - 5 = 3x + 10, find x.

Accepted Answer

Step 1: Start with 6x - 5 = 3x + 10. Step 2: Subtract 3x from both sides: 3x - 5 = 10. Step 3: Add 5 to both sides: 3x = 15. Step 4: Divide by 3: x = 5. Therefore, the answer is 5.

Question 30

Solve for m: 7m + 4 = 32

Accepted Answer

Step 1: Start with 7m + 4 = 32. Step 2: Subtract 4 from both sides: 7m = 28. Step 3: Divide both sides by 7: m = 4. Therefore, the answer is 4.

Question 31

Solve for y: 3(y - 4) = 15

Accepted Answer

Step 1: Divide both sides by 3: y - 4 = 15 ÷ 3 = 5. Step 2: Add 4 to both sides: y = 5 + 4 = 9. Therefore, y = 9.

Question 32

If 3x + 7 = 22, find the value of x.

Accepted Answer

Step 1: Start with 3x + 7 = 22. Step 2: Subtract 7 from both sides: 3x = 15. Step 3: Divide both sides by 3: x = 5. Therefore, the answer is 5.

Question 33

If 5x + 12 = 37, what is the value of x?

Accepted Answer

Step 1: Subtract 12 from both sides: 5x = 37 - 12 = 25. Step 2: Divide both sides by 5: x = 25 ÷ 5 = 5. Therefore, x = 5.

Question 34

Solve: (x + 5)/2 = 8

Accepted Answer

Step 1: Multiply both sides by 2: x + 5 = 8 × 2 = 16. Step 2: Subtract 5 from both sides: x = 16 - 5 = 11. Therefore, x = 11.

Question 35

If 2x - 7 = x + 3, find the value of x.

Accepted Answer

Step 1: Subtract x from both sides: 2x - x - 7 = 3. Step 2: Simplify: x - 7 = 3. Step 3: Add 7 to both sides: x = 3 + 7 = 10. Therefore, x = 10.

Question 36

Solve: 2x + 3 = x + 8

Accepted Answer

Step 1: Start with 2x + 3 = x + 8. Step 2: Subtract x from both sides: x + 3 = 8. Step 3: Subtract 3 from both sides: x = 5. Therefore, the answer is 5.

Question 37

If 4(x - 3) = 20, what is the value of x?

Accepted Answer

Step 1: Start with 4(x - 3) = 20. Step 2: Divide both sides by 4: x - 3 = 5. Step 3: Add 3 to both sides: x = 8. Therefore, the answer is 8.

Question 38

If 3(2x + 5) = 4(x + 8) + 7, find the value of x.

Accepted Answer

Step 1: Expand both sides: 6x + 15 = 4x + 32 + 7. Step 2: Simplify right side: 6x + 15 = 4x + 39. Step 3: Subtract 4x from both sides: 2x + 15 = 39. Step 4: Subtract 15 from both sides: 2x = 24. Step 5: Divide by 2: x = 12.

Question 39

The sum of two consecutive integers is 47. What is the larger integer?

Accepted Answer

Step 1: Let the two consecutive integers be x and x+1. Step 2: Set up the equation: x + (x+1) = 47. Step 3: Simplify: 2x + 1 = 47. Step 4: Solve for x: 2x = 46, so x = 23. Step 5: The larger integer is x + 1 = 24. Therefore, the answer is 24.

Question 40

A man is 4 times as old as his son. After 8 years, he will be 2.5 times as old as his son. What is the son's current age?

Accepted Answer

Step 1: Let son's current age = x, father's current age = 4x. Step 2: After 8 years, son's age = x + 8, father's age = 4x + 8. Step 3: Set up equation: 4x + 8 = 2.5(x + 8). Step 4: Expand: 4x + 8 = 2.5x + 20. Step 5: Subtract 2.5x: 1.5x + 8 = 20. Step 6: Subtract 8: 1.5x = 12. Step 7: Divide by 1.5: x = 8.

Question 41

A man is 4 times as old as his son. If the sum of their ages is 60 years, how old is the son?

Accepted Answer

Step 1: Let the son's age be x years. Then the man's age is 4x years. Step 2: Set up the equation: x + 4x = 60. Step 3: Simplify: 5x = 60. Step 4: Solve for x: x = 12. Therefore, the son is 12 years old.

Question 42

If 3x + 7 = 2x + 12, then the value of x is:

Accepted Answer

Step 1: Rearrange the equation by moving all x terms to the left and constants to the right. 3x - 2x = 12 - 7. Step 2: Simplify both sides. x = 5. Therefore, the answer is 5.

Question 43

A man is 6 years older than his wife. Five years ago, the product of their ages was 391. If the man's current age is M years, which of the following equations correctly represents this situation?

Accepted Answer

Step 1: Let the wife's current age = W years. Then man's current age = W + 6 years. Step 2: Five years ago, wife's age = W − 5 and man's age = W + 6 − 5 = W + 1. Step 3: Product of their ages five years ago = (W − 5)(W + 1) = 391. Step 4: Expanding: W² + W − 5W − 5 = 391 → W² − 4W − 5 = 391 → W² − 4W − 396 = 0. Step 5: Since M = W + 6, we have W = M − 6. Substituting: (M − 6)² − 4(M − 6) − 396 = 0 → M² − 12M + 36 − 4M + 24 − 396 = 0 → M² − 16M − 336 = 0. Therefore, the correct equation is M² − 16M − 336 = 0.

Question 44

The sum of two consecutive even numbers is 86. What is the larger of the two numbers?

Accepted Answer

Step 1: Let the two consecutive even numbers be x and x + 2. Step 2: Set up equation: x + (x + 2) = 86. Step 3: Simplify: 2x + 2 = 86. Step 4: Subtract 2: 2x = 84. Step 5: Divide by 2: x = 42. Step 6: The two numbers are 42 and 44. The larger number is 44.

Question 45

If 5x - 3 = 2x + 9, then 3x + 2 equals:

Accepted Answer

Step 1: Solve for x from the given equation: 5x - 3 = 2x + 9. Step 2: Rearrange: 5x - 2x = 9 + 3. Step 3: Simplify: 3x = 12, so x = 4. Step 4: Now find 3x + 2: 3(4) + 2 = 12 + 2 = 14. Therefore, the answer is 14.

Question 46

A number is divided into two parts such that one part is 5 more than the other. If the number is 23, what is the smaller part?

Accepted Answer

Step 1: Let the smaller part be x. Then the larger part is x + 5. Step 2: Set up the equation: x + (x + 5) = 23. Step 3: Simplify: 2x + 5 = 23. Step 4: Solve for x: 2x = 18, so x = 9. Therefore, the smaller part is 9.

CTET Paper II Linear Equations

CTET Paper II Linear Equations — Past Exam Questions

Linear Equations— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Linear Equations — Practice Questions

60-Second Revision — Linear Equations