Problems on Ages: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 99. Fact 2: A is 10 years older than D, so A – D = 10. Fact 3: R + D = 62. From facts 1 and 3: A = 99 – 62 = 37. From fact 2: D = A – 10 = 37 – 10 = 27. From fact 3: R = 62 – D = 62 – 27 = 35. Check: A + R + D = 37 + 35 + 27 = 99, confirming the answer. So Rohan’s age is 35.

The present age is rx, built from ratio multiplier r and variable x. After n years pass, every person’s age increases by exactly n. This n is a fixed number of years, added once, not multiplied. So the new age is rx + n, with n added directly. It is not r(x+n), which would wrongly multiply n by r too. For example, if r=3, x=5, n=2: the age is 3(5)+2 = 17, not 3(5+2) = 21.

Represent both present ages using one variable x. Let the ages be r1x and r2x, based on the given ratio. After n years, each age increases by exactly n. So the ages become r1x + n and r2x + n. Set this equal to the new ratio: (r1x + n)/(r2x + n) = r3/r4. Cross-multiply, then collect the x terms on one side. Solving this equation gives the value of x directly.

Statement I alone gives a relationship between A and B only, not B’s actual age. Statement II alone gives a relationship between B and C only, not B’s actual age either. Together, Statement I says A – B = 6, and Statement II says B + C = 40. That is two equations, but three unknowns: A, B, and C. This system cannot be solved for B’s specific value. So both statements together are still not sufficient.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 12, so A + B = 2 x 12 = 24. Fact 2: 5 years from now, their ages are in the ratio 23:11. So let their future ages be 23y and 11y, for some number y. Fact 3: present age plus 5 equals future age, so A = 23y – 5 and B = 11y – 5. Substituting into fact 1: (23y – 5) + (11y – 5) = 24. This simplifies to 34y – 10 = 24, so y = 1. So B = 11 x 1 – 5 = 6. Check: A = 23 x 1 – 5 = 18, and A + B = 18 + 6 = 24, confirming the answer. So Sophie’s present age is 6.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 63, so A + B = 2 x 63 = 126. Fact 2: 6 years from now, their ages are in the ratio 97:41. So let their future ages be 97y and 41y, for some number y. Fact 3: present age plus 6 equals future age, so A = 97y – 6 and B = 41y – 6. Substituting into fact 1: (97y – 6) + (41y – 6) = 126. This simplifies to 138y – 12 = 126, so y = 1. So B = 41 x 1 – 6 = 35. Check: A = 97 x 1 – 6 = 91, and A + B = 91 + 35 = 126, confirming the answer. So Sophie’s present age is 35.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 28, so A + B = 2 x 28 = 56. Fact 2: 5 years from now, their ages are in the ratio 15:7. So let their future ages be 15y and 7y, for some number y. Fact 3: present age plus 5 equals future age, so A = 15y – 5 and B = 7y – 5. Substituting into fact 1: (15y – 5) + (7y – 5) = 56. This simplifies to 22y – 10 = 56, so y = 3. So B = 7 x 3 – 5 = 16. Check: A = 15 x 3 – 5 = 40, and A + B = 40 + 16 = 56, confirming the answer. So Sophie’s present age is 16.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 16, so A + B = 2 x 16 = 32. Fact 2: 3 years from now, their ages are in the ratio 23:15. So let their future ages be 23y and 15y, for some number y. Fact 3: present age plus 3 equals future age, so A = 23y – 3 and B = 15y – 3. Substituting into fact 1: (23y – 3) + (15y – 3) = 32. This simplifies to 38y – 6 = 32, so y = 1. So B = 15 x 1 – 3 = 12. Check: A = 23 x 1 – 3 = 20, and A + B = 20 + 12 = 32, confirming the answer. So Sophie’s present age is 12.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 28, so A + B = 2 x 28 = 56. Fact 2: 6 years from now, their ages are in the ratio 41:27. So let their future ages be 41y and 27y, for some number y. Fact 3: present age plus 6 equals future age, so A = 41y – 6 and B = 27y – 6. Substituting into fact 1: (41y – 6) + (27y – 6) = 56. This simplifies to 68y – 12 = 56, so y = 1. So B = 27 x 1 – 6 = 21. Check: A = 41 x 1 – 6 = 35, and A + B = 35 + 21 = 56, confirming the answer. So Sophie’s present age is 21.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 13, so A + B = 2 x 13 = 26. Fact 2: 4 years from now, their ages are in the ratio 9:8. So let their future ages be 9y and 8y, for some number y. Fact 3: present age plus 4 equals future age, so A = 9y – 4 and B = 8y – 4. Substituting into fact 1: (9y – 4) + (8y – 4) = 26. This simplifies to 17y – 8 = 26, so y = 2. So B = 8 x 2 – 4 = 12. Check: A = 9 x 2 – 4 = 14, and A + B = 14 + 12 = 26, confirming the answer. So Sophie’s present age is 12.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 27, so A + B = 2 x 27 = 54. Fact 2: 6 years from now, their ages are in the ratio 17:16. So let their future ages be 17y and 16y, for some number y. Fact 3: present age plus 6 equals future age, so A = 17y – 6 and B = 16y – 6. Substituting into fact 1: (17y – 6) + (16y – 6) = 54. This simplifies to 33y – 12 = 54, so y = 2. So B = 16 x 2 – 6 = 26. Check: A = 17 x 2 – 6 = 28, and A + B = 28 + 26 = 54, confirming the answer. So Sophie’s present age is 26.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 33, so A + B = 2 x 33 = 66. Fact 2: 5 years from now, their ages are in the ratio 47:29. So let their future ages be 47y and 29y, for some number y. Fact 3: present age plus 5 equals future age, so A = 47y – 5 and B = 29y – 5. Substituting into fact 1: (47y – 5) + (29y – 5) = 66. This simplifies to 76y – 10 = 66, so y = 1. So B = 29 x 1 – 5 = 24. Check: A = 47 x 1 – 5 = 42, and A + B = 42 + 24 = 66, confirming the answer. So Sophie’s present age is 24.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 21, so A + B = 2 x 21 = 42. Fact 2: 6 years from now, their ages are in the ratio 5:4. So let their future ages be 5y and 4y, for some number y. Fact 3: present age plus 6 equals future age, so A = 5y – 6 and B = 4y – 6. Substituting into fact 1: (5y – 6) + (4y – 6) = 42. This simplifies to 9y – 12 = 42, so y = 6. So B = 4 x 6 – 6 = 18. Check: A = 5 x 6 – 6 = 24, and A + B = 24 + 18 = 42, confirming the answer. So Sophie’s present age is 18.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 9, so A + B = 2 x 9 = 18. Fact 2: 3 years from now, their ages are in the ratio 5:3. So let their future ages be 5y and 3y, for some number y. Fact 3: present age plus 3 equals future age, so A = 5y – 3 and B = 3y – 3. Substituting into fact 1: (5y – 3) + (3y – 3) = 18. This simplifies to 8y – 6 = 18, so y = 3. So B = 3 x 3 – 3 = 6. Check: A = 5 x 3 – 3 = 12, and A + B = 12 + 6 = 18, confirming the answer. So Sophie’s present age is 6.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 14, so A + B = 2 x 14 = 28. Fact 2: 4 years from now, their ages are in the ratio 7:2. So let their future ages be 7y and 2y, for some number y. Fact 3: present age plus 4 equals future age, so A = 7y – 4 and B = 2y – 4. Substituting into fact 1: (7y – 4) + (2y – 4) = 28. This simplifies to 9y – 8 = 28, so y = 4. So B = 2 x 4 – 4 = 4. Check: A = 7 x 4 – 4 = 24, and A + B = 24 + 4 = 28, confirming the answer. So Sophie’s present age is 4.

Let A = Karan’s present age, B = Sophie’s present age. Fact 1: their average age is 24, so A + B = 2 x 24 = 48. Fact 2: 5 years from now, their ages are in the ratio 47:11. So let their future ages be 47y and 11y, for some number y. Fact 3: present age plus 5 equals future age, so A = 47y – 5 and B = 11y – 5. Substituting into fact 1: (47y – 5) + (11y – 5) = 48. This simplifies to 58y – 10 = 48, so y = 1. So B = 11 x 1 – 5 = 6. Check: A = 47 x 1 – 5 = 42, and A + B = 42 + 6 = 48, confirming the answer. So Sophie’s present age is 6.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 109. Fact 2: A is 12 years younger than D, so A – D = -12. Fact 3: R + D = 77. From facts 1 and 3: A = 109 – 77 = 32. From fact 2: D = A – -12 = 32 – -12 = 44. From fact 3: R = 77 – D = 77 – 44 = 33. Check: A + R + D = 32 + 33 + 44 = 109, confirming the answer. So Rohan’s age is 33.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 74. Fact 2: A is 9 years younger than D, so A – D = -9. Fact 3: R + D = 59. From facts 1 and 3: A = 74 – 59 = 15. From fact 2: D = A – -9 = 15 – -9 = 24. From fact 3: R = 59 – D = 59 – 24 = 35. Check: A + R + D = 15 + 35 + 24 = 74, confirming the answer. So Rohan’s age is 35.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 114. Fact 2: A is 14 years younger than D, so A – D = -14. Fact 3: R + D = 83. From facts 1 and 3: A = 114 – 83 = 31. From fact 2: D = A – -14 = 31 – -14 = 45. From fact 3: R = 83 – D = 83 – 45 = 38. Check: A + R + D = 31 + 38 + 45 = 114, confirming the answer. So Rohan’s age is 38.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 86. Fact 2: A is 16 years younger than D, so A – D = -16. Fact 3: R + D = 68. From facts 1 and 3: A = 86 – 68 = 18. From fact 2: D = A – -16 = 18 – -16 = 34. From fact 3: R = 68 – D = 68 – 34 = 34. Check: A + R + D = 18 + 34 + 34 = 86, confirming the answer. So Rohan’s age is 34.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 102. Fact 2: A is 10 years older than D, so A – D = 10. Fact 3: R + D = 60. From facts 1 and 3: A = 102 – 60 = 42. From fact 2: D = A – 10 = 42 – 10 = 32. From fact 3: R = 60 – D = 60 – 32 = 28. Check: A + R + D = 42 + 28 + 32 = 102, confirming the answer. So Rohan’s age is 28.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 76. Fact 2: A is 5 years older than D, so A – D = 5. Fact 3: R + D = 51. From facts 1 and 3: A = 76 – 51 = 25. From fact 2: D = A – 5 = 25 – 5 = 20. From fact 3: R = 51 – D = 51 – 20 = 31. Check: A + R + D = 25 + 31 + 20 = 76, confirming the answer. So Rohan’s age is 31.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 103. Fact 2: A is 23 years older than D, so A – D = 23. Fact 3: R + D = 60. From facts 1 and 3: A = 103 – 60 = 43. From fact 2: D = A – 23 = 43 – 23 = 20. From fact 3: R = 60 – D = 60 – 20 = 40. Check: A + R + D = 43 + 40 + 20 = 103, confirming the answer. So Rohan’s age is 40.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 76. Fact 2: A is 26 years younger than D, so A – D = -26. Fact 3: R + D = 59. From facts 1 and 3: A = 76 – 59 = 17. From fact 2: D = A – -26 = 17 – -26 = 43. From fact 3: R = 59 – D = 59 – 43 = 16. Check: A + R + D = 17 + 16 + 43 = 76, confirming the answer. So Rohan’s age is 16.

Let A = Anita’s age, R = Rohan’s age, D = Divya’s age. We are given three facts. Fact 1: A + R + D = 98. Fact 2: A is 6 years older than D, so A – D = 6. Fact 3: R + D = 59. From facts 1 and 3: A = 98 – 59 = 39. From fact 2: D = A – 6 = 39 – 6 = 33. From fact 3: R = 59 – D = 59 – 33 = 26. Check: A + R + D = 39 + 26 + 33 = 98, confirming the answer. So Rohan’s age is 26.