2.9 Average Age Problems (Present / Past / Future Shifts)

Key rule: When “x years pass,” every existing member’s age increases by exactly x years, so the group’s average age also increases by exactly x years — provided the group size stays the same and no member joins/leaves. When a new member (e.g., a newborn child) is added at a later time, this rule must be applied carefully, because the new member did not exist x years ago.

Solved Example 24: The present average age of a father and his son is 27 years. Find their average age 5 years hence.

Since both individuals age normally (no one joins or leaves), average age after 5 years = 27 + 5 = 32 years.

Solved Example 25: Five years ago, the average age of a husband and wife was 20 years. Find their present average age.

Present average = 20 + 5 = 25 years.

Solved Example 26: The present average age of a family of 4 members is 24 years. A baby is born, and 3 years later, the average age of the family (now 5 members) is again 24 years. Find the age of the baby when it was born… [typical variant]: Actually — find the baby’s present age.

3 years later, sum of original 4 members’ ages = (24×4) + 4×3 = 96+12 = 108. Total sum for 5 members (present) = 24×5 = 120. Baby’s present age = 120 − 108 = 12 years? — this doesn’t fit a “3-year-old baby,” so let’s re-examine: this shows why average age problems must track who existed when. A cleaner version: The average age of a family of 4 is 24 years. If the present age of the youngest member is 8 years, find the average age of the family at the time of the birth of the youngest member.

Sum of present ages of 4 members = 24×4 = 96. At the time of birth of the youngest, that member’s age was 0, and the other 3 members were each younger by 8 years, so their combined age was 8 years less each, i.e., total reduction = 3×8 = 24. Sum of ages of the 3 (other) members at that time = (96 − 8) − 24 = 88 − 24 = 64. Average age of the family (3 members, since youngest was just born, age 0, often excluded or counted as 0) — commonly this question asks for the average of the family excluding the newborn: 64/3 = 21.33 years.

Solved Example 26(a): The average age of a family of 6 members is 22 years. If the age of the youngest member is 8 years, find the average age of the family at the time of the birth of the youngest member (considering the remaining 5 members only, since the youngest did not exist then).

Sum of present ages of 6 members = 22×6 = 132. Excluding the youngest, sum of ages of the other 5 members presently = 132 − 8 = 124. At the time of the youngest’s birth (8 years ago from now), each of these 5 members was 8 years younger, so total reduction = 5×8 = 40. Sum of ages of the 5 members at that time = 124 − 40 = 84. Average age of the family at that time (5 members) = 84/5 = 16.8 years.

Solved Example 26(b): The average age of a couple was 25 years at the time of their marriage. After 10 years of marriage, the couple has a child, whose present age is 2 years. Find the present average age of the family (couple + child, i.e., 3 members).

At marriage, sum of couple’s ages = 25×2 = 50. After 10 years, each partner is 10 years older, so present sum of couple’s ages = 50 + 2×10 = 70. Child’s present age = 2. Present total = 70+2 = 72. Present average age of the family (3 members) = 72/3 = 24 years.

Key exam pattern to internalise: Whenever a “family/group + newborn/new joinee at a later date” scenario appears, split the problem into two clean steps: (i) age the existing members forward using simple addition of years elapsed, and (ii) add the new member’s age (which is not shifted by the same number of years, since they didn’t exist earlier) separately. Never apply a blanket “+x years to everyone” rule when the group composition itself has changed during that period.

Solved Example 26(c): The average age of a family of 5 members is 24 years. A member aged 40 years leaves the family (say, due to marriage), and at the same time a newborn baby (age 0) joins the family, keeping the family size at 5. Find the new average age of the family.

This is exactly the scenario the task calls for: a member leaving and a member joining simultaneously. Old sum = 24×5 = 120. Remove the member aged 40: sum = 120−40 = 80 (4 members remain). Add the newborn’s age (0): sum = 80+0 = 80 (5 members). New average = 80/5 = 16 years.

Solved Example 26(d): Five years ago, the average age of a family of 4 members was 22 years. Since then, one of those four members (whose present age is 45 years) has left the family, and a child was born to the family 2 years ago. Find the present average age of the family (still 4 members: the 3 original members who remained, plus the newborn child).

Five years ago, sum of the original 4 members’ ages = 22×4 = 88. The member who has since left is presently 45, so five years ago that member was 45−5 = 40 years old. Sum of the ages of the remaining 3 members, five years ago = 88 − 40 = 48. Ageing these 3 members forward by 5 years to the present: sum = 48 + 3×5 = 48+15 = 63. The child, born 2 years ago, is presently 2 years old. Present total (3 original + 1 child) = 63 + 2 = 65. Present average = 65/4 = 16.25 years.

This combines three ideas from this section in one question — a past average, a member leaving partway through the elapsed period, and a new member born partway through — exactly the kind of layered scenario that appears in the toughest RRB NTPC CBT-2 age questions.