AFCAT Numerical Ability · Average

Ages Over Time, and a New Baby

Every member of a family gets a year older each year, so the family's total age grows by its size every year; a baby adds a member but little or nothing to the total.

All Average notes

Why this matters

One AFCAT past-year question (2019) is here, and nearly a quarter of the page is HARD, mostly births at different times that each have to be carried forward to today. The slip is to age the family but not the children, or the children but not the family.

Concept 1 of 4: Moving a family through time

If 6 people each get 4 years older, their total age rises by 24 and their average by 4.

Definition

After $t$ years, every age is $t$ more: the average rises by $t$ and the total by $nt$ , as long as nobody joins or leaves.
$t$ years ago: subtract $t$ from the average.
The difference between two people's ages never changes.
Years and months: turn everything into months, work, then turn back (12 months = 1 year).

A family's total after t years

\text{total after } t \text{ years} = \text{total now} + n\,t

Worked example

A husband and wife averaged 26 years when they married 6 years ago. Their son is now 5. Find the family's average age now.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 02-202Moderate

Example 1 · Average · Ages

B

was born when

A

was 4 yr 7 months old and

C

was born when

B

was 3 yr 4 months old. When

C

was 5 yr 2 months old, their average age was

The gap stays, the ratio does not

A mother 24 years older than her daughter is always 24 years older, but 'twice as old' is true for only one year.

Drill 4 more on moving a family through time

Concept 2 of 4: A child or a baby joins the family

A newborn adds a member and nothing to the total, so the average drops. An older child adds its own age to the total.

Definition

Newborn: new average = family total ÷ $(n + 1)$ .
Born $t$ years later: first add $nt$ to the family total, then let the baby join.
Twins or triplets add 2 or 3 to the count and 0 to the total.
A child of known age joins: add its age to the total and 1 to the count.

A child joins

\text{new average} = \dfrac{\text{family total} + \text{child's age}}{n + 1}

Worked example

A family of 5 has an average age of 30. Three years later a baby is born. Find the average age just after the birth.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Average · Ages

A family of

4

has average age

42

years. After

4

years, a baby is born. What is the average age immediately after baby's birth?

Age the family first

If the baby arrives 3 years later, the five older members are 15 years older in total. Adding the baby to the old total misses those 15 years.

Drill 14 more on a child or a baby joins the family

Concept 3 of 4: Finding the baby's age

Bring the older members' total up to today, then compare it with today's total for the whole family. What is left over is the baby's age.

Definition

Older members' total now = their past total + $n$ × the years since.
Family total now = $(n + 1)$ × today's average.
Baby's age = the difference. Two babies: the difference is the sum of their ages.
Check: a baby born after the old average was taken must be younger than the years since. A larger answer means the question is inconsistent.

The baby's age

\text{baby} = (n + 1)A_{\text{now}} - \left(nA_{\text{then}} + n\,t\right)

Worked example

Six years ago the average age of 6 family members was 23. A baby has been born since, and today the average of all 7 is 25.5. Find the baby's age.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Average · Ages

3

years ago, the average age of a family of

5

members was

16

years. A baby having been born, the average age of the family is the same today. What is the present age of the baby?

Age the older members first

If 6 members totalled 120 four years ago, they total 144 now. Forgetting those 24 years makes the baby 24 years too old.

A just-born baby is 0

If a question says a baby is born 'today' and the arithmetic gives the baby an age of 3, the question is inconsistent. A newborn adds nothing to the total.

Drill 17 more on finding the baby's age

Concept 4 of 4: Births at different times

Bring everyone to today. The parents age by the full gap; each child has only the years since its own birth.

Definition

Step 1: the older members' total now = past total + their number × the years since.
Step 2: each child's age now = the years since that child was born.
Step 3: add everything and divide by everyone alive today.
With three or more events, draw a short time line first.

Today's total

\text{total now} = \text{past total} + n\,t + (\text{each child's age now})

Worked example

Eight years ago a couple's average age was 24. Their first child was born 6 years ago and their second 3 years ago. Find the family's average age now.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Average · Ages

9

years ago, average age of

2

parents was

25

years.

5

years ago their first child was born,

2

years ago their second child was born. What is the current average age of the family?

Children age from their own birth

A child born 6 years ago adds 6 to today's total, not the 8 years since the old average was taken.

Drill 13 more on births at different times

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Moving a family through time
A family's total after t years
$\text{total after } t \text{ years} = \text{total now} + n\,t$
A child or a baby joins the family
A child joins
$\text{new average} = \dfrac{\text{family total} + \text{child's age}}{n + 1}$
Finding the baby's age
The baby's age
$\text{baby} = (n + 1)A_{\text{now}} - \left(nA_{\text{then}} + n\,t\right)$
Births at different times
Today's total
$\text{total now} = \text{past total} + n\,t + (\text{each child's age now})$

Watch out for (5)

The gap stays, the ratio does not
→ Moving a family through time
Age the family first
→ A child or a baby joins the family
Age the older members first
→ Finding the baby's age
A just-born baby is 0
→ Finding the baby's age
Children age from their own birth
→ Births at different times

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