Question

The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that the age of the youngest boy was 8 years, and the ages of the participants, when seated in order of age, have a common difference of 4 months. If the sum of the ages of all the participants is 168 years, find the age of the eldest participant in the painting competition.

Answer 1

We are given that:

The youngest participant is 8 years old.

The ages of participants form an arithmetic progression (AP) with a common difference of 4 months (i.e., `1/3` years).

The sum of all participants' ages is 168 years.

We need to find the age of the eldest participant.

Step 1: Define the AP

Let:

a = 8a years (first term of the AP)

d = `1/3` (common difference)

n = total number of participants

Sn = 168 years (sum of all ages)

The sum of an AP is given by: 

`Sn = n/2 xx (2a + (n − 1)d)`

Substituting the given values:

`168 = n/2 xx (2(8) + (n - 1) xx 1/3)`

`168 = n/2 xx (16 + (n − 1)/3)`

`336 = n × (16 + (n − 1)/3)`

Multiply by 3 to eliminate the fraction:

`1008 = 3n × (16 + (n − 1)/3)`

1008 = n(48 + n − 1)

1008 = n(n + 47)

n² + 47n − 1008 = 0 

Step 2: Solve for n

Solving the quadratic equation: n2 + 47n − 1008 = 0

`n = (−b ± sqrt (b^2 − 4ac))/(2a)`        ...[Using the quadratic formula] 

where a = 1, b = 47, and c = −1008

`n = (−47 ± sqrt (47^2 − 4(1)(− 1008)))/(2(1))`

`n = (−47 ± sqrt (2209 + 4032))/(2)`

`n = (−47 ± sqrt 6241)/(2)`

`n = (−47 ± 79)/(2)`

`n = (−47 + 79)/(2)`     ...[Since n = `(− 47 − 79)/2` is negative, we discard it.]

`n = 32/2 `

`n = 16`

Step 3: Find the Eldest Participant’s Age

a_n ​= a + (n − 1)d

`a_16​ = 8 + (16 − 1) × 1/3`

`= 8 + 15 xx 1/3`

= 8 + 5 = 13 years.