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 o

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.

Accepted Answer

Step 1: The ages of the participants form the following AP:
$8, 8\frac{1}{3}, 8\frac{2}{3}, 9, \ldots$
where first term $= 8$ and common difference $= \frac{1}{3}$.
Step 2: Let the number of participants be $n$.
$S_n = \frac{n}{2}\left[2 	imes 8 + (n-1)\frac{1}{3}ight] = 168$
Step 3: $n^2 + 47n - 1008 = 0$
Step 4: $\Rightarrow n = 16$ (rejecting negative value)
Step 5: $	herefore$ the age of the eldest participant $= 8 + 15 	imes \frac{1}{3} = 13$ years

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