14. Ayeesha had 12 guests aged $6,7,8,9$ and 10 at her birthday party. Four of the guests were 6 years old. The most common age was 8 years old. What was the mean age of the guests?
A 6
В 6.5
C 7
D 7.5
E 8
Solution
D

Since there were four children aged 6 and the most common age was 8 , there were at least five children aged 8 at the party. Hence, as there were 12 children in total, there were at most three other children at the party. We are told that there were children aged 7, 9 and 10 at the event so we can deduce that there were exactly five children aged 8 , one child aged 7 , one child aged 9 and one child aged 10 at Ayeesha's party as well as the four children aged 6. Therefore the total age of the children was $4 \times 6+7+5 \times 8+9+10=90$. Hence the mean age of the children was $90 \div 12=7.5$.