Competition problem — Algebra — Maths Olympiad Prep

Library / /20 of 20

Algebra Difficulty 5.0 AIME, harder Prove it South Africa

Exactly two years ago the Benson family had 4 members, and their average age was $19$ . The Bensons then adopted another child. If the average age of the family today is still $19$ , what is the present age of the adopted child?

Solution

2 years ago the sum of all the family's ages was $4 \times 19 = 76$ . That should have increased by $2 \times 4 = 8$ , but has actually become $5 \times 19 = 95$ , i.e. increased by $19$ . So the new person is now $19 - 8 = 11$ years old.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.