AMC 10 · 2017 · #8
Grade 4 arithmeticPick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Counting handshakes directly is fiddly, because strangers shake hands across two different groups. But every pair of people does exactly one thing: hug or shake hands. So Tool #16 (Count the Complement) says handshakes = all pairs minus the hugs. Tool #2 (Make a Systematic List) gives the clean way to count pairs: each person meets every other person, then halve to undo the double count. Tool #7 (Identify Subproblems) splits the job into two easy counts — total pairs, then hugging pairs — before one subtraction finishes it.
Hug or handshake, never both
Any two people either hug or shake hands — exactly one — so the handshakes are all the pairs left once the hugs are taken out.
If each pair does exactly one of two things, counting one kind means subtracting it from the total.
4.OA.A.3Count The ComplementCount every pair in the room
Each of the 30 people meets the other 29, giving 30 × 29; halve it to undo the double count, so there are 435 pairs.
Every person shakes or hugs 29 others, but you and I share one pair, so halve the total.
4.NBT.B.5Make A Systematic ListCount the hugs
Hugs happen only inside the 20 who know each other, so count that group the same way: (20 × 19)/2 = 190 pairs.
The same pair-counting trick works on any group, so the hugs are just the pairs inside the 20.
4.NBT.B.6Identify SubproblemsSubtract to get the handshakes
Take the hugs out of every pair; the leftover pairs are the handshakes: 435 - 190 = 245, which is choice (B).
Whatever pairs are not hugs must be handshakes, so one subtraction finishes the count.
3.NBT.A.2Count The ComplementWhen every pair does exactly one of two things, count all the pairs and subtract the kind you do not want.
- Hug or handshake, never both
- Count every pair in the room
- Count the hugs
- Subtract to get the handshakes