AMC 10 · 2017 · #5
Grade 4 countingPick an answer.
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
Each pair does exactly one greeting.
If each pair does exactly one of two things, counting one kind means subtracting it from the total.
4.OA.A.3Change Focus Count The ComplementCount every pair in the room
All pairs number 435.
Every person shakes or hugs 29 others, but you and I share one pair, so halve the total.
Every person greets everyone else, but you and I share one greeting, so the total has to be halved.
▸ Why?
Each greeting belongs to exactly two people, so counting per person visits it twice.
▸ Why?
Each pair does one thing or the other and never both, so subtracting one kind leaves the other.
Count the hugs
The easier kind numbers 190.
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
Subtracting gives 245, choice (B).
Whatever pairs are not hugs must be handshakes, so one subtraction finishes the count.
3.NBT.A.2Change Focus Count 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