AMC 10 · 2004 · #4
Grade 4 arithmeticPick an answer.
The count of 30 mixes two generations, so first split off the fixed 6 daughters to find the number of granddaughters, then use that to find how many daughters became mothers. Once those pieces are known, the women with no daughters are just the granddaughters plus the childless daughters, so the whole task breaks into a few small counting steps.
Count the granddaughters
Removing the six daughters from the total leaves 24 granddaughters.
The total is two groups stacked together, so peeling off the group you already know leaves the other group.
3.NBT.A.2Identify SubproblemsFind how many daughters are mothers
Each mother has exactly six, so dividing gives 4 mothers.
If each mother contributes the same size batch, the number of batches tells you the number of mothers.
Since each mother has the same size batch of daughters, the number of batches gives the number of mothers.
▸ Why?
Equal batches repeated is a count of equal groups, which is exactly what multiplying records.
▸ Why?
Dividing undoes multiplying, so sharing the granddaughters into equal batches hands back how many mothers there were.
Find the childless daughters
That leaves 2 daughters with no children of their own.
The whole group minus the part that has children leaves the part that does not.
3.NBT.A.2Identify SubproblemsAdd up everyone with no daughters
Every granddaughter is childless too, so the total is 26, choice (E).
The women with no daughters are exactly the whole bottom generation plus the daughters who never had children.
4.OA.A.3Change Focus Count The ComplementWhen a count mixes two groups, peel off the group you already know, then the pieces are easy to add back up.
- Count the granddaughters
- Find how many daughters are mothers
- Find the childless daughters
- Add up everyone with no daughters