AMC 10 · 2004 · #6
Grade 4 arithmeticPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
The 30 women are Bertha's 6 daughters plus every granddaughter, so peeling off the 6 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 6 daughters, so splitting the 24 granddaughters into groups of 6 gives 4 mothers.
If each mother contributes the same size batch, the number of batches tells you the number of mothers.
If each mother contributes the same size batch, the number of batches tells you the number of mothers.
▸ Why?
Equal batches repeated is a count of equal groups, which multiplying records.
▸ Why?
Dividing undoes multiplying, so sharing the total into equal batches hands back how many there were.
Find the childless daughters
Of Bertha's 6 daughters, 4 are mothers, so 2 are childless.
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
No great-granddaughters means all 24 granddaughters are childless, so add the 2 childless daughters to get 26.
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