AMC 10 · 2004 · #6

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

multi-digit-arithmetic identify-subproblemscomplementary-counting ↑ Prerequisites: mental-arithmetic

📏 Medium solution 💡 1 insight

Problem

Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: Bertha has 6 daughters. Some of those daughters each have exactly 6 daughters of their own, and the rest have none. Counting daughters and granddaughters together gives 30 women, and no one in the next generation exists. Find how many of these 30 women have no daughters.

Givens: Bertha has 6 daughters and no sons.; Each daughter who has children has exactly 6 daughters; the others have 0.; Total number of daughters and granddaughters is 30.; There are no great-granddaughters.

Unknowns: How many of Bertha's daughters and granddaughters have no daughters.

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #16 Change Focus / Count the Complement

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.

Execute — Answer: E

#7 Identify Subproblems 3.NBT.A.2 Step 1

Count the granddaughters

The 30 women are Bertha's 6 daughters plus all of her granddaughters.
Since the 6 daughters are always there, subtract them from 30 to find how many granddaughters there are.

$$30 - 6 = 24 \text{ granddaughters}$$

💡 The total is two groups stacked together, so peeling off the group you already know leaves the other group.

#7 Identify Subproblems 3.OA.A.3 Step 2

Find how many daughters are mothers

Every daughter who has children has exactly 6 of them, and all 24 granddaughters come from these mothers.
Divide the granddaughters into groups of 6 to count how many of Bertha's daughters are mothers.

$$24 \div 6 = 4 \text{ daughters are mothers}$$

💡 If each mother contributes the same size batch, the number of batches tells you the number of mothers.

#7 Identify Subproblems 3.NBT.A.2 Step 3

Find the childless daughters

Bertha has 6 daughters in all, and 4 of them are mothers, so the rest have no daughters.

$$6 - 4 = 2 \text{ daughters with no daughters}$$

💡 The whole group minus the part that has children leaves the part that does not.

#16 Change Focus / Count the Complement 4.OA.A.3 Step 4

Add up everyone with no daughters

All 24 granddaughters have no daughters, because there are no great-granddaughters.
Add them to the 2 daughters who have no daughters to get the total number of women with no daughters.
That total matches choice (E).

$$24 + 2 = 26$$

💡 The women with no daughters are exactly the whole bottom generation plus the daughters who never had children.

[1] #7 3.NBT.A.2 The 30 women are Bertha's 6 daughters plus all of her granddaughters. Since the

[2] #7 3.OA.A.3 Every daughter who has children has exactly 6 of them, and all 24 granddaughters

[3] #7 3.NBT.A.2 Bertha has 6 daughters in all, and 4 of them are mothers, so the rest have no da

[4] #16 4.OA.A.3 All 24 granddaughters have no daughters, because there are no great-granddaughte

Review

Reasonableness: Check the count both ways. Women who DO have daughters are only the 4 mother-daughters, and 30 - 4 = 26, which agrees with the direct count of 24 + 2 = 26. Also 26 is less than 30 and larger than the 24 granddaughters alone, exactly as expected.

Alternative: Instead of counting the women with no daughters directly, count the women who DO have daughters. Only the daughters who are mothers have children, and there are 4 of them, so 30 - 4 = 26 women have no daughters.

CCSS standards used (min grade 4)

3.NBT.A.2 Fluently add and subtract within 1000 (Subtracting the 6 daughters from 30 to find the granddaughters, and subtracting the 4 mothers from 6 to find the childless daughters.)
3.OA.A.3 Solve multiplication and division word problems within 100 (Dividing 24 granddaughters into groups of 6 to count how many daughters are mothers.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Combining the subproblem results to total the women who have no daughters.)

⭐ When a count mixes two groups, peel off the group you already know, then the pieces are easy to add back up.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Count the granddaughters

Find how many daughters are mothers

Find the childless daughters

Add up everyone with no daughters

Review

CCSS standards used (min grade 4)

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