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?
- A)
- B)
- C)
- D)
- E)
Answer
E
Key insight
There are 24 granddaughters, so exactly 4 daughters had children; the childless women are the other 2 daughters plus all 24 granddaughters.
Solution
Of the women counted, are Bertha's daughters, so are granddaughters.
Each daughter who has children has exactly of them, so the number of daughters with children is . The remaining daughters have no children.
No granddaughter has children (there are no great-granddaughters). So the women with no daughters are the childless daughters together with all granddaughters:
The answer is .
Why this works
Organize the people by generation and count each generation separately. The key structural fact is that every mother in the second generation has the same number of children, which turns "how many daughters had kids" into a division. Then the final answer is simply the total minus the number of mothers.
Alternative approach
Count the mothers instead: the only women with daughters are the daughters of Bertha who had children. Out of women, have none.
The trap
Answering 24 (only the granddaughters) and forgetting the daughters who have no children, choice (C).
Common mistakes
- Answering 24 (only the granddaughters) and forgetting the daughters who have no children, choice (C).
- Including Bertha herself in the count, or counting the mothers as if they had no daughters.
Techniques
Set up the equation/formula and compute; no special trick needed