Difference between revisions of "2004 AMC 12A Problems/Problem 4"

(added category)
m (fixed typo in answer choice)
Line 2: Line 2:
 
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?
 
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?
  
<math> \mathrm{(A) \ } 22 \qquad \mathrm{(B) \ } 23 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 25 {2}\qquad \mathrm{(E) \ } 26  </math>
+
<math> \mathrm{(A) \ } 22 \qquad \mathrm{(B) \ } 23 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 25 \qquad \mathrm{(E) \ } 26  </math>
  
 
==Solution==
 
==Solution==

Revision as of 12:08, 5 November 2006

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?

$\mathrm{(A) \ } 22 \qquad \mathrm{(B) \ } 23 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 25 \qquad \mathrm{(E) \ } 26$

Solution

Since Bertha has 6 daughters, Bertha has $30-6=24$ granddaughters, of which none have daughters. Of Bertha's daughters, $\frac{24}6=4$ have daughters, so $6-4=2$ do not have daughters.

Therefore, of Bertha's daughters and granddaughters, $24+2=26$ do not have daughters $\Rightarrow\mathrm{(E)}$.

See Also

Invalid username
Login to AoPS