Difference between revisions of "2004 AMC 12A Problems/Problem 4"
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{{duplicate|[[2004 AMC 12A Problems|2004 AMC 12A #4]] and [[2004 AMC 10A Problems/Problem 6|2004 AMC 10A #6]]}} | {{duplicate|[[2004 AMC 12A Problems|2004 AMC 12A #4]] and [[2004 AMC 10A Problems/Problem 6|2004 AMC 10A #6]]}} | ||
| − | + | == Problem == | |
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| − | ==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? | 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 \qquad \mathrm{(E) \ } 26 | + | <math>\mathrm{(A) \ } 22 \qquad \mathrm{(B) \ } 23 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 25 \qquad \mathrm{(E) \ } 26</math> |
| − | ==Solution== | + | == Solutions == |
| + | === Solution 1 === | ||
Since Bertha has <math>6</math> daughters, she has <math>30-6=24</math> granddaughters, of which none have daughters. Of Bertha's daughters, <math>\frac{24}6=4</math> have daughters, so <math>6-4=2</math> do not have daughters. Therefore, of Bertha's daughters and granddaughters, <math>24+2=26</math> do not have daughters. <math>\boxed{\mathrm{(E)}\ 26}</math> | Since Bertha has <math>6</math> daughters, she has <math>30-6=24</math> granddaughters, of which none have daughters. Of Bertha's daughters, <math>\frac{24}6=4</math> have daughters, so <math>6-4=2</math> do not have daughters. Therefore, of Bertha's daughters and granddaughters, <math>24+2=26</math> do not have daughters. <math>\boxed{\mathrm{(E)}\ 26}</math> | ||
| − | + | === Solution 2 (From Alcumus) === | |
Bertha has <math>30 - 6 = 24</math> granddaughters, none of whom have any daughters. The granddaughters are the children of <math>24/6 = 4</math> of Bertha's daughters, so the number of women having no daughters is <math>30 - 4 = \boxed{26}</math>. | Bertha has <math>30 - 6 = 24</math> granddaughters, none of whom have any daughters. The granddaughters are the children of <math>24/6 = 4</math> of Bertha's daughters, so the number of women having no daughters is <math>30 - 4 = \boxed{26}</math>. | ||
| − | ==Solution | + | === Solution 3 === |
Draw a tree diagram and see that the answer can be found in the sum of <math>6+6</math> granddaughters, <math>5+5</math> daughters, and <math>4</math> more daughters. Adding them together gives the answer of <math>\boxed{\mathrm{(E)}\ 26}</math>. | Draw a tree diagram and see that the answer can be found in the sum of <math>6+6</math> granddaughters, <math>5+5</math> daughters, and <math>4</math> more daughters. Adding them together gives the answer of <math>\boxed{\mathrm{(E)}\ 26}</math>. | ||
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| + | == Video Solution == | ||
| + | https://youtu.be/HfOIeW-TGhc | ||
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| + | Education, the Study of Everything | ||
== See also == | == See also == | ||
Latest revision as of 12:56, 19 January 2021
- The following problem is from both the 2004 AMC 12A #4 and 2004 AMC 10A #6, so both problems redirect to this page.
Contents
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?
Solutions
Solution 1
Since Bertha has 
daughters, she has 
granddaughters, of which none have daughters. Of Bertha's daughters, 
have daughters, so 
do not have daughters. Therefore, of Bertha's daughters and granddaughters, 
do not have daughters. 
Solution 2 (From Alcumus)
Bertha has 
granddaughters, none of whom have any daughters. The granddaughters are the children of 
of Bertha's daughters, so the number of women having no daughters is 
.
Solution 3
Draw a tree diagram and see that the answer can be found in the sum of 
granddaughters, 
daughters, and 
more daughters. Adding them together gives the answer of .
Video Solution
Education, the Study of Everything
See also
| 2004 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
| 2004 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
