Eating Beans
The Bean family are very particular about beans. At every meal all Beans eat some beans... At their last meal they ate 23 beans altogether. How many beans did Pa Bean eat?
Age
11 to 14
Challenge level
Problem
The Bean family are very particular about beans. At every meal all Beans eat some beans. Pa Bean always eats more beans than Ma Bean but never eats more than half the beans.
Ma Bean always eats the same number of beans as both children together, and the two children always eat the same number of beans as each other.
At their last meal they ate 23 beans altogether.
How many beans did Pa Bean eat?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 11
Using a table
2 children together eat the same as Ma Bean
| Ma Bean | 2 | 6 | 8 | ||
|---|---|---|---|---|---|
| Ma + 2 children | 4 | 8 | 12 | 16 | 20 |
| Pa Bean | 19 | 15 | 11 | 7 | 3 |
| Comment | Pa eats more than half | Good | Ma eats more than Pa | ||
Using reasoning
Pa Bean does not eat more than half the beans, so he eats at most 11 beans. Ma Bean eats the same number of beans as both children together, so she eats an even number of beans that is at least one quarter of the total number of beans eaten. Therefore she eats at least 6 beans. If she does eat 6 beans, then Pa Bean eats 11 beans, which is consistent with the information given. However, if Ma Bean eats 8 or more beans, then Pa Bean eats at most 7 beans and this is impossible as we are told that Pa Bean eats more beans than Ma Bean. So Pa Bean eats 11 beans.