Cherries Come in Twos
Problem
Susie had a large bowl of cherries.
They were all in pairs.
Susie took out a pair, she ate one cherry and put the other one
back. She took out another pair and did the same again.
Then she helped herself to one of the single cherries in the
bowl.
Susie continued helping herself to the cherries in this way (pair, pair, single - pair, pair, single - ...)
After she had done this lots of times, there were just $14$ single cherries left.
How many cherries had there been in the bowl to start with?
Getting Started
How many cherries does Susie eat during each round of helping herself to pair, pair, single cherries?
How many extra single cherries are there in the bowl after each round?
Student Solutions
Abigail from Histon and Impington Infants School sent a very clear solution to this problem:
I used some counters to represent the cherries. I did what Suzie did and worked out that if you started with $4$ cherries, you would end up with $1$ left after doing pair, pair, single. If you started with $8$ cherries, you would end up with $2$. If you started with $12$, you would end up with $3$, and if you started with $16$, you would end up with $4$.
I spotted that the end numbers went up by one each time, and the start numbers went up by four. Then I did a table:
| Start | End |
| 4 | 1 |
| 8 | 2 |
| 12 | 3 |
| 16 | 4 |
| 20 | 5 |
| 24 | 6 |
| 28 | 7 |
| 32 | 8 |
| 36 | 9 |
| 40 | 10 |
| 44 | 11 |
| 48 | 12 |
| 52 | 13 |
| 56 | 14 |
So there were $56$ cherries in the bowl at the start.
Well reasoned, Abigail.
Teachers' Resources
Cherries Come in Twos
Susie had a large bowl of cherries.
They were all in pairs.
Susie took out a pair, she ate one cherry and put the other one back. She took out another pair and did the same again.
Then she helped herself to one of the single cherries in the bowl.
Susie continued helping herself to the cherries in this way (pair, pair, single - pair, pair, single - ...)
After she had done this lots of times, there were just $14$ single cherries left.
How many cherries had there been in the bowl to start with?