Adding plus
Problem
If you write plus signs between each of the digits $1$ to $9$, this is what you get:
$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45$
However, if you alter where the plus signs go, you could also get:
$12 + 3 + 45 + 6 + 7 + 8 + 9 = 90$
Can you put plus signs in so this is true?
$1\;$ $2\;$ $3\;$ $4\;$ $5\;$ $6\;$ $7\;$ $8\;$ $9\;$ $= 99$
How many ways can you do it?Getting Started
How about trying to make $90$ using the digits $1-8$ and leaving the $9$ for adding on at the end?
A good place to start may be to push the $1$ and $2$ together to make $12$. What will you still need to add onto $12$ to make $90$?
Can you make $90$ simply by adding the remaining digits together? What could you do?
Perhaps you could try pushing the $2$ and $3$ together to make $23$? What do you need this time to make $90$?
Student Solutions
We had lots and lots of replies to this problem - it seems to have caused a great deal of interest!
There were three ways to make $99$ by putting addition signs in between the digits. John from Finningley Church of England School says:
Well, what I did was get a piece of paper and wrote down all of
the possible sums. Then I added up all the sums and finally found
the solutions below:
$1+23+45+6+7+8+9=99$
$12+3+4+56+7+8+9=99$
$1+2+3+4+5+67+8+9=99$
Hanna and Brian from Windsor Hill Primary School in Northern Ireland sent us this explanation of how they did the problem:
We used an approach of trial and error to help us find different solutions. We began by choosing bigger numbers to find an approximate answer and then checked the rest of the calculation to see how close we were to $99$. We continued this process adjusting our guess until we hit the target.
Many of you found all three solutions including:
Adam, Ryan, Benjamin, Rebecca, Luke, Alex, Ryo and Jake from
Moorfield Junior School
Ali, Nat, Craig, Lucy, Megan, Rachel, Ryan and Toni from South
Parade Junior School
Jessica, Morgan and James from Aldermaston Primary School
Khoo Kian Koon, Ong Kai Loon and Teo Mei Ting from Corporation
Primary School, Singapore
Isobel, David, James, Liam, Alfie and Joe from Longwick C of E
Combined School
Ã?dil from ErenkÃ? ¶y
IÃ? ¾Ãƒ? ½k Elementary
School in Istanbul
Charlotte and Joanna from Tattingstone School
MinJung from Hanoi International School
Prachi from Canadian Academy
Very well done. Do remember, we want to know HOW you go about solving our problems, not just your answers!