Make 100
2 out of 3
Find at least one way to put in some operation signs to make these digits come to 100.
Problem
Find at least one way to put some operations signs ($+$, $-$, $\times$, $\div$) between these individual digits to make the digits come to 100.
1 2 3 4 5 6 7 8 9 = 100
Getting Started
How close can you get with just adding?
Which sorts of calculations make the most difference?
Student Solutions
Well we had a lot of solutions sent in and here are just some that represent the kind of answers submitted.
$(1 + 2 + 3 + 4) \times 5 + 67 - ( 8 + 9 ) = 100$
$1+2+3+4+5=15$, $15\times6=90$, $90-7=83$, $83+8=91$, $91+9=100$
$1\times2+3=5$, $5\times4\times5+6= 106$, $106-7=99$, $99-8=91$, $91+9=100$
$(({8\times9}\div4)\times5) + ((7-6)+1)\times(3+2)) = 100$
$9\times8=72$, $1+2+3+4+5+6+7 = 28$, $72+28= 100$
$1+8=9$, $9\times9=81$, $81+6=87$, $87+3=90$, $90+2=92$, $92+7=99$, $99+5=104$, $104-4=100$
$5\times4=20$, $20\times3=60$, $60\times2=120$, $120-(4\times5)=100$
$(1+2+3-4)+5+6+78+9=100$
$1+2+3+4+5+6+7+(8\times9)=100$
$9\times7 = 63$, $63 + 6\times5 = 93$, $93 + 4 + 3= 100$
$(1 + (2 \times3) + (4 \times5) - 6) + 7 + (8 \times9) = 100$
$-1 \times 2 - 3 - 4 - 5 + 6\times7 + 8\times9 = 100$
$9\times6=54$, $54\times2=108$, $108-4-5=99$, $99+7=106$, $106-8=98$, $98-1=97$, $97+3=100$
$9\times8=72$, $72+7+1=80$, $80+4+6=90$, $90+5+2+3=100$
$9+1=10$, $10\times6=60$, $60+8+2=70$, $70+(7\times4)=98$, $98+5-3=100$
$1\times2+3=5$, $5\times4\times5-6=94$, $94+7+8-9=100$
$6\times4=24$, $24+1=25$, $25\times5=125$, $9+7=16$, $16+8=24$, $24-2=22$, $22+3=25$, $125-25=100$
$6+1=7$, $7\times7=49$, $9\times5=45$, $49+45=94$, $7+8=15$, $2+3+4=9$, $15-9=6$, $94+6=100$
$6+5=11$, $11\times7=77$, $4+3+9+8=24$, $24-2=22$, $22+1=23$, $77+23=100$
$6+7=13$, $13\times5=85$, $9+8=17$, $17-4=13$, $13+3=16$, $16-2=14$, $14+1=15$, $85+15=100$
$4+3=7$, $7\times9=63$, $8\times6=48$, $5\times2=10$, $48-10=38$, $38-1=37$, $63+37=100$
$4+5=9$, $9\times6=54$, $7x3=21$, $21+9=30$, $2\times8=16$, $30+16=46$, $46\times1=46$, $46+54=100$
$4+6=10$, $10\times7=70$, $7+8=15$, $15+9=24$, $24+5+2=31$, $31-1=30$, $70+30=100$
$4+7=11$, $11\times3=33$, $9\times6=54$, $8\times1=8$, $54+8=64$, $64+5=69$, $69-2=67$, $67+33=100$
$(1+9)(2+8)((7-3)\div4)(6-5) = 100$
$((1+2 + 3+4) \times (5+ 6)) + 7 - 8 - 9 = 100$
Here are some of the accounts that described the all-important processes that were used.
We started it by picking out two numbers and multiplied them together to get an answer and then we multiplied another pair of numbers to get an answer and added them together to get $87$ and added the remaining digits to get a subtotal of $100$. (Courtney and Michelle from Denfield Park Junior School)
We needed to get the biggest possible number so we multiplied the biggest numbers ($9$ and $8$). Then we added all the other numbers up in random order. We found we reached $100$ using every number.(Nadia and Millie, Greenacre School for Girls)
Our aim was to get to $50$ and double it, so we timesed $7$ by $9$ to get to $63$ and then minused it down to $60$ and then down to $50$. Then we timesed by $2$ to get $100$. (Karla and Gemma, Greenacre School for Girls)
We thought it would be good to start with number bonds to $10$. As we only had one $5$, we decided to use the other four number bonds to $10$ ($6 + 4$, $7 + 3$, $1 + 9$, $2 + 8$), and then use addition and subtraction to make $20$ and then times that by $5$ to get to $100$. (Maddie and Harriet A., Greenacre School for Girls)
Thank you to all involved - a splendid set of results!
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem offers great opportunities for mental arithmetic and estimation. It can also be used as an opening to discussing the order of operations.
Possible approach
Display the numbers 1 - 9 on the board and ask the children to add them up. (They might do this in any order, perhaps noticing that pairs from either end add to 10.) As they explain their working, record it in order on the board, for example:
$1+ 9 +2 + 8 + 3 + 7 + 4 + 6 + 5 = 45$
Ask if they can suggest a way to make the answer bigger, but still only using the numbers 1 - 9. Again, record the calculations on the board in the order that the children say them. This is likely to involve a multiplication sign. Ask if they can make it even bigger. Again, record the calculations.
Then offer the problem. Allow some time for children to work, possibly in pairs, and provide calculators for them to use to check their arithmetic if necessary. Provide a central wall space for children to record their solutions. This would make an ideal 'simmering' activity that could go on for a week or more.
Key questions
How close can you get to 100 with just adding?
What operation might you use to make the result bigger?
Which sorts of calculations make the most difference to the total?
Which numbers less than 100 is it possible to make?
What other questions can you suggest?
Do you get the same answer every time from your string of calculations? If not, why don't you?
Possible extension
An additional challenge would be for the children to decide on their own target number and see if they can make it using 1 - 9.
If your children know and use the convention of the order of operations this can be an opportunity to ask whether the order that they have written the calculations in is the same as the order in which they would do them. Is there a better way they could write the same calculation, using the correct order of operations?
Possible support
The numbers could be written on separate pieces of paper, together with several $+$, $-$, $\times$ and $\div$ signs. Being able to rearrange the numbers can sometimes help to see patterns or number bonds that help with calculations. Digit and operation cards could also be used. And whilst the problem offers great opportunities for mental arithmetic and estimation, pupils who are less confident at these could use a calculator. This would help to support their estimation skills and include them in a whole class activity.