Can you make this equation true?

Logic Level 2

9 9 9 9 = 100 \LARGE 9 \ \ \ \ \ 9 \ \ \ \ \ 9 \ \ \ \ \ 9 \ \ = \ \ 100

What is the fewest number of operators ( + , , × , ÷ ) (+, -, \times, \div) needed on the left hand side of the equation to make the statement true?

Details and Assumptions :

  • For example, if the equation 9 + 9 + 9 ÷ 9 = 100 9 + 9 + 9 \div{9}= 100 is true, then you have used 3 operators: one division and two additions.

  • You are allowed to concatenate the digits, but it does not count as an operation. (That is, you can treat 9 9 as 99.)

Want an additional challenge? Try this harder version !


The answer is 2.

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43 solutions

Nihar Mahajan
Jun 12, 2015

99 + 9 ÷ 9 = 99 + 1 = 100 \Large 99 + 9 \div 9 = 99 + 1 = 100

Moderator note:

Correct.

where in the RULES OF BODMAS does this is being considered? you can not just combine two 9's as 99, 99=9(10)+9 not any 9 9 that's totally different!

Caeo Tan - 5 years, 12 months ago

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I totally agree with you. This is rather guessing than math reasoning.

Louis-Marie Gaulin - 5 years, 12 months ago

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Please read details and assumptions. If you have proper vision , you can see that it is written "You are allowed to combine the digits."

Nihar Mahajan - 5 years, 12 months ago

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@Nihar Mahajan concatenate* then. not concate. that's the correct term

Abhinav Verma - 5 years, 9 months ago

@Nihar Mahajan Then why not 999÷9?

Danny Turner - 4 years, 11 months ago

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@Danny Turner You should prob make sure you can do arithmetic before posting that

Gabe Castelli - 4 years, 11 months ago

@Nihar Mahajan You're right, but I don't change my mind. This is not real math reasoning. I'm not a member of Brilliant for this kind of problem. I won't try to answer to any similar question in the future. Of course, this is OK for people who have interest and agree to answer. Thank you for your attention. By the way, are you really 14? On your photo, you don't look as 14. I suppose that you use the photo of another and anonymous or famous person. Regards.

Louis-Marie Gaulin - 5 years, 12 months ago

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@Louis-Marie Gaulin This problem is not in the math category; it is in logic. All of these problems are about thinking outside of the box.

Andy Luce - 5 years, 7 months ago

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@Andy Luce Being logical isn't thinking outside the box. Thinking outside the box is saying 2+3=5, but if the 2 ='d 3 then 2+3=6. Logically speaking is saying, 2+3=5. It's black and white.

Mayson Mowbray - 5 years, 6 months ago

@Louis-Marie Gaulin If you have any problem with the combining of digit , you are free to express your thought to the person who "posted" this problem rather than telling it to me. I am really 14 and I am a fan of Robert Downey Jr. I have intentionally kept him as my profile pic.

Nihar Mahajan - 5 years, 12 months ago

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@Nihar Mahajan My intention wasn't to express my thought "against" you as one of the guys who found the answer. I simply commented on a post by Caeo Tan, not "against you". Please don't take it personal. I now identify Robert Downey jr. on your photo. Thanks.

Louis-Marie Gaulin - 5 years, 12 months ago

@Louis-Marie Gaulin I don't know what your perception of "Math reasoning" is, but i suggest you improve your logical reasoning first, you don't really make any sense. Concatenate the digits, Whassa pwablem?

Aarush Kumbhakern - 5 years, 2 months ago

Please read details and assumptions. If you have proper vision , you can see that it is written "You are allowed to combine the digits."

Nihar Mahajan - 5 years, 12 months ago

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You sir are a man of great common sense and savageness.

Rico Lee - 5 years ago

the question said that you could concatenate the digits if you want. So it's perfectly correct.

Naman Bhalla - 5 years, 7 months ago

The question description says you can. But yes, it is weird that you have to do that in this case, as you're NEVER allowed to do that in real life.

Ranjeev Grewal - 5 years, 9 months ago

9x9=81+9=90+9=99x9=891+9=900/9=100

Junior Correia - 5 years, 11 months ago

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what a legend

Wenhao Tan - 5 years, 7 months ago

The answer is really close :P

Harsh Shrivastava - 6 years ago

Should be 9 ÷ 9 + 99 or 99 + (9 ÷ 9). Otherwise this could be considered (99 + 9) ÷ 9...

Kieran Uddin - 5 years, 7 months ago

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The statement is correct, because we follow the Order of Operations .

Calvin Lin Staff - 5 years, 7 months ago

No...Order of operation mult/div first then addition/subtraction... 99+9/9 is the same as 99+ (9/9 )

Mindy Leslie - 5 years, 6 months ago

I tried to input the answer but it wouldnt let me, it kept saying I had to have an integer, I had 9÷9+99=100

Eli Dutko - 5 years, 6 months ago

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The answer is how many operators are at least needed.

Niko Nodaros - 5 years, 2 months ago

I do exactly the same as you do.

Chi Tran - 5 years, 12 months ago

can't I do 999 : 9?

Sawaragi Sawaragi - 5 years, 6 months ago

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That would equal 111

Carlos Cruz - 5 years, 6 months ago

its true because the rule allow as to concatenate the numbers.. haleer!

Mise Laine - 5 years, 6 months ago

There's no plus 1

Addison Maddox - 4 years, 10 months ago

the solution might be right if you can insert brackets- (9/9). i like the thinking

Micah Mzumara - 5 years, 12 months ago

you cannot combine two nines like this

Ishaan Panda - 5 years, 12 months ago

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Please read details and assumptions. If you have proper vision , you can see that it is written "You are allowed to combine the digits."

Nihar Mahajan - 5 years, 12 months ago
Barbara Chiu
Jun 14, 2015

99- (9/9) = 100 is an answer, but following PEMDAS, the parenthesis is not needed, so we can rewrite the problem as 99- 9/9 = 100. Thus, the least amount of operators needed would be two ( the subtraction and division ).

Moderator note:

Good job. To ensure that you have indeed found the minimum (and not just a possible bound), you should explain why the answer is not 1.

It must be addition, not subtraction. It's like you're saying: 99- 9/9 = 98

Vlicu Ramirez - 5 years, 7 months ago

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For me it must like this 99-(9/-9) or 99+(9/9) your right.. !!!

jonard adame - 5 years, 7 months ago

it is in bracket so subtraction will change to addition ! 99-(9/9) = 100

Rounak Banerjee - 5 years, 7 months ago

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that makes no sense whatsoever. even with the brackets, the answer is going to be 98

Marcos Real - 5 years, 7 months ago

it will not change the equation even it has bracket!!!!!!! it will fall like this 99-1 =98

Joniel Codoy - 5 years, 7 months ago

+, not -. With -, it seems like you're saying 99 minus 1 is 100.

John Doe - 5 years, 8 months ago

The above replies are right, you probably don't realize you are subtracting instead of adding. Your answer would turn out at 98 instead of 100 :O

Bryan Gonzalez - 5 years, 7 months ago

I assumed adding a decimal was allowed and submitted 1 as an answer: (99/.99=100). Disappointed I am.

Mike Gerber - 5 years, 7 months ago

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It is not allowed. You cannot randomly add decimals to your calculations.

Calvin Lin Staff - 5 years, 7 months ago

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If this problem allows you to combine two numbers, you can add decimals.

Anthony Landati - 5 years, 7 months ago

I have submitted the solution as 99 + 9/9 = 100. it is refusing to accept telling that the answer must be an integer. But answer is given as 100 and i have use minimum operator, right!

Tahir Choudhury - 5 years, 7 months ago

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The answer is 2

Kyle Gillett - 5 years, 7 months ago

You actually gotta answer whats the fewest number of operations necessary, not which operations. Thus, the answer is 2.

Mariana Fiorin - 5 years, 7 months ago

it should be 99+(9/9)

John Bhabe Capili - 5 years, 7 months ago

I'm sorry, I don't actually get it... Wouldn't 99-9/9 be 98? So, Why would that be the answer?

EnDray White - 5 years, 7 months ago

the above answer is 98, not 100

Mindy Leslie - 5 years, 6 months ago
Sravanth C.
Jun 12, 2015

99 + 9 9 \huge 99+\dfrac99

Moderator note:

Why is 2 the minimum? Why not 1 (or Zoidberg)?

This is because 9/9 also contains one operation which is 9÷9

Ayush Gupta - 6 years ago

Sravanth, please be responsible when setting the level of a problem. This problem is clearly not a level 5. If you think that this is at the difficulty of a level 5 problem, that will make me question if you do deserve to be at level 5.

Calvin Lin Staff - 5 years, 12 months ago

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I'm sorry for that sir. In future I'll set the level carefully. I just wanted to see how good the people will be at solving it. I am sorry for inconvenience caused by me

Sravanth C. - 5 years, 12 months ago

This is an improper way to write a fraction. You should only write the proper fraction a b c a \frac bc when 0 < b < c 0 <b<c .

It would be right if you write 99 + ( 9 ÷ 9 ) 99 + (9 \div 9) .

Pi Han Goh - 6 years ago

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Thanks! So is it correct to write the solution as @Nihar Mahajan had written?

Actually @Mehul Arora posted a report saying, 99 + 9 9 99+\dfrac99 will become, 99 ( 9 9 ) 99(\dfrac99)

Sravanth C. - 6 years ago

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Yes, Nihar's one is correct. It's only true if the fraction is a proper fraction, that is the numerator is less than the denominator.

Pi Han Goh - 6 years ago

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@Pi Han Goh Thanks! I had written the correct solution earlier, but due to Mehul's report, I got confused and . . . ;P

Sravanth C. - 6 years ago
Malekai Zagorski
Nov 7, 2015

I know I'm going to get yelled at for this, but I think it's funny I got the correct answer incorrectly.

I understand you are able to CONCATENATE, but I didn't realize you are not able to UNCONCATENATE (i.e. Split the 100 into 1 00.

My solution was:

99 - 99 = 10 * 0

I know it's incorrect. I just think it's interesting.

It's not incorrect. That truly is thinking outside of the box. I don't see why one would say that's incorrect. 👏👏👏

Jeff Sackett - 5 years, 5 months ago

It's incorrect as operations can only be used on the left-hand side.

Kenny O. - 3 years, 8 months ago
Noel Lo
Jun 19, 2015

99 + 9 9 99+\frac{9}{9} HAHAAAHH!!!

Angelo Curcio
Nov 7, 2015

9÷9+99=100

99+(9-9)! =100 ergo god

Uttkarsh Kohli
Jul 18, 2015

99+9/9=99+1=100

[99 is combined as written that you can combine]

Jingyang Tan
Jun 16, 2015

9/9+99=100. 9/9=1,1+99=100. Easy Peeasy Lemon Squezy. im 11 years old by the way...

Sonia Grover
Jun 13, 2015

99+9/9=100

(999 - 99 ) : 9 = 100

99 + 9 ÷ 9 = 100

99+(9-9)! = 100

Kenny O.
Aug 26, 2016

99+9/9=100 (2 operations) It cannot be one as none of the possible solutions are 100;
999+9=1008
999-9=990
999/9=111
999x9=8991
99+99=198
99-99=0
99/99=1
99x99=9801
9-999=-990
9/999=1/111









Tom Bottomley
Jan 4, 2016

9 9 - 9 9 = 1 x 0 0? That'd be two then? :)

Allen Hanscom
Dec 4, 2015

( 99 - 9 ) / .9

The answer is 2. => 99+9/9

Karthik Raj
Nov 30, 2015

99+(9/9) - 2 operators

99+1=100

Logan Bridgham
Nov 23, 2015

an easier solution is 999 ÷9= 100 so it would be 1

999 ÷ 9 = 111.

Sab Samuel - 5 years, 5 months ago
Krishna Kumar
Nov 23, 2015

99 + 9/9 = 99 +1 = 100 using 2 signs.

Abhishek Acharya
Nov 21, 2015

((9/9) +9)*((9/9) +9)= 10 *10 =100

Michael Soroczak
Nov 20, 2015

9 divided by 9 = 1 + 99 = 100

9/9+9 9=1+99=100 by PEMDAS

Prasit Sarapee
Nov 4, 2015

99+9/9 =100 that is two operations.

Harrison Rudston
Nov 2, 2015

9/9=1+99=100 so the operator is 2

Bharat Bhosale
Oct 31, 2015

9/9 + 99 = 100

Debashish Saha
Oct 30, 2015

99 + 9/9 =100

99 + (9 / 9) = 99 +1 = 100

Billy Zottnick
Oct 23, 2015

This would be simpler if we just address the problem as if it were code in Java. Ex: 9 9 9 9 = 100 we could solve this quite easily if we concatenate the digits into 9999 and used only one operator. 9999 != 100. Simple.

Jacco Rubens
Sep 21, 2015

9/9 = 1

1+99 = 100

2 operations.

Hadia Qadir
Sep 8, 2015

99+9/9 = 100

Hirak Ray
Sep 8, 2015

First, we could do 9/9 on the two left-most nines to get 1. Then we add with the remaining concatenated two nines (99). 1 + 99 = 100. In other words, 9/9 + 9 9 =100

Atika Samiha
Sep 8, 2015

9/9+99=100

Blake Edwards
Jun 29, 2015

9÷9 +99=100. Easier expression as the division is first in the equation and there was no need to change to a fraction.

Ellie Justice
Jun 15, 2015

9/9 + 99 = 100

two operators used

AwaIs Ali
Jun 15, 2015

99+9/9=100

Nomrota Anushka
Jun 14, 2015

99+9/9 = 99+1 = 100

Tanvi Bharadwaj
Jun 15, 2015

99+(9/9)=99+1=100

this solution makes sense.

Micah Mzumara - 5 years, 12 months ago
Kryon Krenoteve
Jun 15, 2015

Written in the question: With your answer, define N as the least amount of mathematical operators used. Find N.

I think, the value of N is not 'least', it is 'only'.

Rickie Williams
Jun 15, 2015

PEMDAS = Please excuse my dear aunt Sally. BODMAS = Billy only does math after supper ??? Or is there already a better memory aid?

Vikram Venkat
Jun 14, 2015

99 + 99+ 9 9 \frac{9}{9} = 100 =100

99+9/9= 100

Abdullah Qureshi
Jun 13, 2015

=99+9/9 =99+1 =100

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