M07-37

Question

A company will use 1212 plates to number all of the houses in a new development. The numbers on the houses will be consecutive integers, beginning from 1. If each plate displays a single digit (so the number '97' requires two plates, and the number '104' requires three plates) how many houses are in the development?

A. 260
B. 440
C. 556
D. 792
E. 1200

A. 260
B. 440
C. 556
D. 792
E. 1200

Bunuel · Accepted Answer

Official Solution:

A company will use 1212 plates to number all of the houses in a new development. The numbers on the houses will be consecutive integers, beginning from 1. If each plate displays a single digit (so the number '97' requires two plates, and the number '104' requires three plates) how many houses are in the development?

A. 260
B. 440
C. 556
D. 792
E. 1200

A. 260
B. 440
C. 556
D. 792
E. 1200

There will be 9 single-digit numbered houses (ranging from 1 to 9). As a result, only 9 plates are required for these houses.
 
There will be 90 double-digit numbered houses (ranging from 10 to 99). As a result, only 90*2 = 180 plates are required for these houses.
 
Therefore, out of the total 1212 plates, 189 have been used for the 99 houses mentioned above, leaving 1023 plates. These plates will be used for triple-digit numbered houses (since they are not enough for both triple and four-digit numbered houses). 
 
Since 1023 plates are sufficient for 1023/3 = 341 houses, the total number of houses must be 341 + 99 = 440.

Answer: B

chetan2u · Answer

Hi,
The Q is correct as it is, and I would say "a good Question'
what you are referring to is not single digit but a number, as a number can have 2,3 etc digits....
if we do not talk of the base, we are working on, it is supposed to be decimal system..
Decimal system has 10 digits, 0 to 9..
Binary system has two digits, 0 and 1..
and so on..
so the moment we say digits plate, it means single digit plate..

the houses can have 1,2,3,.. digit numbers..
since it is consecutive and starts from 1..
single digit platyes required are 9, 1 to 9..
2 digit numbers, thereafter, are 2*90, 10 to 99...

now lets see how many plates are left..
1212-9-180=1023..
if these are 3 digit numbers, number of digit plates required=1023/3=341..
total number of houses= 9+180+341=440, which is a three digit number and so OK..

buffaloboy · Answer

I think this is a  poor-quality  question. Pl mention that digit plates contain single digit each.

buffaloboy · Answer

Hi , 
I don't understand how to understand that digit plates contain single digits only ? Only information available is that there are 1212 digit plates. Even if I understand that digit plates contain only single digits then it is likely possible that all the plates contain only 1s or 2s , though it may sound irrational . Also I don't understand why would we prefer more number of 1s, 2s,3s,4s, over 5s,6s,7s,9s,? Hope you address what my confusion is ! I think Question can be tweaked a bit by adding that each digit plate contain contain single digit and no number is preferred over other. In that case ans would be less than 441 and some of the plates would be left unused.

chetan2u · Answer

Hi,
i would believe that you are asking two Qs..
 1)how to understand that digit plates contain single digits only .. 
digits are supposed to be single and are from 0 to 9 in decimal system we follow...
WHY? because that is what is the defination of digit is.  .any of the numerals from 0 to 9, especially when forming part of a number.  ..
we say 159 has three digits 1,5, and 9..

2)we prefer more number of 1s, 2s,3s,4s, over 5s,6s,7s,9s,? 
NO, we don't prefer any number over any other number....
it is given that the houses are numbered consecutively starting from 1..
so the numbering will be 1, then2,3,4,5.....88,89,90....410,402,.....439,440 and that is where all the 1212 digits plates are used up..

Hope it helped

Hawkmoon · Answer

You mean 9 + 90 + 341 :wink:

NatC · Answer

Here's how I solved it quickly:

The question talks about numbering houses with digit plates only. A digit is an integer havng only a unit's place. 
Thus, the first 9 houses would need a total of 9 digit plates.

The houses numbered from 10 to 99 would each need need 2 digit plates. Total number of houses in this range = 99-10+1= 90 houses. Total number of plates= 90*2=180 digit plates
So far, we have already used up 9+180 plates for a total of 9+90 houses.

We are left with 1212-189 plates now, i.e. 1023 digit plates.

Now, all houses numbered from 100 to 999 would each need 3 digit plates. Total number of houses in this range= 999-100+1= 900 houses. Total number of plates needed= 900*3= 2700 digit plates.
But we only have 1023 plates. So we can only number a few houses from the total of 900 in this range with the remaining plates.

Remaining plates can be used to number these many houses= 1023/3 = 341 houses

Therefore, total number of houses numbered using all of 1212 plates = 9+90+341= 440 houses.

alexwines24 · Answer

I think this is a poor-quality question. What is a digit plate? :lol:

IanStewart · Answer

Mathematically, I think it's quite a good question, but I also needed to spend a minute pondering what a 'digit plate' might be, and had to deduce what was meant based on my familiarity with similar problems I've seen in the past. If I were writing the question, I'd be more clear about what 'digit plate' means.

alexwines24 · Answer

Fair, but if 'the company puts an order for 1212 plates', this insinuates that the company is going to throw a ton of plates away. Seems that either the company needs to hire a new manager or the question should state that '1212' is the largest number on any of the plates ordered by the company and clarify what it means by 'digit plates. I agree that this is a good question mathematically, but I doubt that a question with this degree of ambiguity would show up on the GMAT. Maybe I'm just bitter because I got it wrong.

Bunuel · Answer

Thank you, Ian. Can you please suggest how it should be to avoid confusion? Thank you.

IanStewart · Answer

I prefer when writing questions to be 'too clear', rather than allow any possibility of misunderstanding, so off the top of my head, I'd probably write this question like I've done below. I'm sure this can be improved with some editing though:

A company will use 1212 plates to number all of the houses in a new development. The numbers on the houses will be consecutive integers, beginning from 1. If each plate displays a single digit (so the number '97' requires two plates, and the number '104' requires three plates) how many houses are in the development?

A. 260
B. 440
C. 556
D. 792
E. 1200

Bunuel · Answer

Edited as suggested. Thank you very much!

Kidbeat · Answer

Hello,
Can anyone explain the role of the word consecutive here?
I interpreted it as the numbers should only be of the form 12, 23, 34, 123, 234, etc.

Bunuel · Answer

"Consecutive integers" ALWAYS mean integers that follow each other in order with common difference of 1:  ... x-3, x-2, x-1, x, x+1, x+2, ... .

"Consecutive integers, beginning from 1" are 1, 2, 3, 4, ...

RRJ12 · Answer

An excellent question ! Thanks GMAT team

Paok · Answer

my easy 30 sec approach

1-9 houses 1*9=2 
10-99 houses 2*90=180
100-x houses 3*(100-x)

LETS MAKE A EQUATION 9+2*90+3*H= 1212. 
Solving this gives us H(houses)=341 
so 341+90+9= 440

Wish you guys all the best and luck with your studies

Rabab36 · Answer

TOTAL NO OF HOUSES ARE 540 INSTEAD OF 440

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M07-37

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