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


Run the numbers for houses. For the first 9 houses we need 9 signs.

For the next 90 we need 2 signs each or 180.

For the next 900 houses we need 3 signs each.

If we assume 300 homes, we will need 189 signs for the first 99 homes and then 603 signs for the 201 homes. So, we will use 792 signs for the first 300 homes and will have 420 signs remaining, which is enough for \(\frac{420}{3}\) or another 140 homes.

The total number of homes is 440.

Alternative Explanation:

The first 9 houses will require 9 plates. The next 90 houses will require \(2*90 = 180\) plates. The rest \(x - 99\) houses will require \((x - 99)*3\) plates. The total number of plates will be \(9 + 180 + 3x - 297 = 1212\). From this equation \(x = 440\).


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

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..
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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.

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
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You mean 9 + 90 + 341

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.

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

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.
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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.

Thank you, Ian. Can you please suggest how it should be to avoid confusion? Thank you.
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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?
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Edited as suggested. Thank you very much!
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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.

"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, ...
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An excellent question ! Thanks GMAT team