Nihal was asked to write all the numbers from 1 to 100 (both included)

Question

Nihal was asked to write all the numbers from 1 to 100 (both included) on board. The ratio of the number of digits which he wrote even number of times to odd number of times is?

A. 4:1
B. 1:4
C. 2:3
D. 3:2
E. 5:2

Husky28 · Answer

When you write from 1 to 100, 1 is written 21 times(1,10,11,12,13,14,15,16,17,18,19,21,31,41,51,61,71,81,91,100) i.e. odd number of times and 2 is written 20 times(2,12,20,21,22,23,24,25,26,27,28,29,32,42,52,62,72,82,92) i.e even number of times. Similar to 2, 3-9 are written 20 times i.e. even number of times. However, O is written 11 times(0,10,20,30,40,50,60,70,80,90,100) i.e. odd number of times. Hence, the ratio of number of digits written even number of times(8 numbers-2,3,4..9) to odd number of times( 2 numbers-0,1) is 8:2 i.e. 4:1

hrushij00 · Answer

I understand how to solve this question just by counting the individual digits from 1-100 and figuring out that digits 2-9 all have the same even frequency of 20 and that only the digits 0 and 1 have an odd frequency. But is there a faster mathematical way to solve this?

Bunuel · Answer

One way or another, you still have to count digit occurrences. Here is another way:

In a full 00 to 99 cycle (treating every number as two digits), each digit 0 to 9 appears exactly 10 times in the ones place and 10 times in the tens place, so 20 times total, which is even.

Now adjust because the actual list is 1 to 100, with no leading zeros.

Compared to 00 to 99, the block 00 to 09 contributes 11 zeros that are not actually written (00 has two zeros, and 01 through 09 each has one leading zero). So the number of zeros drops from 20 to 9 for 1 to 99. Then 100 adds two more zeros, making 11 zeros total.

100 also adds one more 1, so digit 1 becomes odd.

Thus, digits 2 to 9 occur an even number of times, while digits 0 and 1 occur an odd number of times, giving 8:2 = 4:1.

egmat · Answer

There's a useful pattern to speed things up.

For any range 1 to (10^n - 1), like 1-99:

All digits 1-9 appear with EQUAL frequency.
 
This is because they're completely symmetric - each appears the same number of times in each place value.

So you only need to:
1. Calculate frequency for ONE digit (say, 5) → 20
2. Instantly know digits 1-9 all have frequency 20

The only "special" digit is 0, which appears less often (no leading zeros).
For 1-99: digit 0 appears 9 times.

Then just adjust for 100:
- Adds two 0s → 9 + 2 = 11 (odd)
- Adds one 1 → 20 + 1 = 21 (odd)
- Digits 2-9 unchanged → 20 (even)

Result: 8 even, 2 odd → 4:1

There's no closed-form formula that skips the counting entirely, but recognizing the symmetry of digits 1-9 cuts your work significantly.

Answer: A (4:1)

Nihal was asked to write all the numbers from 1 to 100 (both included)

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Nihal was asked to write all the numbers from 1 to 100 (both included)

Prep Toolkit

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