# A book has 400 pages numbered 1,2,3 and so on. How many times does the digit 3 appear in the page numbers?

The way of representing the numbers using digits, symbols are called the number system. It is also called the set of values that are used to represent quantity and things. There are different types of Number systems, the most commonly known and used is the Decimal Number system, the digits involved in the decimal numbers are from 0-9 with the base 10. There are mainly four types of number system that is used for representing are:

- Decimal Number (0, 1, 2, 3, 4, 5 .. (so on) with base 10).
- Binary Numbers ( set of 0s and 1s with base 2).
- Octal Numbers (numbers with base 8 i.e., 55
_{8 }) - Hexadecimal Numbers (numbers with base 16, represent using A, B, C, D, E, F)

### Decimal Number system

Decimal systems are the number systems that have base 10, that is, there are 10 digits (from 0-9) used in decimal systems. Some examples of numbers in the decimal system are, (89)_{10}, (117)_{10}, (598)_{10}, etc. The digits in this system will have a difference of 10 times, for instance, (89)_{10}, here 8 has its value 10 times more than 9. Therefore, the value at tens place is ten times the value at units place. Below is a detailed explanation of how numbers are formed,

- (34)
_{10}= 3× 10^{1}+ 4× 10^{0} - (965)
_{10}= 9× 10^{2}+ 6× 10^{1}+ 5× 10^{0}

### A book has 400 pages numbered 1,2,3 and so on. How many times does digit 3 appear in the page numbers?

**Solution:**

Let us, calculate number of 3 in the first 99 pages.

1 3s in the first 9 pages.

1 3s in the pages from 10 – 19.

1 3s in the pages from 20 – 29.

11 3s in the pages from 30- 39 (one each for 30- 32 and 34- 39 and 2 for 33).

6 3s in the pages from 40- 99 ( 43, 53, 63, 73, 83, 93).

So, we have 20 3s in the first 99 pages.

Now,

We have 20 3s in pages 100- 199, 20 3s in pages 200-299 and 0 3s on page 400.

Now,

For pages, 300-399, there are 100 3s for the hundred digit in addition to the 20 3s found in the tens and ones digit.

Conclusion:

So, we have a total of 20 + 20 + 20 + 120 = 180 3s on the page numbers.

**Similar Questions**

**Question 1: A book has 500 pages numbered 1,2,3 and so on. How many times does digit 2 appear in the page numbers?**

**Solution:**

Let us, calculate number of 2 in the first 99 pages.

1 2s in the first 9 pages.

1 2s in the pages from 10 – 19 pages.

1 2s in the pages from 30 – 39 pages.

11 2s in the pages from 20 – 29 (one each for 20-21 and 23 – 29 and 2 for 22).

6 2s in the pages from 40- 99 ( 42, 52, 62, 72, 82, 92).

So, we have 20 2s in the first 99 pages.

Now,

We have 20 2s in pages from 100- 199, 20 2s in pages from 300-399 and 20 2s in pages from 400 – 499 and 0 2s on page 500.

Now,

For pages, 200-299, there are 100 2s for the hundred digit in addition to the 20 2s found in the tens and ones digit.

Conclusion:

So, we have a total of 20 + 20 + 20 + 20 + 120 = 200 2s on the page numbers.

**Question 2: A book has 600 pages, numbered 1, 2, 3…and so on. Find out how many times does 7 appear in the book.**

**Solution:**

As explained in the above questions, there will be 20 7s in the first 99 pages, 20 more in the pages from 100- 199, 20 more from 200- 299, and so on.

Hence, the number of 7s present in the book,

20 (0-99)+ 20 (100-199)+ 20 (200- 299)+ 20 (300- 399)+ 20 (400- 499) + 20 (500- 600)=

120 7s

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