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Tricks with numbers

Number of times a particular digit (except 0) appears in a certain range:

The concept, the title of the post talks about, is best understood with the help of examples. This is our way of practical learning and using problem types to understand concepts.
Example 1: Calculate how many times digit 3 appears in first hundred natural numbers?
Solution: Well, there are two series of three’s here:
1. Those set of numbers, which have 3 at unit place. 3, 13,……..,83 ,93, Here we have Ten 3s at the unit’s place
2. Those set of numbers, which have 3 at tens place. 30,31…39.Here we have ten 3s at the ten’s place. So a total of 20 threes make an appearance here.

Example 2: Calculate how many times digit 3 appears in natural numbers, which are less than one thousand?
Solution: In previous example we calculated number of 3 s present till 100 are 20.
Now we look at three digit numbers and calculate how many times digit 3 appears in numbers between 100 and 1000 Well, there are three types of series here:
1. Those set of numbers, which have 3 at unit place.
103, 113, 123, ……..,193
203, 213 ,………………., 293
………………………………….
903,913,………………….993

Here we have 90 numbers having 3 at the unit’s place.
2. Those set of numbers, which have 3 at tens place.
130, 131, 132, ……..,139
230, 231, 232, ……..,239
………………………………….
930,931,………………….939
Here we have 90 numbers having 3 at the ten’s place

3. Those set of numbers, which have 3 at hundred`s place.
301, 301, 302, ……..,399
Here we have hundred 3s at the hundred’s place.
Thus total number of times digit 3 appears in natural numbers, which are less than one thousand are 20+90+90+100=300

Key Observations:
From 1-9, the digit 3 is used 1 time
From 1 – 99, the digit 3 is used 20 times.
From 1 – 999, the digit 3 is used 300 times
From 1 – 9999, the digit 3 is used 4000 times and so on

Note: Above is true for any digit except 0.