How to count number between two given numbers

Counting

Let’s illustrate it with an example.
Find the number of numbers between 100 to 200 if
i. Both 100 and 200 are counted.
ii. Only one of 100 and 200 is counted.
iii. Neither 100 nor 200 are counted.

Solution:
i. Number of numbers when both ends included: (200 - 100) + 1 = 101.
ii. Number of numbers when one of 100 and 200 is counted: 200 - 100 = 100.
iii. Number of numbers when neither 100 nor 200 is counted = 200 - 100 - 1 = 99.

Example 2:
How many multiples of 17 are there between 300 and 500?

Solution:
Required number
$$\eqalign{ & = \frac{{500}}{{17}} - \frac{{300}}{{17}} \cr & = 29\frac{7}{{17}} - 17\frac{{11}}{{17}} \cr & = 29 - 17 \cr & = 12 \cr} $$

Example 3:
Find the number of those numbers which are between 200 and 600 and divisible by 4, 5 and 6.

Solution:
Here the LCM of 4,5 and 6 = 60.
Hence, required numbers,
$$\eqalign{ & = \frac{{600}}{{60}} - \frac{{200}}{{60}} \cr & = 10 - 3\frac{{20}}{{60}} \cr & = 10 - 3 \cr & = 7 \cr} $$

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