Question

How many numbers between 400 and 600 begin with or end with a digit of 5?
(a) 40
(b) 100
(c) 110
(d) 120

Answer 1

Hint: First of all, we will find the numbers between 400 and 600 which begin with digit 5 so the numbers which begin with the digit 5 are the numbers which are from 500 to 599. Now, we are going to find the numbers which end with digit 5. The counting of these numbers are done as from 400 to 410, 405 will end with 5. Similarly, you can find a number which ends with digit 5 between 411 to 420 and which is equal to 415. Now, count all the numbers which end or start with digit 5.

Complete step by step solution:
The numbers starting with digit 5 between 400 and 600 are:
500, 501, 502, 503………599
The counting of the above numbers which start with digit 5 are 100.
Now, we are going to find the numbers which end with the digit 5.
Let us count the numbers from 400 to 499 which end with the digit 5. From 400 to 410, 405 is the number which will end with the digit 5. After that, from 410 to 420, 405 is the number which end with the digit 5. Similarly, you can find the numbers which end with the digit 5 from 400 to 499 as follows:
405, 415, 425, 435………495
As you can see the above series is the sequence of A.P. in which the first term $\left( a \right)$ is equal to 405 and the common difference (d) is 10. Now, using the general term of an A.P. we can find the number of terms present in the above sequence.
${{T}_{n}}=a+\left( n-1 \right)d$
Substituting ${{T}_{n}}=495,a=405,d=10$ in the above equation and we get,
$495=405+\left( n-1 \right)\left( 10 \right)$
Subtracting 405 on both the sides of the above equation we get,
$\begin{align}
  & 495-405=\left( n-1 \right)10 \\
 & \Rightarrow 90=10n-10 \\
\end{align}$
Adding 10 on both the sides of the above equation we get,
$\begin{align}
  & 90+10=10n \\
 & \Rightarrow 100=10n \\
 & \Rightarrow n=10 \\
\end{align}$
Hence, the number of numbers between 400 and 499 which end with digit 5 is 10.
And we have already found the number of numbers which start with digit 5 from 500 to 599 so we have covered all the possibilities.
Now, adding the number of numbers start with 5 and end with digit 5 we get,
$\begin{align}
  & 100+10 \\
 & =110 \\
\end{align}$
Hence, 110 numbers from 400 to 600 start or end with digit 5.

Note: You might be thinking why we haven’t counted the numbers from 500 to 600 which will end with digit 5. The reason is that we have already counted those numbers when we have counted the numbers from 500 to 600 which start with digit 5.
For e.g. 525, in this number, digit 5 is in the starting and end also so counting this number in the name of the number which starts with digit 5 makes us already count the number which ends with digit 5.