Question

In a three digit number, the digit in the units place is four times the digit in the hundreds place. If the digits in the units place and tens place are interchanged, the new number so formed is 18 more than the original number. If the digit in the hundreds place is one third of the digit in the tens place, what is 25% of the original number?
a. 67
b. 84
c. 73
d. 64

Answer 1

Hint: In order to solve the question, we will consider the three digit number as 100x + 10y + z where x, y and represents the hundreds, tens and ones digit place respectively. So, we will form all the possible equations from the given conditions and then we will find the number to find its 25% and get the answer.

Complete step-by-step answer:
In this question, we have been asked to find the 25% of a three digit number and we have been given a few conditions for the digit of the number. To solve this question, we will first consider a 3 digit number as 100x + 10y + z where, x, y and z represents the hundreds, tens and ones place.
So, according to the conditions given in the question, that is, the digit in the units place is four times the digit in the hundreds place. So, we can say that,
z = 4x ……… (i)
Also, we have been given that if the digits in the units place and tens place are interchanged, the new number so formed is 18 more than the original number. So, we can say that,
100x + 10z + y = 18 + 100x + 10y + z
Now, taking all the terms with x, y and z to one side and the constant term, that is, 18 to the other side, we will get,
10z + y + 100x – 100x – 10y – z = 18
On simplifying, we get,
9z – 9y = 18
Now, dividing the above expression by 9, we get,
z – y = 2
z = 2 + y ……… (ii)
We have also been given that hundreds digit is one third of the tens digit. So, we can write,
$x=\dfrac{y}{3}.........\left( iii \right)$
From equation (i) and equation (ii), we can say,
4x = 2 + y
And if we put the value of x from equation (iii) in the above equation, we get,
$\begin{align}
  & 4\times \dfrac{y}{3}=2+y \\
 & \dfrac{4y}{3}-y=2 \\
 & y\left( \dfrac{4}{3}-1 \right)=2 \\
 & y\left( \dfrac{1}{3} \right)=2 \\
 & y=2\times 3 \\
 & y=6.........\left( iv \right) \\
\end{align}$
Now, we will put the value of y in equation (iii) and (ii). So, we get,
$x=\dfrac{6}{3}$ and z = 2 + 6
x = 2 ……… (v) and z = 8 ……… (vi)
Hence, we get the value of x, y and z as x = 2, y = 6 and z = 8. Therefore, we get the three digit number as,
100 (2) + 10 (6) + 8
=200 + 60 + 8
=268
Now, we have been asked to find the 25% of the original number. So, we get,
25% of 168
$\begin{align}
  & \dfrac{25}{100}\times 268 \\
 & \dfrac{268}{4}=67 \\
\end{align}$
Hence, we get the 25% of the original number as 67. Therefore, option (a) is the correct answer.

Note: One can think of solving this question with the help of the options, considering the original number as x and then by considering each option as 25% of x, one by one and then finding the value of x and looking for that value of x which satisfies all the given conditions of question. It can give us the answer, but it is a very time consuming method.