Question

In a division operation the divisor is 5 times the quotient and twice the remainder. If the remainder is 15, then what is the dividend?
(a) 175
(b) 185
(c) 195
(d) 205

Answer 1

Hint:first of all, assume the quotient be x. The divisor is equal 5 times of quotient and also 2 times of remainder. The value of the remainder is 15. Now, get the numerical value of the divisor with the help of the value of the remainder. Then, get the value of the quotient by using the relation that the divisor is 5 times of quotient. Now, use the formula, \[\text{Dividend=Divisor }\!\!\times\!\!\text{ Quotient+Remainder}\] . Put the numerical values of the divisor, the quotient and the remainder in the formula. Then, solve it further and get the dividend.

Complete step-by-step answer:
According to the question, we have that a division operation the divisor is 5 times the quotient and twice the remainder and the remainder is 15.
Remainder = 15 ………………………(1)
Let us assume the quotient be x.
Since the divisor is 5 times the quotient so, the divisor is 5x.
Divisor = 5x ………………(2)
We also have that the divisor is twice the remainder.
\[\text{Divisor =2 }\!\!\times\!\!\text{ Remainder}\] ……………………….(3)
From equation (1), we have the value of the remainder which is equal to 15.
Now, putting the value of the remainder from equation (1) in equation (3), we get
\[\text{Divisor =2 }\!\!\times\!\!\text{ Remainder}\]
\[\Rightarrow \text{Divisor =2 }\!\!\times\!\!\text{ 15}\]
\[\Rightarrow \text{Divisor =30}\] …………………………..(4)
Now, putting the value of divisor in equation (2), we get
\[\begin{align}
  & \Rightarrow 30=5x \\
 & \Rightarrow 6=x \\
\end{align}\]
The value of the quotient is 6 ………………(5)
We know the relation between dividend, divisor, quotient, and remainder.
\[\text{Dividend=Divisor }\!\!\times\!\!\text{ Quotient+Remainder}\] ……………………………(6)
From equation (1), equation (4), and equation (5) we have the values of the divisor, the quotient, and the remainder.
Now, putting the value of the divisor, the quotient, and the remainder in equation (6), we get
\[\text{Dividend=Divisor }\!\!\times\!\!\text{ Quotient+Remainder}\]
\[\begin{align}
  & \Rightarrow \text{Dividend=30 }\!\!\times\!\!\text{ 6+15} \\
 & \Rightarrow \text{Dividend=180+15} \\
\end{align}\]
\[\Rightarrow \text{Dividend=195}\]
Therefore, the value of the dividend is 195.
Hence, the correct option is (c).

Note: In this question, one might make a mistake in getting the equations. Like, in the question it is given that the divisor is 5 times of quotient. In mathematical form, one may write it as \[\text{Divisor }\!\!\times\!\!\text{ 5=Quotient}\] , which is wrong. The divisor is equal to 5 times of quotient. So, the mathematical equation should be \[\text{Divisor=5}\times \text{Quotient}\] .\[\text{Dividend=Divisor }\!\!\times\!\!\text{ Quotient+Remainder}\] ……………………………(6)
This equation is vital to solve this question as well as used in many areas of mathematics.