Question

If dividend and divisor have unlike signs then the quotient will be
(a) Positive
(b) Negative
(c) Zero
(d) None of the above.

Answer 1

Hint: We solve this problem by assuming all possibilities such that dividend and divisor have unlike signs.
We have only two signs that are positive and negative.
(1) Dividend is positive and divisor is negative
(2) Dividend is negative and divisor is positive
We take examples in each possibility and conclude the answer.

Complete step by step answer:
We are given that the dividend and the divisor have unlike signs.
We know that there are only two signs possible for a number: they are positive and negative.
Let us take the possibilities such that the dividend and divisor have unlike signs.
(1) Dividend is positive and divisor is negative
Let us take an example in this possibility that is dividend is 6 and divisor is -3
Now let us divide the number 6 with -3 then we get
\[\Rightarrow \dfrac{6}{-3}=-2\]
Here, we can see that the quotient has negative sign
Now, let us generalise the equation
Let us assume that there are three positive numbers \[x,y,p\] such that \[\dfrac{x}{y}=p\]
Now let us take the dividend as \[x\] and divisor as \[-y\]
Now, by dividing the number \[x\] with \[-y\] then we get
\[\begin{align}
  & \Rightarrow \dfrac{x}{-y}=-\left( \dfrac{x}{y} \right) \\
 & \Rightarrow \dfrac{x}{-y}=-p \\
\end{align}\]
Here, we can see that the quotient is negative.
Therefore we can conclude that the quotient in this case is negative.
Now, let us take the second possibility
(2) Dividend is negative and divisor is positive
Let us take an example in this possibility that is dividend is -6 and divisor is 3
Now let us divide the number -6 with 3 then we get
\[\Rightarrow \dfrac{-6}{3}=-2\]
Here, we can see that the quotient has negative sign
Now, let us generalise the equation
Let us assume that there are three positive numbers \[x,y,p\] such that \[\dfrac{x}{y}=p\]
Now let us take the dividend as \[-x\] and divisor as \[y\]
Now, by dividing the number \[-x\] with \[y\] then we get
\[\begin{align}
  & \Rightarrow \dfrac{-x}{y}=-\left( \dfrac{x}{y} \right) \\
 & \Rightarrow \dfrac{-x}{y}=-p \\
\end{align}\]
Here, we can see that the quotient is negative.
Therefore we can conclude that the quotient in this case is negative.
So, we can say that when the dividend and divisor have an unlike sign then the quotient is negative.

So, the correct answer is “Option b”.

Note: We have a shortcut explanation for this problem.
We have a standard condition of numbers for multiplication and division that is
When we apply the multiplication or division of two numbers having opposite signs then the result will be negative.
By using the above condition we can say that if the dividend and divisor have opposite or unlike signs then the quotient will be negative.
So, option (b) is the correct answer.