Question

How do you define a variable and write an expression for each phrase: the quotient of 6 times a number and 16?

Answer 1

Hint: We recall what quotient means when we divide the dividend by the divisor and recall Euclid’s division lemma $n=dq+r$. We assume the remainder is zero and we assume the variable as any number $x$, we multiply 6 to it to get the dividend as $6x$ and the divisor as 16 to express $q$ in terms of $n$ and $d$.

Complete step-by-step solution:
We know that in arithmetic operation of division the number we are going to divide is called dividend, the number by which divides the dividend is called divisor. We get a quotient which is the number of times the divisor is of dividend and also remainder obtained at the end of division. If the number is $n$, the divisor is $d$, the quotient is $q$ and the remainder is $r$, they are related by the following equation,
\[n=dq+r\]
Here the divisor can never be zero. The above relation is called Euclid’s division Lemma. If remainder is zero which means the divisor exactly divides the number , then we shall have
\[\begin{align}
  & n=dq \\
 & \Rightarrow q=\dfrac{n}{d} \\
\end{align}\]
 We are given the question that we have to define a variable and write an expression for the phrase “the quotient of 6 times a number and 16” . So we need an algebraic expression for the quotient. So let us assume the number as $x$ and 6 times the number is $6\times x=6x$. The word ‘and’ tells us that there is a division operation between $6x$ and 16. So we can express the quotient as
\[q=\dfrac{6x}{16}=\dfrac{3x}{8}\]

Note: We note that an algebraic expression is a finite mathematical expression involving constants and variables. The variables are unknown to us and constants are known numbers. Here in this problem in $\dfrac{3x}{8}$ the variable is $x$ and constants are $3,8$. The algebraic expression most of the time represents quantity.