Question

How do you write: The quotient of a number and $ - 5$ increased by one is at most $7$?

Answer 1

Hint: Here we take the quotient of a number to be $x$, and first write the expression by analyzing the question they have given. By analyzing the question we can write the expression as $\dfrac{x}{{ - 5}} + 1 \leqslant 7$ and simplifying this will give us the answer.

Complete step by step answer:
Whenever we have this type of problem saying the quotient of a number and $ - 5$ or in the general quotient of $a$ and $b$ is nothing but $a$ will be the dividend and the $b$ will be the divisor, which can represent as $\dfrac{a}{b}$ or $a \div b$.
Now we need to analyze the question in such a way that we can get an equation that gives the required number.
The first part of the question is the quotient of a number and $ - 5$, let the number be $x$ . Therefore we can say $a$ as $x$ and $b$ as $ - 5$ and can be written as $\dfrac{x}{{ - 5}}$. Now analyze the next half that is, increased by one is at most $7$which means $\dfrac{x}{{ - 5}}$ is increased by one, which can be written as $\dfrac{x}{{ - 5}} + 1$ and next it is at most $7$, which means the value $\dfrac{x}{{ - 5}} + 1$ can be $7$ or less than $7$. Which can be represented as $\dfrac{x}{{ - 5}} + 1 \leqslant 7$. We need to find a dividend which is $x$.
Hence the given equation is $\dfrac{x}{{ - 5}} + 1 \leqslant 7$ . Now simplify this equation to find the number $x$.
$ \Rightarrow \dfrac{{x - 5}}{{ - 5}} \leqslant 7$ this can be written as
$ \Rightarrow \dfrac{{x - 5}}{{ - 5}} \leqslant \dfrac{7}{1}$
Now, by cross multiplication we get
$ \Rightarrow (x - 5) \times 1 \leqslant 7 \times 5$
$ \Rightarrow x - 5 \leqslant 35$
On simplification,
$ \Rightarrow x \leqslant 35 + 5$
$ \Rightarrow x \leqslant 40$

Hence, $x$ is a real number that is less than or equal to $40$.

Note:
Sometimes they may ask to find a quotient by giving both the dividend and divisor value, or they may ask for a remainder. Whatever they ask for, if you know how to write or form the equation by reading the question then you can easily find the answer correctly.