Question

If $y$ varies directly with $x$, if $y=-36$ when $x=6$, how do you find $x$ when $y=42$?

Answer 1

We explain the process to get a ratio of two numbers $x$ and $y$ from the given values of $y= -36$ when $x=6$. As the values of $x$ and $y$ are directly proportional, we try to describe the relation between the denominator and the numerator. We use the G.C.D of the denominator and the numerator to divide both of them. We get the simplified form as the G.C.D is 1.

Complete step by step solution:
If $y$ varies directly with $x$, the ratio of $\dfrac{y}{x}$ has to remain constant for any values of $x$ and $y$.
The ratio is used to find the unitary value of a particular number with respect to the other number. Therefore, for the ratio of any two numbers $y$ and $x$, we can express it as $\dfrac{y}{x}$. Ratios work like fractions. Simplified form is achieved when the G.C.D of the denominator and the numerator is 1.
For the fraction $\dfrac{y}{x}$, we first find the G.C.D of the denominator and the numerator. If it’s 1 then it’s already in its simplified form and if the G.C.D of the denominator and the numerator is any other number d then we need to divide the denominator and the numerator with d and get the simplified fraction form as $\dfrac{{}^{y}/{}_{d}}{{}^{x}/{}_{d}}$.
For the values $y=-36$ when $x=6$, we get $\dfrac{y}{x}=\dfrac{-36}{6}=-6$.
Let assume for $y=42$ the value of $x$ will be $a$. Then the ratio is $\dfrac{y}{x}=\dfrac{42}{a}$.
So, $\dfrac{y}{x}=\dfrac{42}{a}=-6$ which gives $a=\dfrac{42}{-6}=-7$.
The value of $x$ is $-7$.

Note: The process is similar for both proper and improper fractions or ratios. In case of mixed fractions, we need to convert it into an improper fraction and then apply the case. Also, we can only apply the process on the proper fraction part of a mixed fraction.