Question

If y varies directly with x, if y=12 when x=18, how do you find x when y=-16?

Answer 1

Hint: The given question will be solved by the concept of ratio and proportion. Since y varies directly with x, we can infer that ratio of x and y will be a constant. We will write the given data in terms like this $\dfrac{a}{b}=\dfrac{c}{d}$ , where $\dfrac{a}{b},\dfrac{c}{d}$ are the two ratios. We will substitute a as 18, b as 12, c as unknown and d as -16 and then get the answer.

Complete step by step solution:
We can solve these questions using the fact that x and y can be written as fraction and equated to a constant since they vary directly.
We will consider all values of x and y as below,
${{y}_{1}}=12,\text{for }{{\text{x}}_{1}}\text{=18 and }{{\text{y}}_{2}}\text{=-16}$
Then we will write these as ratios since x and y vary directly. So, whatever values of x and y are given, we can write them as a ratio and equate them since they will be equal to a constant and in each other in return. We will get
$\dfrac{18}{12}=\dfrac{{{x}_{2}}}{-16}$
Now, we will solve for the unknown quantity. We will compare the ratio by cross multiplying and finding out the value of the unknown. On cross multiplying we get :
\[\Rightarrow {{x}_{2}}=\dfrac{18\times (-16)}{12}\]
We know that the product of positive and negative integers results in negative integers. So here we have $18$ and $-16$ as numerators and its products come to $-288$ which is a negative integer.
\[\Rightarrow {{x}_{2}}=\dfrac{-288}{12}\]
Here the negative number is getting divided by the positive number which is $12$, so again the value comes to be negative number which is $-24$ .
\[\Rightarrow {{x}_{2}}=-24\]
${{x}_{2}}=-24$
$\therefore $ The value of $x$ becomes $-24$ when the value of $y$ is $-16$ .

Note: The problem can even be solved using unitary method (word problem), but this method takes more time than ratio. The proportion is used to express the relation of two ratios. The ratio should exist between quantities of the same kind while comparing two things the unit should be similar. The comparison of two ratios can be formed if the ratio is equivalent like the fraction. While comparing two things the unit should be similar.
We can check the value of $x$ whether it is correct or not. Consider the ${{1}^{st}}$ fraction to be $y=12$ and $x=18$ and ${{2}^{nd}}$ fraction to be $y=-16$ and $x=-24$ . Let's check the fraction , so for ${{1}^{st}}$ fraction turns to be $\dfrac{12}{18}\Rightarrow \dfrac{2}{3}$, and ${{2}^{nd}}$ fraction turns to $\dfrac{-16}{-24}\Rightarrow \dfrac{2}{3}$. Since both the fractions are equal,the value of $x$ is correct.