Question

If y varies directly with x, and if \[y=45\] when \[x=15\], how do you find x when \[y=15\]?

Answer 1

Hint: If b is directly proportional to a, algebraically this is expressed as \[b\propto a\]. Let k is the constant of proportionality, then we can also express the above expression as, \[b=ka\]. Using this equation, we can find the value of one variable for a particular value of another variable. We can do this by finding the constant of proportionality. If we are given one ordered pair value of the variables using it we can find the constant of proportionality.

Complete step-by-step solution:
We are given that y directly varies with x, this can be algebraically expressed as \[y\propto x\]. Let the constant of proportionality be k, then the proportionality equation can also be expressed as \[y=kx\].
We are given one ordered pair as \[y=45\] and \[x=15\], using this we can find the value of k, as follows
\[\Rightarrow 45=k\times 15\]
Solving the above equation, we get \[k=3\]. Hence, the proportionality equation is \[y=3x\].
We are asked to find the value of x, when \[y=15\]. Substituting the values in the equation, we get
\[\Rightarrow 15=3x\]
On solving the above equation, we get \[x=5\].

Note: We can also use the unitary method to solve the above problem. It is a method in which we find the value of a single unit and then multiply the value of a single unit by the number of units to get the necessary value.
For this problem, the value of one unit can be found as \[\dfrac{45}{15}=3\]. We need to find the number of units for which the value equals \[15\]. We can do it as \[\dfrac{15}{3}=5\].