Answer :
To solve the problem, we need to understand the ratio and proportion given: [tex]6 : x :: 15 : y[/tex]. This can be expressed as the equation [tex]\frac{6}{x} = \frac{15}{y}[/tex].
This implies:
[tex]6y = 15x[/tex]
Simplifying the equation by dividing both sides by 3 gives:
[tex]2y = 5x[/tex]
From this, we can express the ratio of [tex]x[/tex] to [tex]y[/tex] as:
[tex]\frac{y}{x} = \frac{5}{2}[/tex]
This can be interpreted as [tex]y = \frac{5}{2}x[/tex]. Hence, [tex]x[/tex] and [tex]y[/tex] must be in the simplest form of [tex]2k[/tex] and [tex]5k[/tex] for some integer [tex]k[/tex].
We need to find the Least Common Multiple (LCM) of [tex]x[/tex] and [tex]y[/tex], which are in the forms [tex]2k[/tex] and [tex]5k[/tex].
The LCM of [tex]2k[/tex] and [tex]5k[/tex] is calculated as:
- The LCM of the coefficients [tex]2[/tex] and [tex]5[/tex] is [tex]10[/tex] since they are relatively prime numbers.
- The common factor is [tex]k[/tex].
Hence, the LCM will be [tex]10k[/tex].
Since [tex]k[/tex] is any integer, [tex]10k[/tex] can be any multiple of 10.
Therefore, the LCM of [tex]x[/tex] and [tex]y[/tex] will be a multiple of [tex]10[/tex].
Correct option: B. 10
