The sides of a triangle are in the ratio of $\frac{1}{4}:\frac{1}{5}:\frac{1}{6}$. If the perimeter of the triangle is 37 cm, then find the length of the greatest side of the triangle?

The triangle has sides in the ratio $\dfrac{1}{4}:\dfrac{1}{5}:\dfrac{1}{6}$.

To simplify, find the LCM of the denominators 4, 5, and 6.

$4 = 2^2$, $5 = 5$, $6 = 2 \times 3$

$\text{LCM} = 2^2 \times 3 \times 5 = 60$

Multiply each fraction by 60:

$\dfrac{1}{4} \times 60 = 15$

$\dfrac{1}{5} \times 60 = 12$

$\dfrac{1}{6} \times 60 = 10$

The simplified ratio of the sides is $15:12:10$.

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Let the common multiplier be $x$.

First side $= 15x$

Second side $= 12x$

Third side $= 10x$

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The perimeter is $37$ cm.

$15x + 12x + 10x = 37$

$37x = 37$

$x = 1$

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The greatest side corresponds to the largest ratio value, which is $15x$.

Greatest side $= 15 \times 1 = 15$ cm