If l : m = $2\dfrac{1}{2} : 1\dfrac{2}{3}$ and m : n = $1\dfrac{1}{4} : 3\dfrac{1}{2}$, find l : m : n.

Given:

l : m = $2\dfrac{1}{2} : 1\dfrac{2}{3}$

m : n = $1\dfrac{1}{4} : 3\dfrac{1}{2}$

Convert mixed to improper fractions:

l : m = $\dfrac{5}{2} : \dfrac{5}{3}$

m : n = $\dfrac{5}{4} : \dfrac{7}{2}$

Let us simplify the ratios into whole numbers.

l : m = $\dfrac{5}{2} : \dfrac{5}{3}$

Let us find L.C.M. of 2 and 3:

$\begin{array}{l|rr} 2 & 2, & 3 \ \hline 3 & 1, & 3 \ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 3 = 6

l : m = $\Big(\dfrac{5}{2} \times 6\Big) : \Big(\dfrac{5}{3} \times 6\Big)$ = 15 : 10

l : m = 3 : 2 $\quad$[Dividing by 5]

m : n = $\dfrac{5}{4} : \dfrac{7}{2}$

Let us find L.C.M. of 4 and 2:

$\begin{array}{l|rr} 2 & 4, & 2 \ \hline 2 & 2, & 1 \ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 2 = 4

m : n = $\Big(\dfrac{5}{4} \times 4\Big) : \Big(\dfrac{7}{2} \times 4\Big)$

m : n = 5 : 14

We make m, the same in both the ratios.

Let us find L.C.M. of 2 and 5:

$\begin{array}{l|rr} 2 & 2, & 5 \ \hline 5 & 1, & 5 \ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 5 = 10

So, we make m equal to 10 in each case.

Now, multiply the first ratio by 5:

$l : m = 3 : 2 \\[1em] = \dfrac{3 \times 5}{2 \times 5} \\[1em] = \dfrac{15}{10} \\[1em] = 15 : 10$

Multiply the second ratio by 2:

$m : n = 5 : 14 \\[1em] = \dfrac{5 \times 2}{14 \times 2} \\[1em] = \dfrac{10}{28} \\[1em] = 10 : 28$

Now we have,

l : m = 15 : 10 and m : n = 10 : 28

∴ l : m : n = 15 : 10 : 28