If \(p : q = \frac{2}{3} : \frac{5}{6}\) and \(q : r = \frac{3}{4} : \frac{1}{2}\), find \(p : q : r\)

Answer Details

If p : q = $\frac{2}{3}$ : $\frac{5}{6}$, then the sum S1 of ratio = $\frac{2}{3}$ + $\frac{5}{6}$ = $\frac{9}{6}$
If q : r = $\frac{3}{4}$ : $\frac{1}{2}$, then the sum S2 of ratio = $\frac{3}{4}$ + $\frac{1}{2}$ = $\frac{5}{4}$
Let p + q = T1, then
q = ($\frac{5}{6}÷\frac{9}{6}$)T1 = ($\frac{5}{6}\times \frac{6}{9}$)T1 = $\frac{5}{9}$T1
Again, let q + r = T2, then
q = ($\frac{3}{4}÷\frac{5}{4}$)T2 = ($\frac{3}{4}\times \frac{4}{5}$)T2 = $\frac{3}{5}$T2
Using q = q
$\frac{5}{9}$T1 = $\frac{3}{5}$T2
5 x 5T1 = 9 x 3T2
$\frac{T_1}{T_2}$ = $\frac{9\times 3}{5x5}$ = $\frac{27}{5}$
Giving that, T1 = 27 and T2 = 25
P = ($\frac{2}{3}÷S_1$)T1 = ($\frac{2}{3}÷\frac{9}{6}$)T1
= ($\frac{2}{3}\times \frac{6}{9}$)27 = 12
q = ($\frac{5}{6}÷S_1$)T1 = ($\frac{5}{6}÷\frac{9}{6}$)T1
= ($\frac{5}{6}\times \frac{6}{9}$)27 = 15
and r = ($\frac{1}{2}÷S_2$)T2 = ($\frac{1}{2}÷\frac{5}{4}$)T2
= ($\frac{1}{2}\times \frac{4}{5}$)25 = 10
Hence p : q : r = 12: 15 : 10