Recall your repeated identity concept from PSLE

Given :

p : q
¾ : 2

p : r
⅓ : ½


Make p the same for both ratios.


①

p : q
¾ : 2
↓
(Multiply by the LCM of the denominators, 4. Recall that 2 = 2/1)
↓
3 : 8


②
p : r
⅓ : ½
↓
(Multiply by the LCM of the denominators, 6)
↓
2 : 3


③
Now multiply the numbers for p in both ratios to get to their LCM. The LCM of 2 and 3, is 6.

p : q
3 : 8
↓
(Multiply by 2)
↓
6 : 16

p : r
2 : 3
↓
(Multiply by 3)
↓
6 : 9



Now you can combine them.

p : q : r
6 : 16 : 9



So q : r = 16 : 9

Second way

p : q
¾ : 2

p : r
⅓ : ½


Make p the same for both ratios. Multiply both p's by each other.


①
p : q
¾ : 2
↓
(Multiply by ⅓)
↓
¼ : ⅔


②
p : r
⅓ : ½
↓
(Multiply by ¾)
↓
¼ : ⅜


③
Combine all three together.

p : q : r
¼ : ⅔ : ⅜


④
Multiply by the LCM of the denominators 4,3 and 8. Their LCM is 24


p : q : r
¼ : ⅔ : ⅜
↓
(Multiply by 24)
↓
6 : 16 : 9


We get the same result.