Home » » If p : q = $$\frac{2}{3}$$ : $$\frac{5}{6}$$ and q : r = $$\frac{3}{4}$$ : $$\frac{1}{2}$$, find p : q : r...

# If p : q = $$\frac{2}{3}$$ : $$\frac{5}{6}$$ and q : r = $$\frac{3}{4}$$ : $$\frac{1}{2}$$, find p : q : r...

### Question

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

A) 12 : 15 : 10

B) 12 : 15 : 16

C) 10 : 15 : 24

D) 9 : 10 : 15

### Explanation:

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 = ($$\frac{5}{6} \div \frac{9}{6}$$)T1 = ($$\frac{5}{6} \times \frac{6}{9}$$)T1 = $$\frac{5}{9}$$T1
Again, let q + r = T2, then
q = ($$\frac{3}{4} \div \frac{5}{4}$$)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
$$\frac{T_1}{T_2}$$ = $$\frac{9 \times 3}{5 x 5}$$ = $$\frac{27}{5}$$
Giving that, T1 = 27 and T2 = 25
P = ($$\frac{2}{3} \div S_1$$)T1 = ($$\frac{2}{3} \div \frac{9}{6}$$)T1
= ($$\frac{2}{3} \times \frac{6}{9}$$)27 = 12
q = ($$\frac{5}{6} \div S_1$$)T1 = ($$\frac{5}{6} \div \frac{9}{6}$$)T1
= ($$\frac{5}{6} \times \frac{6}{9}$$)27 = 15
and r = ($$\frac{1}{2} \div S_2$$)T2 = ($$\frac{1}{2} \div \frac{5}{4}$$)T2
= ($$\frac{1}{2} \times \frac{4}{5}$$)25 = 10
Hence p : q : r = 12: 15 : 10

## Dicussion (1)

• 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 = ($$\frac{5}{6} \div \frac{9}{6}$$)T1 = ($$\frac{5}{6} \times \frac{6}{9}$$)T1 = $$\frac{5}{9}$$T1
Again, let q + r = T2, then
q = ($$\frac{3}{4} \div \frac{5}{4}$$)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
$$\frac{T_1}{T_2}$$ = $$\frac{9 \times 3}{5 x 5}$$ = $$\frac{27}{5}$$
Giving that, T1 = 27 and T2 = 25
P = ($$\frac{2}{3} \div S_1$$)T1 = ($$\frac{2}{3} \div \frac{9}{6}$$)T1
= ($$\frac{2}{3} \times \frac{6}{9}$$)27 = 12
q = ($$\frac{5}{6} \div S_1$$)T1 = ($$\frac{5}{6} \div \frac{9}{6}$$)T1
= ($$\frac{5}{6} \times \frac{6}{9}$$)27 = 15
and r = ($$\frac{1}{2} \div S_2$$)T2 = ($$\frac{1}{2} \div \frac{5}{4}$$)T2
= ($$\frac{1}{2} \times \frac{4}{5}$$)25 = 10
Hence p : q : r = 12: 15 : 10