If p q frac 2 3 frac 5 6 and q r frac 3 4 frac ... - JAMB Mathematics 2010 Question

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