Three friends A, B and C invest in a business in the ratio 6 : 4: 10 respectively. They share the profit in the ratio 8 : 7 : 9 respectively. Find the ratio of time periods of their investment.

We know that Ratio of Investment x Time = Ratio of Profit
∴ To find the ratio of time just divide respective profits ratios by respective investment ratios
So if the profit ratio is P₁: P₂ : P₃ and the investment ratio is I₁ : I₂ : I₃ then,
Time ratio is found by = (P₁/I₁) : (P₂/I₂) : (P₃/I₃)

6 : 4 : 10 can be reduced to 3 : 2 : 5. So, their investments are actually in the ratio 3 : 2 : 5.
The ratio of time periods for A, B, and C is found by = 8/3 : 7/2 : 9/5
Making denominators common by multiplying by 30
∴ The ratio of Time periods for A, B, and C = (30 × 8)/3 : (30 × 7)/2 : (30 × 9)/5
= 80 : 105 : 54