1. **Problem statement:** Three partners A, B, and C invest in a business in the ratio 4:5:6. The total profit is 120000, and A's share of the profit is 16000. We need to find the total investment. 2. **Understanding the problem:** The profit is divided in the ratio of their investments. If the investments are in ratio 4:5:6, then the profits are also divided in the same ratio. 3. **Formula:** If investments are in ratio $a:b:c$, and total profit is $P$, then the profit shares are: $$\text{Profit of A} = \frac{a}{a+b+c} \times P$$ $$\text{Profit of B} = \frac{b}{a+b+c} \times P$$ $$\text{Profit of C} = \frac{c}{a+b+c} \times P$$ 4. **Apply the formula:** Given ratio $a:b:c = 4:5:6$, total profit $P = 120000$, and A's profit share is 16000. 5. **Calculate total parts:** $$4 + 5 + 6 = 15$$ 6. **Calculate A's profit share from ratio:** $$\frac{4}{15} \times 120000 = 32000$$ 7. **Compare with given A's profit:** Given A's profit is 16000, but calculated is 32000. This means the actual profit is scaled down by a factor. 8. **Find the scaling factor:** $$\text{Scaling factor} = \frac{16000}{32000} = 0.5$$ 9. **Calculate total investment:** Since the ratio of investments is 4:5:6, total investment parts = 15. Let the total investment be $I$, then A's investment is: $$\frac{4}{15} \times I$$ Given A's investment is 16000, so: $$\frac{4}{15} \times I = 16000$$ 10. **Solve for $I$:** $$I = \frac{16000 \times 15}{4} = 60000$$ **Final answer:** The total investment is 60000.