Orange juice, a raisin, and a cup of coffee cost a total of $\$ 2.90$. Billy posts an announcement that the price of orange juice will increase $50 \%$ and the price of bagels will increase by $20 \%$. After the increase, the same purchase will cost $\$ 3.60$, and the orange juice will cost twice as much as the coffee. Find the price of each item before the increase

Step 1 ：Let's denote the price of orange juice as O, the price of a raisin as R, and the price of coffee as C. We know from the problem that O + R + C = 2.90. This is the total cost of the items before the price increase.

Step 2 ：We also know that after the price increase, the total cost of the items is 1.5O + R + 1.2C = 3.60. This is because the price of orange juice increased by 50% and the price of coffee increased by 20%.

Step 3 ：Finally, we know that after the price increase, the price of orange juice is twice the price of coffee. This gives us the equation 1.5O = 2C.

Step 4 ：We can solve these three equations simultaneously to find the values of O, R, and C.

Step 5 ：The solution to these equations is O = 1.08, R = 1.02, and C = 0.81.

Step 6 ：So, the price of orange juice before the increase is \(\boxed{1.08}\) dollars, the price of a raisin is \(\boxed{1.02}\) dollars, and the price of coffee is \(\boxed{0.81}\) dollars.