Joe had R150. He spent 2/5 of the money on a shirt, and then he spent 2/3 of the money that was left to buy a cap. How much money did he have left?

$$
\text{Shirt cost} = \frac{2}{5} \times \text{R}150
$$
$$
\text{Shirt cost} = 2 \times \frac{150}{5}
$$
$$
\text{Shirt cost} = 2 \times 30 = \text{R}60
$$
The shirt cost $\boxed{\text{R}60}$.

$$
\text{Money left after shirt} = \text{R}150 - \text{R}60
$$
$$
\text{Money left after shirt} = \text{R}90
$$
He had $\boxed{\text{R}90}$ left after buying the shirt.

$$
\text{Cap cost} = \frac{2}{3} \times \text{R}90
$$
$$
\text{Cap cost} = 2 \times \frac{90}{3}
$$
$$
\text{Cap cost} = 2 \times 30 = \text{R}60
$$
The cap cost $\boxed{\text{R}60}$.

$$
\text{Money left after cap} = \text{R}90 - \text{R}60
$$
$$
\text{Money left after cap} = \text{R}30
$$
He had $\boxed{\text{R}30}$ left after buying the cap.