1. **State the problem:** We need to find how many ice cream sticks Peter, Paul, and Gernard each have given these conditions: - Peter and Paul together have 115 sticks. - Paul and Gernard together have 145 sticks. - Peter has $\frac{2}{3}$ as many sticks as Gernard. 2. **Define variables:** Let: - $P$ = number of sticks Peter has - $Pa$ = number of sticks Paul has - $G$ = number of sticks Gernard has 3. **Write equations from the problem:** - $P + Pa = 115$ - $Pa + G = 145$ - $P = \frac{2}{3}G$ 4. **Substitute $P$ in the first equation:** $$\frac{2}{3}G + Pa = 115$$ 5. **Express $Pa$ from the above:** $$Pa = 115 - \frac{2}{3}G$$ 6. **Use the second equation $Pa + G = 145$ and substitute $Pa$:** $$115 - \frac{2}{3}G + G = 145$$ 7. **Simplify the equation:** $$115 + \left(1 - \frac{2}{3}\right)G = 145$$ $$115 + \frac{1}{3}G = 145$$ 8. **Isolate $G$:** $$\frac{1}{3}G = 145 - 115$$ $$\frac{1}{3}G = 30$$ $$G = 30 \times 3 = 90$$ 9. **Find $P$ using $P = \frac{2}{3}G$:** $$P = \frac{2}{3} \times 90 = 60$$ 10. **Find $Pa$ using $P + Pa = 115$:** $$60 + Pa = 115$$ $$Pa = 115 - 60 = 55$$ **Final answer:** - Peter has 60 ice cream sticks. - Paul has 55 ice cream sticks. - Gernard has 90 ice cream sticks.