1. **State the problem:** We are given that Jiahao and Raju have stamps in the ratio 3 : 1. Jiahao's stamps are \(\frac{2}{3}\) of Ken's stamps. (a) Find the ratio of Jiahao : Raju : Ken. (b) Ken has 45 stamps. Find how many more stamps Ken has than Jiahao. 2. **Use the given ratios and relationships:** Let Jiahao's stamps be \(J\), Raju's stamps be \(R\), and Ken's stamps be \(K\). From the problem: \[ J : R = 3 : 1 \implies R = \frac{1}{3}J \] Also, \[ J = \frac{2}{3}K \] 3. **Express all in terms of \(J\) and \(K\):** Since \(R = \frac{1}{3}J\), the ratio \(J : R : K\) is: \[ J : \frac{1}{3}J : K \] Replace \(J\) with \(\frac{2}{3}K\): \[ \frac{2}{3}K : \frac{1}{3} \times \frac{2}{3}K : K = \frac{2}{3}K : \frac{2}{9}K : K \] 4. **Simplify the ratio by dividing all terms by \(K\):** \[ \frac{2}{3} : \frac{2}{9} : 1 \] 5. **Clear denominators by multiplying all terms by 9:** \[ 9 \times \frac{2}{3} : 9 \times \frac{2}{9} : 9 \times 1 = 6 : 2 : 9 \] So, the ratio \(J : R : K = 6 : 2 : 9\). 6. **Answer part (b):** Given \(K = 45\) stamps. Since \(J = \frac{2}{3}K\), \[ J = \frac{2}{3} \times 45 = 30 \] Difference between Ken and Jiahao: \[ 45 - 30 = 15 \] **Final answers:** (a) The ratio of Jiahao : Raju : Ken is \(6 : 2 : 9\). (b) Ken has 15 more stamps than Jiahao.