1. **Problem:** Find the ratios between the number of pencils in Group A and Group B. Group A has 3 bundles with 6 pencils each, so total pencils in Group A = $3 \times 6 = 18$. Group B has 4 bundles with 6 pencils each, so total pencils in Group B = $4 \times 6 = 24$. The ratio of the number of pencils in Group A to Group B is $$\frac{18}{24}$$. 2. **Simplify the ratio:** $$\frac{18}{24} = \frac{\cancel{6} \times 3}{\cancel{6} \times 4} = \frac{3}{4}$$. So, the ratio of pencils in Group A to Group B is $3:4$. 3. **Ratio of pencils in Group B to Group A:** $$\frac{24}{18} = \frac{\cancel{6} \times 4}{\cancel{6} \times 3} = \frac{4}{3}$$. So, the ratio of pencils in Group B to Group A is $4:3$. --- 4. **Problem:** Find the ratios for P, Q, and R based on the bar representations. - P has 10 filled sections. - Q has 8 filled sections. - R has 15 empty sections. (a) Ratio of P to Q: $$\frac{10}{8} = \frac{\cancel{2} \times 5}{\cancel{2} \times 4} = \frac{5}{4}$$. So, the ratio of P to Q is $5:4$. (b) Ratio of R to P: $$\frac{15}{10} = \frac{\cancel{5} \times 3}{\cancel{5} \times 2} = \frac{3}{2}$$. So, the ratio of R to P is $3:2$. --- 5. **Problem:** Simplify the given ratios. (a) Simplify $6 : 16$: $$\frac{6}{16} = \frac{\cancel{2} \times 3}{\cancel{2} \times 8} = \frac{3}{8}$$. So, $6 : 16 = 3 : 8$. (b) Simplify $10 : 35 : 45$: Find the greatest common divisor (GCD) of 10, 35, and 45, which is 5. Divide each term by 5: $$\frac{10}{5} : \frac{35}{5} : \frac{45}{5} = 2 : 7 : 9$$. So, $10 : 35 : 45 = 2 : 7 : 9$. **Final answers:** 1. Ratio of Group A to Group B: $3:4$ 2. Ratio of Group B to Group A: $4:3$ 3. (a) Ratio of P to Q: $5:4$ 3. (b) Ratio of R to P: $3:2$ 4. (a) Simplified ratio $6 : 16 = 3 : 8$ 4. (b) Simplified ratio $10 : 35 : 45 = 2 : 7 : 9$