The radar graph below displays the total number of buckets (plastic + wooden) sold by five shops (P, Q, R, S, and T) in the months of May and June. In May, the total wooden buckets sold by shop R were 200\% more than those sold by shop S. Also, the number of plastic buckets sold by R in May is greater than the number of plastic buckets sold by S in May. Based on this information, which of the following cannot be the number of plastic buckets sold by R in May?
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**Solution:** 1. From the radar graph, the total buckets sold by R in May is given as 210 (assumed from the graph data). 2. The total buckets sold by S in May is 150. 3. Given that wooden buckets sold by R are 200\% more than wooden buckets sold by S in May. - Let wooden buckets sold by S be $ W_S $. - Then wooden buckets sold by R, $ W_R = W_S + 200\% \times W_S = 3W_S $. 4. Total buckets sold by S in May = wooden + plastic = 150. - So, plastic buckets sold by S in May = $ 150 - W_S $. 5. Total buckets sold by R in May = wooden + plastic = 210. - So, plastic buckets sold by R in May = $ $210 - W_R = 210 - 3W_S$ $. 6. Since plastic buckets sold by R are more than plastic buckets sold by S: $$ 210 - 3W_S > 150 - W_S $$ 7. Simplify the inequality: $$ 210 - 3W_S > 150 - W_S \ 210 - 150 > 3W_S - W_S \ 60 > 2W_S \ W_S < 30 $$ 8. Since $ W_S < 30 $, wooden buckets sold by S are less than 30. 9. Now, plastic buckets sold by R = $ 210 - 3W_S $. - For $ W_S = 29 $, plastic buckets by R = $ 210 - 87 = 123 $. - For $ W_S = 0 $, plastic buckets by R = 210. 10. Plastic buckets sold by R must be between 123 and 210. 11. Check options: - (a) 90 (less than 123) - Not possible - (b) 117 (less than 123) - Not possible - (c) 96 (less than 123) - Not possible - (d) 120 (less than 123) - Not possible 12. However, the question states plastic buckets by R are more than plastic buckets by S, so options less than 123 are invalid. 13. Among given options, 90 and 96 are definitely impossible. **Answer:** Both (a) 90 and (c) 96 cannot be the number of plastic buckets sold by R in May.

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