If the cat and the rat land on the same number, the cat will catch the rat. The cat is now on 6 and the rat on 12. When the cat jumps 3 steps forward, the rat jumps 2 steps forward. Will the cat catch the rat? If yes, at which number?
Step 1: Cat starts at 6, jumps 3 steps: positions after n jumps = 6 + 3n
Step 2: Rat starts at 12, jumps 2 steps: positions after n jumps = 12 + 2n
Step 3: Set positions equal to find when they land on the same number at the same time: 6 + 3n = 12 + 2n
Step 4: Solve for n: 3n - 2n = 12 - 6 → n = 6
Step 5: Calculate the position at n = 6: Cat is at 6 + 3(6) = 24, Rat is at 12 + 2(6) = 24
Step 6: Both meet at 24 after 6 jumps, so cat catches rat at 24.
Answer: Yes, the cat will catch the rat at number 24.
Explanation
The cat starts at 6 and jumps 3 steps: positions after n jumps are 6 + 3n. The rat starts at 12 and jumps 2 steps: positions after n jumps are 12 + 2n. Since they jump simultaneously, we need to find when they land on the same number at the same time. Setting the equations equal: 6 + 3n = 12 + 2n. Solving for n gives n = 6. After 6 jumps, the cat will be at 6 + 3(6) = 24 and the rat will be at 12 + 2(6) = 24. Thus, the cat catches the rat at number 24.
Solution Steps
Step 1: Cat starts at 6, jumps 3: positions after n jumps = 6 + 3n
Step 2: Rat starts at 12, jumps 2: positions after n jumps = 12 + 2n
Step 3: Set positions equal to find when they land on the same number at the same time: 6 + 3n = 12 + 2n
Step 4: Solve for n: 3n - 2n = 12 - 6 → n = 6
Step 5: Calculate the position at n = 6: Cat is at 6 + 3(6) = 24, Rat is at 12 + 2(6) = 24
Step 6: Both meet at 24 after 6 jumps, so cat catches rat at 24