What is the missing digit in the middle number, given the numbers 350, ?, and 410 that satisfy certain conditions?

Answer

To find the missing digit in the middle number, we need to analyze the conditions given and the possible options. 1. We know that the smallest number is 350 and the largest is 410. The middle number must be between these two. 2. The conditions state: - The difference between the middle number and the smallest number (350) is more than 10. - The difference between the largest number (410) and the middle number is more than 30. Let's denote the middle number as 3X0, where X is the missing digit. First, let's apply the first condition: - 3X0 - 350 > 10 This simplifies to: 3X0 > 360 3X0 must be greater than 360. Now for the second condition: - 410 - 3X0 > 30 This simplifies to: 3X0 < 380 3X0 must be less than 380. Now, we have the inequalities: - 360 < 3X0 < 380 This means that possible values for 3X0 are 360, 361, 362, ..., 379. From this range, we can analyze the candidates based on the digit X: - If X = 6: Middle number is 360 (not valid, as it is not > 360) - If X = 7: Middle number is 370 (valid) - If X = 8: Middle number is 380 (not valid, as it is not < 380) - If X = 9: Middle number is 390 (not valid, as it is not < 380) Thus, the only digit that satisfies both conditions above is: - X = 7, with the middle number being 370. Therefore, the missing digit in the middle number is 7.