Mira found an interesting relationship between the two digits and the difference obtained. Extend this table by choosing appropriate digits so that the resulting differences are 2, 3, 5, and 7 respectively. What do the differences between the digits indicate?
Pattern: Difference = 9 × (difference of digits)
For difference 18 (digits differing by 2): (1,3): 31 - 13 = 18 (2,4): 42 - 24 = 18 (3,5): 53 - 35 = 18
For difference 27 (digits differing by 3): (1,4): 41 - 14 = 27 (2,5): 52 - 25 = 27
For difference 45 (digits differing by 5): (1,6): 61 - 16 = 45 (2,7): 72 - 27 = 45
For difference 63 (digits differing by 7): (1,8): 81 - 18 = 63 (2,9): 92 - 29 = 63
Conclusion: The difference between the digits indicates what to multiply 9 by to get the final difference.
Explanation
The table in the chapter shows that the difference between the two-digit number and its reverse equals 9 times the difference between the digits. This relationship allows us to predict the difference for any pair of digits.
Solution Steps
Step 1: Identify the pattern: Difference = 9 × (difference of digits)
Step 2: For difference 18: Choose digits differing by 2, e.g., (1,3), (2,4), (3,5)
Step 3: For difference 27: Choose digits differing by 3, e.g., (1,4), (2,5)
Step 4: For difference 45: Choose digits differing by 5, e.g., (1,6), (2,7)
Step 5: For difference 63: Choose digits differing by 7, e.g., (1,8), (2,9)
Step 6: Verify that reversing and subtracting gives the expected difference in each case