The ratio of a two-digit natural number to a number formed by reversing its digits is 4: 7. The number of such pairs is

Let the two-digit natural number be 10a + b, where 'a' is the tens digit and 'b' is the units digit. Since it's a two-digit number, 'a' must be an integer from 1 to 9, and 'b' must be an integer from 0 to 9. The number formed by reversing its digits is 10b + a. According to the question, the ratio of the original number to the reversed number is 4:7. So, (10a + b) / (10b + a) = 4 / 7. Cross-multiplying gives: 7(10a + b) = 4(10b + a) 70a + 7b = 40b + 4a Rearranging the terms to group 'a' and 'b': 70a - 4a = 40b - 7b 66a = 33b Dividing both sides by 33: 2a = b Now we need to find the possible integer values for 'a' and 'b' that satisfy this equation and the digit constraints: 'a' belongs to {1, 2, 3, 4, 5, 6, 7, 8, 9} 'b' belongs to {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Let's test values for 'a': 1. If a = 1, then b = 2 * 1 = 2. This is a valid digit. The number is 12, its reverse is 21. Ratio = 12/21 = 4/7. (Pair: 12, 21) 2. If a = 2, then b = 2 * 2 = 4. This is a valid digit. The number is 24, its reverse is 42. Ratio = 24/42 = 4/7. (Pair: 24, 42) 3. If a = 3, then b = 2 * 3 = 6. This is a valid digit. The number is 36, its reverse is 63. Ratio = 36/63 = 4/7. (Pair: 36, 63) 4. If a = 4, then b = 2 * 4 = 8. This is a valid digit. The number is 48, its reverse is 84. Ratio = 48/84 = 4/7. (Pair: 48, 84) 5. If a = 5, then b = 2 * 5 = 10. This is not a valid single digit for 'b'. Any value of 'a' greater than 4 would result in 'b' being a two-digit number, which is not allowed. So, there are 4 such pairs of numbers: (12, 21), (24, 42), (36, 63), and (48, 84). The number of such pairs is 4. The final answer is B