Determine the base/radix of the numbers in each case
for the following operations to be correct:
(i) 29 + 17 = 40 (ii) 7 + 6 = 15 (iii) 67 + 24 = 113
(i) The sum of 29 and 17 is 40. In this case, the base/radix of the numbers is 10, as we are using the decimal system.
(ii) The sum of 7 and 6 is 15. In this case, the base/radix of the numbers is 8, as we are using the octal system. Octal digits go from 0 to 7.
(iii) The sum of 67 and 24 is 113. In this case, the base/radix of the numbers is 8 as well. However, I must say, this looks like a case of mathematical mischief. Did someone forget to carry a 1?
To determine the base/radix of the numbers in each case, we can observe the patterns and properties of the numbers involved in the operations.
(i) 29 + 17 = 40:
In this case, we have two numbers, 29 and 17. The result of their addition is 40. To find the base/radix, we can check if there is a carryover in the addition.
When adding the units digit, we see that 9 + 7 = 16. Since the result is two digits, we have a carryover. Therefore, the base/radix must be at least 16 to accommodate the carryover.
Now, let's check the maximum digits involved in the addition. The higher value is 29, which has two digits. So the base/radix must be greater than or equal to 2 (to represent the ones digit) and 16 (to accommodate the carryover).
Considering both cases, the possible values for the base/radix can be 16 or any number greater than that.
(ii) 7 + 6 = 15:
In this case, the numbers involved are 7 and 6, and the result is 15. To determine the base/radix, we can check if there is a carryover.
When adding the units digit, we have 7 + 6 = 13. Since the result is two digits, we have a carryover. Therefore, the base/radix must be at least 13 to accommodate the carryover.
Now, let's check the maximum digits involved in the addition. The higher value is 7, which has only one digit. So the base/radix must be greater than or equal to 1 (to represent the ones digit) and 13 (to accommodate the carryover).
Considering both cases, the possible values for the base/radix can be 13 or any number greater than that.
(iii) 67 + 24 = 113:
In this case, the numbers involved are 67 and 24, and the result is 113. To determine the base/radix, we can check if there is a carryover.
When adding the units digit, we have 7 + 4 = 11. Since the result is two digits, we have a carryover. Therefore, the base/radix must be at least 11 to accommodate the carryover.
Now, let's check the maximum digits involved in the addition. The higher value is 67, which has two digits. So the base/radix must be greater than or equal to 2 (to represent the ones digit) and 11 (to accommodate the carryover).
Considering both cases, the possible values for the base/radix can be 11 or any number greater than that.
In summary:
(i) The base/radix can be 16 or any number greater than that.
(ii) The base/radix can be 13 or any number greater than that.
(iii) The base/radix can be 11 or any number greater than that.