The digits of a two-digit number are reversed. When both these two-digit numbers are added, the result is 88, and when subtracted, the result is 54. The original two-digit number contains which of the following digits?
2
3
5
7
Explain your reasoning.
How did you come to your conclusion?
original number: 10x+y
reversed number: 10y+x
their sum: 11x + 11y = 88
x+y = 8
their difference: 9x-9y=54
x-y=6
add these two:
2x = 14
x=7
sub back in:
7+y=8
y=1
thus ....
thank you, Mr. Reiny.
To solve this problem, we need to set up a system of equations based on the information given.
Let's assume the original two-digit number is written as AB, where A represents the tens digit and B represents the ones digit. Since we're given that the digits are reversed, the reversed number would be BA.
According to the problem, when the two-digit numbers AB and BA are added, the result is 88. This can be expressed as the equation:
AB + BA = 88
When these two numbers are subtracted, the result is 54. This can be expressed as the equation:
AB - BA = 54
To solve this system of equations, we can rearrange the equation AB + BA = 88 to get 2AB = 88, and then simplify it to AB = 44. Similarly, rearranging AB - BA = 54 gives us 10A - 10B = 54.
Now let's consider the possible values for A and B:
1. A = 4 and B = 4: In this case, the original two-digit number is 44, but if we reverse the digits, we still obtain 44. So, this doesn't meet the given condition.
2. A = 5 and B = 4: In this case, the original two-digit number is 54, and if we reverse the digits, we get 45. The sum of these numbers is 99, which is not equal to 88. Therefore, this is also not the correct answer.
3. A = 6 and B = 4: In this case, the original two-digit number is 64, and if we reverse the digits, we get 46. The sum of these numbers, 64 + 46, is indeed 88, which satisfies the first condition. Now, if we subtract 46 from 64, we get 18, which is not equal to 54. Thus, this is not the correct answer as well.
4. A = 7 and B = 4: In this case, the original two-digit number is 74, and if we reverse the digits, we get 47. The sum of these numbers, 74 + 47, is indeed 88, which satisfies the first condition. Moreover, if we subtract 47 from 74, we get 27, which is equal to 54. Therefore, this is the correct answer.
Thus, based on the given information, the original two-digit number contains the digit 7.