two 2-digit numbers have the same digits. Their sum is 77 and their difference is 27. What are the numbers? What stratagey did u use?
To solve this problem, we can use a strategic approach. Let's break down the given information step by step:
1. Two 2-digit numbers have the same digits: This means that both numbers have identical digits.
2. Their sum is 77: Let's assume that the two numbers are XY and XY, where X and Y represent digits. So, the sum of these two numbers would be XY + XY = 77.
3. Their difference is 27: The difference between two numbers is obtained by subtracting one number from another. In this case, we have XY - XY = 27.
Now, let's find the numbers by solving the equations we obtained:
1. XY + XY = 77: Combining the two numbers leads to 2XY = 77. Dividing both sides by 2, we get XY = 38.
2. XY - XY = 27: Subtracting the numbers, we have 0 = 27, which is not possible. This means that our assumption in step 2 is incorrect.
Since our assumption was not correct, let's try a different approach to solve the problem:
1. Let the two numbers be AB and BA. Here, A and B represent digits, and the first number is formed by placing A in the tens place and B in the units place.
2. Their sum is 77: So, AB + BA = 77.
3. Their difference is 27: Therefore, AB - BA = 27.
Now let's find the numbers by solving the equations we obtained:
1. AB + BA = 77: Adding the numbers, we get 11(A + B) = 77. Dividing both sides by 11, we have A + B = 7.
2. AB - BA = 27: Subtracting the numbers, we get 9(A - B) = 27. Dividing both sides by 9, we have A - B = 3.
Now we have a system of equations:
A + B = 7 (Equation 1)
A - B = 3 (Equation 2)
By solving this system of equations, we can find the values of A and B, which will give us the two numbers.
Adding Equation 1 and Equation 2, we have:
(A + B) + (A - B) = 7 + 3
2A = 10
A = 5
Substituting the value of A into Equation 1, we find:
5 + B = 7
B = 2
Therefore, the two numbers are 52 and 25, respectively.
x1 + x2 = 77
x2 - x1 = 27
x2 = 27 + x1
x1 + (27 + x1) = 77
2 x1 = 50
x1 = 25
x2 = 52