The sum of the digits of a two -digit number is 7. If 27 is subtracted from the number the digit interchange their places. Find the number.
10 a + b
a + b = 7
so a = (7-b)
10 a + b - 10*2 - 7 = 10 b + a
10(7-b)+b-20-7=10b+(7-b)
70 -10b+b-27=9 b+7
36 = 18b
b = 2
then a = 5
so
52
To find the number, let's break down the problem and solve it step by step:
Step 1: We'll assume the two-digit number is written as "10x + y", where x and y are the digits. The digit x represents the tens place digit, and digit y represents the ones place digit.
Step 2: Given that the sum of the digits is 7, we can write an equation:
x + y = 7
Step 3: It is also given that if we subtract 27 from the number, the digits interchange their places. So we can write another equation:
10y + x - 27 = 10x + y
Step 4: Simplifying the equation obtained in step 3, we have:
9y - 9x = 27
Step 5: Dividing both sides of the equation by 9, we get:
y - x = 3
Step 6: Now we have two equations:
x + y = 7
y - x = 3
Step 7: Solving the system of equations by adding the two equations together:
(x + y) + (y - x) = 7 + 3
2y = 10
y = 5
Step 8: Now that we have the value of y, we can substitute it back into one of the equations to solve for x. Let's use the first equation:
x + 5 = 7
x = 2
Step 9: Therefore, the two-digit number is 10x + y = 10(2) + 5 = 20 + 5 = 25.
So, the number is 25.