In a positive number of two digits, the sum of the digits is 15. If the digits are interchanged, the number is increased by 9. Find the number
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To solve this problem, let's break it down step by step.
Let's assume the original number with two digits is represented as "ab," where 'a' represents the tens digit and 'b' represents the ones digit.
From the given information, we know that the sum of the digits is 15.
This can be represented as:
a + b = 15 (Equation 1)
We are also given that when the digits are interchanged, the number increases by 9.
So, the new number can be represented as "ba," which is obtained by swapping the positions of 'a' and 'b.'
The new number is increased by 9 compared to the original number.
This can be represented as:
10b + a = 10a + b + 9 (Equation 2)
Now, we have two equations:
a + b = 15 (Equation 1)
10b + a = 10a + b + 9 (Equation 2)
To solve this system of equations, we can use any suitable method, such as substitution or elimination.
Let's solve using the elimination method:
Multiply Equation 1 by 10:
10a + 10b = 150 (Equation 3)
Now, subtract Equation 2 from Equation 3:
10a + 10b - (10b + a) = 150 - (10a + b + 9)
9a = 141 - 9b
Simplify the equation:
9a = 132 - 9b (Equation 4)
Now, we can rearrange Equation 4 to solve for 'a' in terms of 'b':
9b + 9a = 132
9a = 132 - 9b
a = (132 - 9b)/9
a = 14 - b
Substitute the value of 'a' in Equation 1:
14 - b + b = 15
14 = 15
Since the equation is not true, it means our assumption that the original number is a positive two-digit number is incorrect.
Therefore, it is not possible to find a positive two-digit number that satisfies the given conditions (sum of the digits is 15 and the number increases by 9 when the digits are interchanged).
let the number be ab. Then
a+b = 15
10b+a = 10a+b + 9
Now solve for a and b, and thus the number
Or, just knowing that ba = ab+9 if b=a+1, you have the number 78