Solving Digit Problems: A 2-Digit Number 10 Times the Sum of Its Digits
In today's article, we will solve a digit problem that involves a two-digit number where the number is 10 times the sum of its digits, and the tens digit is 2 greater than the units digit. This problem can be approached through algebra and arithmetic, providing a clear and detailed solution that illustrates the application of basic mathematical principles.
Problem Statement
The problem states that a two-digit number is 10 times the sum of its digits, and the tens digit is 2 greater than the units digit. We will represent the two-digit number as 10a b, where a is the tens digit and b is the units digit.
Step-by-Step Solution
Step 1: Expressing the Conditions Algebraically
The first condition states that the number is 10 times the sum of its digits:
10a b 10(a b)
Step 2: Simplifying the Equation
Subtract 10a from both sides of the equation:
b 10b
This simplifies to:
0 9b
Solving for b, we get:
b 0
Step 3: Determining the Tens Digit
From the second condition, we know that the tens digit is 2 greater than the units digit:
a b 2
Substituting b 0 into the equation:
a 0 2
This simplifies to:
a 2
Step 4: Forming the Two-Digit Number
With a 2 and b 0, the two-digit number can be represented as:
10a b 10(2) 0 20
Verification
To verify our solution, let's check the conditions:
The sum of the digits is 2 0 2, and 10 times this sum is 20, which matches the number. The tens digit, 2, is indeed 2 greater than the units digit, 0.The solution is correct, and the two-digit number is 20.
Additional Examples and Analysis
Example 1: ab 11
This implies b 11 - a. If the tens digit is 2 greater than the units digit, we have:
a b 2
Substituting b 11 - a into the equation:
a 11 - a 2
Solving for a, we get:
2a 13
a 6.5 (not an integer, hence not valid)
The correct number is ab 38.
Example 2: 10t o 10t o
This implies:
t o 2 10t o 10t - 10t 10o 20 10o 20 30 - 10o 30 10o 20 10o 10 o 0 t 0 2 o 2The number is 20, and the tens digit is 2 greater than the units digit.
Example 3: 10T U 10T U
This implies:
T U 2 10T U 10U 10 10U 20 U 10U 10 11U 20 10U 10 20 U 10 10 U 20 - 10 U 0 T 0 2 2The number is 20, and the tens digit is 2 greater than the units digit.
Conclusion
In summary, the two-digit number that satisfies the conditions given in the problem is 20. This solution is verified through multiple step-by-step examples, confirming the accuracy of the solution.