One of the two digits of a two-digit number is three times the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 88. What is the original number?
Question
One of the two digits of a two-digit number is three times the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 88. What is the original number?
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Answer
Let's denote the digit in the units place as $x$. Accordingly, the digit in the tens place will be $3x$ since one digit is three times the other.
Therefore, the original number can be represented as: $$ 10(3x) + x = 31x $$
Next, we interchange the digits. By doing this:
- The new digit in the units place is $3x$.
- The new digit in the tens place is $x$.
Thus, the new number becomes: $$ 10(x) + 3x = 13x $$
According to the problem statement, when you add the original number and the new number, the result is 88: $$ 31x + 13x = 88 $$
Combining like terms, we get: $$ 44x = 88 $$
By solving for $x$: $$ x = \frac{88}{44} = 2 $$
To find the digits, multiply $x$ by 3: $$ 3x = 3 \cdot 2 = 6 $$
Hence, the original number is: $$ 62 $$
The original number is 62.
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