A three-digit number between 600 and 700 is one less than 30 times the sum of the digits. Of the tens digit is one less than the unit digit, what is the number?
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To find the three-digit number, we'll need to analyze the given information step by step.
Step 1: Determine the range
The problem states that the number is between 600 and 700, so the hundreds digit must be 6.
Step 2: Form the equation
Let's use variables to represent the digits of the number:
Hundreds digit = 6
Tens digit = T (unknown)
Units digit = U (unknown)
Forming the equation:
Number = 100 * 6 + 10 * T + U = 610 + 10T + U
Step 3: Translate the given information into an equation
According to the problem, "A three-digit number between 600 and 700 is one less than 30 times the sum of the digits". This can be expressed as:
Number = 30 * (T + U) - 1
Step 4: Solve the equation
Setting the two equations equal to each other:
610 + 10T + U = 30(T + U) - 1
Simplifying the equation:
610 + 10T + U = 30T + 30U - 1
Combining like terms:
611 = 29T + 29U
Divide both sides of the equation by 29 to isolate T and U:
611/29 = T + U
By trying different combinations of numbers that add up to 611/29 (which is approximately 21.07), you'll find that T = 8 and U = 13/29.
Step 5: Confirm the answer
As the problem states that the tens digit is one less than the unit digit, we need to check if T = U - 1:
8 = 13/29 - 1
8 = -16/29
Since 8 ≠ -16/29, the assumed values for T and U are incorrect. Therefore, there is no possible solution to this problem within the given conditions.