A three- digit number between 600 and 700 is one less than 30 times ths sum of the digits. If the tens digits is one less than the units digit, What is the number?
To find the number, let's break down the information given step by step.
1. A three-digit number between 600 and 700: This means the hundreds digit should be 6.
2. The number is one less than 30 times the sum of the digits: Let's assume the three-digit number is XYZ. The sum of the digits would be X + Y + Z. According to the given information, the number can be written as 30(X + Y + Z) - 1.
3. The tens digit is one less than the units digit: This means Y = Z - 1.
Now, let's put all the information together and solve for the number.
Since the hundreds digit is 6, we now have the number in the form 6XY.
Using the second information, the number becomes 30(6 + Y + Z) - 1, or simply 180 + 30Y + 30Z - 1.
Using the third information, we replace Y with Z - 1, giving us 180 + 30(Z - 1) + 30Z - 1.
Simplifying further, we have 180 + 30Z - 30 + 30Z - 1, which can be simplified to 60Z + 149.
We know the number is between 600 and 700, which means it should be more than 600 but less than 700. Therefore, the hundreds digit (6) and the tens digit (Z) must add up to a number less than 10.
To find the value of Z, we can calculate the value of 60Z + 149 for each possible Z value less than 10 and check if it falls between 600 and 700.
Let's calculate the possible values and check:
For Z = 1: 60(1) + 149 = 209 (not between 600 and 700)
For Z = 2: 60(2) + 149 = 269 (not between 600 and 700)
For Z = 3: 60(3) + 149 = 329 (not between 600 and 700)
For Z = 4: 60(4) + 149 = 389 (not between 600 and 700)
For Z = 5: 60(5) + 149 = 449 (not between 600 and 700)
For Z = 6: 60(6) + 149 = 509 (not between 600 and 700)
For Z = 7: 60(7) + 149 = 569 (not between 600 and 700)
For Z = 8: 60(8) + 149 = 629 (between 600 and 700)
Therefore, the tens digit (Z) should be 8. Now, substitute the value of Z back into our original number.
6XY becomes 68Y.
Now we need to find Y, which is one less than Z. So, Y would be 8 - 1 = 7.
Putting it all together, the three-digit number is 687.
Therefore, the number is 687.
Let's solve this step-by-step.
Step 1: Let's represent the three-digit number as ABC, where A represents the hundreds digit, B represents the tens digit, and C represents the units digit.
Step 2: We are given that the number is between 600 and 700. Therefore, A must be 6.
Step 3: We are also given that the tens digit is one less than the units digit. Let's represent this information as B = C - 1.
Step 4: According to the problem, the number is one less than 30 times the sum of the digits. We can write this as ABC = 30 * (A + B + C) - 1.
Step 5: Substituting the values from Step 2 and Step 3, we have 600 + 10B + C = 30 * (6 + B + C) - 1.
Step 6: Simplifying the equation, we have 600 + 10B + C = 180 + 30B + 30C - 1.
Step 7: Combining like terms, we have 10B + C = 179 + 30B + 30C.
Step 8: Rearranging the equation, we have 20B + 29C = 179.
Step 9: Since we are looking for a three-digit number, the values of B and C can only be integers between 0 and 9.
Step 10: We can try different pairs of values for B and C until we find the values that satisfy the equation from Step 8.
If we start by trying C = 0, we have 20B = 179. However, 179 is not divisible by 20, so this is not a valid solution.
If we try C = 1, we have 20B + 29 = 179. Solving this equation, we find that B = 8.
Therefore, the number is 681.